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MINISTRY OF EDUCATION MATRICULATION DIVISION PHYSICS LABORATORY MANUAL DP014 & DP024 5th EDITION MATRICULATION DIVISION MINISTRY OF EDUCATION MALAYSIA PHYSICS LABORATORY MANUAL SEMESTER I & II DP014 & DP024 MINISTRY OF EDUCATION MALAYSIA MATRICULATION PROGRAMME FIFTH EDITION First Printing, 2011 (First Edition) Second Printing, 2015 (Second Edition) Third Printing, 2018 (Third Edition) Fourth Printing, 2020 (Fourth Edition) Fifth Printing, 2022 (Fifth Edition) Copyright © 2022 Matriculation Division Ministry of Education Malaysia ALL RIGHTS RESERVED. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system, without the prior written permission from the Director of Matriculation Division, Ministry of Education Malaysia. Published in Malaysia by Matriculation Division Ministry of Education Malaysia, Level 6 – 7, Block E15, Government Complex Parcel E, Federal Government Administrative Centre, 62604 Putrajaya, MALAYSIA. Tel : 603-88844083 Fax : 603-88844028 Website : http://www.moe.gov.my/ Printed in Malaysia by Malaysia National Library Physics Laboratory Manual Semester I & II DP014 & DP024 Fifth Edition e ISBN 978-983-2604-67-9 NATIONAL EDUCATION PHILOSOPHY Education in Malaysia is an on-going effort towards further developing the potential of individuals in a holistic and integrated manner, so as to produce individuals who are intellectually, spiritually and physically balanced and harmonious based on a firm belief in and devotion to God. Such an effort is designed to produce Malaysian citizens who are knowledgeable and competent, who possess high moral standards and who are responsible and capable of achieving a high level of personal well- being as well as being able to contribute to the betterment of the family, society and the nation at large. NATIONAL SCIENCE EDUCATION PHILOSOPHY In consonance with the National Education Philosophy, science education in Malaysia nurtures a science and technology culture by focusing on the development of individuals who are competitive, dynamic, robust and resilient and able to master scientific knowledge and technological competency. FOREWORD I am delighted to write the foreword for the Laboratory Manual, which aimed to equip students with knowledge, skills, and the ability to be competitive undergraduates. This Laboratory Manual is written in such a way to emphasise students’ practical skills and their ability to read and understand instructions, making assumptions, apply learnt skills and react effectively in a safe environment. Science process skills such as making accurate observations, taking measurement in correct manner, using appropriate measuring apparatus, inferring, hypothesizing, predicting, interpreting data, and controlling variables are further developed during practical session. The processes are incorporated to help students to enhance their Higher Order Thinking Skills such as analytical, critical and creative thinking skills. These 21st century skills are crucial to prepare students to succeed in Industrial Revolution (I.R.) 4.0. The manipulative skills such as handling the instruments, setting up the apparatus correctly and drawing the diagrams can be advanced through practical session. The laboratory experiments are designed to encourage students to have enquiry mind. It requires students to participate actively in the science process skills before, during and after the experiment by preparing the pre- report, making observations, analysing the results and in the science process skills before, during, after the experiment by preparing the pre-report, making observations, analysing the results and drawing conclusions. It is my hope and expectation that this manual will provide an effective learning experience and referenced resource for all students to equip themselves with the skills needed to fulfil the prerequisite requirements in the first-year undergraduate studies. DR HAJAH ROSNARIZAH BINTI ABDUL HALIM Director Matriculation Division iii CONTENTS 1.0 Student Learning Time (SLT) Page 2.0 Learning Outcomes v 3.0 Guidance for Students v 4.0 Significant Figures viii 5.0 Uncertainty in Measurements xi xii Semester I 1 Experiment Title 6 12 1 Physical Measurement 15 2 Plotting and Interpreting Linear Graph 19 3 Free Fall Motion 23 4 Linear Momentum 5 Friction Page 6 Thermal Conduction 26 29 Semester II 32 35 Experiment Title 38 46 1 Simple Harmonic Motion (SHM) 49 2 Standing Waves 50 3 Ohm’s Law 4 Capacitor 5 Magnetic Field 6 Geometrical Optics References Acknowledgements iv Physics Lab Manual 1.0 Student Learning Time (SLT) Students will be performing the experiment within the time allocated for each practical work. Face-to-face Non face-to-face 2 hours 0 2.0 Learning Outcomes 2.1 Matriculation Science Programme Educational Objectives Upon a year of graduation from the programme, graduates are: i. Knowledgeable and technically competent in science disciplines study in-line with higher educational institution requirement. ii. Able to apply information and use data to solve problems in science disciplines. iii. Able to communicate competently and collaborate effectively in group work to compete in higher education environment. iv. Able to use basic information technologies and engage in life- long learning to continue the acquisition of new knowledge and skills. v. Able to demonstrate leadership skills and practice good values and ethics in managing organisations. 2.2 Matriculation Science Programme Learning Outcomes At the end of the programme, students should be able to: i. Acquire knowledge of science and mathematics as a fundamental of higher level education. (MQF LOC i – Knowledge and understanding) ii. Apply logical, analytical and critical thinking in scientific studies and problem solving. (MQF LOC ii – Cognitive skills) iii. Demonstrate manipulative skills in laboratory works. (MQF LOC iii a – Practical skills) Updated: 19/1/2021 v Physics Lab Manual iv. Collaborate in group work with skills required for higher education. (MQF LOC iii b – Interpersonal skills) v. Deliver ideas, information, problems and solution in verbal and written communication. (MQF LOC iii c – Communication skills) vi. Use basic digital technology to seek and analyse data for management of information. (MQF LOC iii d – Digital skills) vii. Interpret familiar and uncomplicated numerical data to solve problems. (MQF LOC iii e – Numeracy skills) viii.Demonstrate leadership, autonomy and responsibility in managing organization. (MQF LOC iii f – Leadership, autonomy and responsibility) ix. Initiate self-improvement through independent learning. (MQF LOC iv – Personal and entrepreneurial skills) x. Practice good values attitude, ethics and accountability in STEM and professionalism. (MQF LOC v – Ethics and professionalism) 2.3 Physics 1 Course Learning Outcome At the end of the course, student should be able to: 1. Describe basic concepts of mechanics and heat. (C2, PLO 1, MQF LOC i) 2. Solve problems related to mechanics and heat. (C3, PLO 2, MQF LOC ii) 3. Apply the appropriate scientific laboratory skills in physics experiments. (P3, PLO 3, MQF LOC iii a) Updated: 19/1/2021 vi Physics Lab Manual 2.4 Physics 2 Course Learning Outcome At the end of the course, student should be able to: 1. Explain basic concepts of waves, electricity, magnetism and optics. (C2, PLO 1, MQF LOC i) 2. Solve problems of waves, electricity, magnetism and optics. (C4, PLO 2, MQF LOC ii) 3. Apply the appropriate scientific laboratory skills in physics experiments. (P3, PLO 3, MQF LOC iii a) 2.5 Physics Practical Learning Outcomes Physics experiment is to give the students a better understanding of the concepts of physics through experiments. The aims of the experiments in this course are to be able to: 1. introduce students to laboratory work and to equip them with the practical skills needed to carry out experiment in the laboratory. 2. determine the best range of readings using appropriate measuring devices. 3. recognise the importance of single and repeated readings in measurement. 4. analyse and interpret experimental data in order to deduce conclusions for the experiments. 5. make conclusions in line with the objective(s) of the experiment which rightfully represents the experimental results. 6. verifying the correct relationships between the physical quantities in the experiments. 7. identify the limitations and accuracy of observations and measurements. 8. familiarise student with standard experimental techniques. 9. choose suitable apparatus and to use it correctly and carefully. Updated: 19/1/2021 vii Physics Lab Manual 10. gain scientific trainings in observing, measuring, recording and analysing data as well as to determine the uncertainties (errors) of various physical quantities observed in the experiments. 11. handle apparatus, measuring instruments and materials safely and efficiently. 12. present a good scientific report for the experiment. 13. follow instructions and procedures given in the laboratory manual. 14. gain confidence in performing experiments. 3.0 Guidance for Students 3.1 Ethics in the laboratory a. Follow the laboratory rules. b. Students must be punctual for the practical session. Students are not allowed to leave the laboratory before the practical session ends without permission. c. Co-operation between members of the group must be encouraged so that each member can gain experience in handling the apparatus and take part in the discussions about the results of the experiments. d. Record the data based on the observations and not based on any assumptions. If the results obtained are different from the theoretical value, state the possible reasons. e. Get help from the lecturer or the laboratory assistant should any problems arise during the practical session. Updated: 19/1/2021 viii Physics Lab Manual 3.2 Preparation for experiment 3.2.1 Planning for the practical a. Before entering the laboratory i) Read and understand the objectives and the theory of the experiment. ii) Think and plan the working procedures properly for the whole experiment. Make sure you have appropriate table for the data. iii) Prepare a jotter book for the data and observations of the experiments during pre-lab discussion. b. Inside the laboratory i) Check the apparatus provided and note down the important information about the apparatus. ii) Arrange the apparatus accordingly. iii) Conduct the experiment carefully. iv) Record all measurements and observations made during the experiment. Updated: 19/1/2021 ix Physics Lab Manual 3.3 Report writing The report must be written properly and clearly in English and explain what has been carried out in the experiment. Each report must contain name, matriculation number, number of experiment, title, date and practicum group. The report must also contain the followings: i) Objective • state clearly ii) Theory • write concisely in your own words • draw and label diagram if necessary iii) Apparatus • name, range, and sensitivity, e.g Voltmeter: 0.0 – 10.0 V Sensitivity: ± 0.1 V iv) Procedure • write in passive sentences about all the steps taken during the experiment v) Observation • data tabulation with units and uncertainties • data processing (plotting graph, calculation to obtain the results of the experiments and its uncertainties). • Calculation of uncertainties using LSM method can refer attachment A vi) Discussion • give comments about the experimental results by comparing it with the standard value • state the source of mistake(s) or error(s) if any as well as any precaution(s) taken to overcome them • answer all the questions given vii) Conclusion • state briefly the results with reference to the objectives of the experiment Reminder: NO PLAGIARISM IS ALLOWED. Updated: 19/1/2021 x Physics Lab Manual 4.0 Significant Figures The significant figures of a number are those digits carry meaning contributing to its precision. Therefore, the most basic way to indicate the precision of a quantity is to write it with the correct number of significant figures. The significant figures are all the digits that are known accurately plus the one estimated digit. For example, we say the distance between two towns is 200 km, that does not mean we know the distance to be exactly 200 km. Rather, the distance is 200 km to the nearest kilometres. If instead we say that the distance is 200.0 km that would indicate that we know the distance to the nearest tenth of a kilometre. More significant figures mean greater precision. Rules for identifying significant figures: 1. Nonzero digits are always significant. 2. Final or ending zeros written to the right of the decimal point are significant. 3. Zeros written on either side of the decimal point for the purpose of spacing the decimal point are not significant. 4. Zeros written between significant figures are significant. Updated: 19/1/2021 xi Physics Lab Manual Example: Value Number of Remarks 0.5 significant figures 0.500 Implies value between 0.45 and 0.050 1 0.55 5.0 3 1.52 Implies value between 0.4995 and 2 0.5005 1.52 × 104 2 3 Implies value between 0.0495 and 150 3 0.0505 2 or 3 Implies value between 4.95 and (ambiguous) 5.05 Implies value between 1.515 and 1.525 Implies value between 15150 and 15250 The zero may or may not be significant. If the zero is significant, the value implied is between 149.5 and 150.5. If the zero is not significant, the value implied is between 145 and 155. 5.0 Uncertainty in Measurements No matter how careful or how accurate are the instruments, the results of any measurements made at best are only close enough to their true values (actual values). Obviously, this is because the instruments have certain smallest scale by which measurement can be made. Chances are, the true values lie within the smallest scale. Hence, we have uncertainties in our measurements. The uncertainty of a measurement depends on its type and how it is done. For a quantity x with uncertainty Dx , the measurement should be recorded as x ± Dx with appropriate unit. The relative uncertainty of the measurement is defined as Dx . x and therefore its percentage of uncertainty, is given by Dx ´100% . x Updated: 19/1/2021 xii Physics Lab Manual 5.1 Single Reading (a) If the reading is taken from a single point or at the end of the scale we use: Dx = 1 ´ (smallest division of the scale) 2 (b) If the readings are taken from two points on the scale: Dx = 2 ´ é 1 ´(smallest division from the scale)ùûú ëê 2 (c) If the apparatus has a vernier scale: Dx = 1 ´ (smallest unit of the vernier scale) 5.2 Repeated Readings For a set of n repeated measurements, the best value is the average value, that is x = n xi å i =1 n where: n is the number of measurements taken xi is the ith measurement value The uncertainty is given by n å x - xi Dx = i=1 n The result should be written in the form of x = x ± Dx Updated: 19/1/2021 xiii Physics Lab Manual 5.3 Combination of uncertainties We adopt maximum uncertainty. (a) Addition or subtraction Dx = Da + Db + Dc x=a+b-c Þ (b) Multiplication with constant k x = ka Þ Dx = kDa (c) Multiplication or division x = ab Þ Dx = çæ Da + Db + Dc ÷ö c x è a b c ø (d) Powers x = an Þ Dx =n æ Da ö x èç a ÷ø 5.4 Uncertainty gradient and y-intercept using Least Square Method ( LSM ) 5.4.1 Formula uncertainty for gradient and y-intercept Straight line graphs are very useful in data analysis for many physics experiments. From straight line equation, that is, y = mx + c we can easily determine the gradient m of the graph and its intercept c on the vertical axis. When plotting a straight line graph, the line does not necessary passes through all the points. Therefore, it is important to determine the uncertainties ∆m and ∆c for the gradient of the graph and the y-interception respectively. Updated: 19/1/2021 xiv Physics Lab Manual Consider the data obtained is as follows: x x1 x2 x3………………..xn y y1 y2 y3………………..yn ( )(a) Find the centroid x , y , where n xi n yi å å x i =1 and y i =1 = = n n (b) Draw the best straight line passing through the centroid and balance. (c) Determine the gradient of the line by drawing a triangle using dotted lines. The gradient is given by m= y2 - y1 x2 - x1 y ´ ´ (x2, y2) ´ Ä ´ ´ c (x´1, y1) 0 x Figure A Updated: 19/1/2021 xv Physics Lab Manual (d) The uncertainty of the slope, ∆ can be calculated using the following equation ∆ = $( ∑!#$%( ! − ) !)" ̅)" − 2) ∑#!$%( ! − where n is the number of readings and ̅ is the average value of x given by # ̅ = 1 1 ! !$% and the estimated value of y, 2 & is given by, 2 & = 2 ! + ̂ (e) The uncertainty of the y-intercept, ∆ can be calculated using the following equation ∆ = 6 1 2 # − ) ! )" 7 1 + ̅ " ̅)"8 − ∑#!$%( ! − 1( ! !$% 5.4.2 Procedure to draw a straight line graph and to determine its gradient with its uncertainty (a) Choose appropriate scales to use at least 80% of the sectional paper. Draw, label, mark the two axes, and give the units. Avoid using scales of 3, 7, 9, and the likes or any multiple of them. Doing so will cause difficulty in plotting the points later on. (b) Plot all points clearly with ´. At this stage you can see the pattern of the distribution of the graph points. If there is a point which is clearly too far-off from the rest, it is necessary to repeat the measurement or omit it. (c) Calculate the centroid and plot it on the graph. Updated: 19/1/2021 xvi Physics Lab Manual Example: Suppose a set of data is obtained as below. Graph of T2 against ! is to be plotted. ! (± 0.1 cm) 10.0 20.0 30.0 40.0 50.0 60.0 T2 (± 0.01 s2) 0.33 0.80 1.31 1.61 2.01 2.26 From the data: ! = 10.0 + 20.0 + 30.0 + 40.0 + 50.0 + 60.0 = 35.0 cm 6 T 2 = 0.33 + 0.80 + 1.31 + 1.61 + 2.01 + 2.26 = 1.39 s2 6 Therefore, the centroid is (35.0 cm, 1.39 s2). (d) Draw a best straight line through the centroid and balance. Points above the line are roughly in equal number and positions to those below the line. (e) Determine the gradient of the line. Draw a fairly large right-angle triangle with part of the line as the hypotenuse. From the graph in Figure B, the gradient of the line is as follows: For the best line: m = (2.10 - 0.00) s2 (53.0 - 0.0) cm = 0.040 s2 cm-1 The gradient of the graph and its uncertainty should be written as follows: m = (0.040 ± ___) s2 cm-1 Take extra precaution so that the number of significant figures for the gradient and its uncertainty are in consistency. Updated: 19/1/2021 xvii Physics Lab Manual T2 (s2) Graph of T2 against ! 2.4 m 2.2 2.0 1.8 Wrong best straight line 1.6 1.4 1.2 1.0 2.10 – 0.00 0.8 0.6 0.4 ! (cm) 0.2 53.0 – 0.0 0 10 20 30 40 50 60 Figure B Updated: 19/1/2021 xviii Physics Lab Manual (f) Calculation of uncertainties Rewrite the data in the form of ! − :! ; − :! T2 > − > ? − > @ 10.0 -25.0 < 0.33 20.0 -15.0 0.80 30.0 -5.0 625.0 1.31 0.4 -0.070 0.0049 40.0 5.0 225.0 1.61 50.0 15.0 25.0 2.01 0.8 0.000 0.0000 60.0 25.0 25.0 2.26 Ʃ=210.0 225.0 1.2 0.110 0.0121 625.0 Ʃ=1750.0 1.6 0.010 0.0001 2.0 0.010 0.0001 2.4 -0.140 0.0196 Ʃ=0.0368 Where, :! is the average of , :! = "%( = 35.0 cm ) Where, > " is the expected value of T2 > " = 0.04 Calculate the uncertainty of slope, Δm ∆ = G(#-∑"!#$) %∑(#!,$!%-(0,.!!-)"0̅)" = G()-(".()(3%)546() = ±0.002 Then, calculate the uncertainty of y-intercept, Δc ∆ = $7∑!#$%( !−−2 ) !)" 8 7 1 + ̅ " ̅)"8 ∑!#$%( ! − ∆ = $H06.0−3628K 761 + 13755"08 ∆ = ±0.09 Updated: 19/1/2021 xix Physics Lab Manual The data given in section 5.4.2(e) was obtained from an experiment to verify the relation between T2 and ! . Theoretically, the quantities obey the following relation, T 2 = æ k ö ! ç p ÷ è ø where k is a natural number equals 39.48 and p is a physical constant. Calculate p and its uncertainty. Solution: From the equation, we know that k = gradient m p p = k m = 39.48 0.040 = 987 cm s-2 Since k is a natural number which has no uncertainties, that is Dk = 0. ∆ = ;∆88 + ∆99< = ;0 + ((..(((:"(< 987 = 49.35 so we write, p = (987 ± 49.35) cm s–2 or p = (1000 ± 50) cm s–2 Updated: 19/1/2021 xx Physics Lab Manual 5.5 Percentage of difference: When comparing an experimental result to a value determined by theory or to an accepted known value, the difference between the experimental value and the theoretical value can be determined by: Percentage of difference = X - XTheory Experiment ´100% X Theory Updated: 19/1/2021 xxi SEMESTER I DP014 Physics Lab Manual EXPERIMENT 1: PHYSICAL MEASUREMENT Objective: To measure and determine the uncertainty of physical quantities. Theory: Measuring some physical quantities is part and parcel of any physics experiment. It is important to realise that not all measured values are exactly the same as the actual values. This could be due to the errors that we made during the measurement, or perhaps the apparatus that we used may not be accurate or sensitive enough. Therefore, as a rule, the uncertainty of a measurement must be taken and it has to be recorded together with the measured value. The uncertainty of a measurement depends on the type of measurement and how it is done. For a quantity x with the uncertainty Dx, its measurement is recorded as below: x ± Dx The relative uncertainty of the measurement is defined as: Dx x and therefore, its percentage of uncertainty is given by Dx ´100% x 1. Single Reading 1.1 If the reading is taken from a single point or at the end of the scale, Dx = 1 ´ (the smallest division from the scale) 2 1.2 If the readings are taken from two points on the scale, Dx = 2 ´ [ 1 ´ (the smallest division from the scale)] 2 1.3 If the apparatus uses a vernier scale, ∆x = 1 ´ (the smallest unit from the vernier scale) Updated: 19/1/2022 1 DP014 Physics Lab Manual 2. Repeated Readings For a set of n repeated measurements of x, the best value is the average value n å xi x= i=1 1.1 n where n = the number of measurements taken xi = the ith measurement The uncertainty is given by Dx = n x - xi 1.2 n å i=1 The result should be written as x = x ± Dx 1.3 Updated: 19/1/2022 2 DP014 Physics Lab Manual 3. Combination of uncertainties We adopt maximum uncertainty. 3.1 Addition or subtraction x=a±b±c Þ Dx = Da + Db + Dc 3.2 Multiplication with constant k x = ka Þ Dx = kDa 3.3 Multiplication or division x = ab Þ Dx = æ Da + Db + Dc ö x c çè a b c ø÷ 3.4 Powers Þ Dx = n æ Da ö x x = an èç a ÷ø Apparatus: A micrometer screw gauge A vernier calipers A metre rule A ball bearing A coin A metal rod A glass rod An electronic balance Updated: 19/1/2022 3 DP014 Physics Lab Manual Procedure: 1. Choose the appropriate instrument for measurement of single reading i. The length of a metal rod. ii. The length and width of a laboratory manual. iii. The mass of a ball bearing. 2. Determine the percentage of uncertainty for each set of readings. 3. Choose the appropriate instrument for measurement repeated reading i. The diameter of a ball bearing. ii. The diameter a coin. iii. The external diameter of a glass rod. 4. For procedure 3, perform the measurement and record the data in a suitable table for at least 5 readings. (Refer to Table 1.1 as an example) Table 1.1 No. The diameter of a ball d - di cm bearing, d (±…….cm) 1 2 n n 3 å di å d - di 4 d i =1 Dd = i=1 n 5 = = = n Average 5. Determine the percentage of uncertainty for each set of readings. Updated: 19/1/2022 4 DP014 Physics Lab Manual 6. Calculate the following derived quantities and its uncertainties. i. Perimeter of the laboratory manual. ii. Circumference of the coin iii. Surface area of the ball bearing. iv. Volume of the ball bearing. v. Density of the ball bearing. Updated: 19/1/2022 5 DP014 Physics Lab Manual EXPERIMENT 2: PLOTTING AND INTERPRETING LINEAR GRAPH Objective: To develop skill in plotting and interpreting linear graph. Theory: Graphs are often used to represent the dependence or relationship of two or more experimental quantities. In experimental work, the independent variable is usually represented by the x-axis while the dependent variable is represented by the y-axis. The shape of the graphs can take many forms, but our focus will only be on linear graphs. Linear graphs obey the following equation y = mx + c 2.1 where m is the gradient of the graph and c is the intercept on the y-axis. 1. Procedure to draw a graph 1.1 Axes scale, label and title a) Choose the scale on each axis such that all data points should be shown and should fill not less than half the size of the plotted area. The scales should be in multiples of 10, 5, 2 or 1 and DO NOT USE odd multiples. In some cases, it may be necessary to adjust the graph with an axis that does not start at zero. b) Label the scales of each axis with simple numbers. In cases where the numbers are very large or very small, use scientific notation. c) Axes title should represent the relevant quantities with appropriate units. Updated: 19/1/2022 6 DP014 Physics Lab Manual 1.2 Plotting the data points and the straight line a) Determine the centroid which is defined as follows: nn å xi å yi i =1 i =1 x = y = 2.2 n n b) The data points should be plotted using the symbol ´. Use the symbol Ä to plot the centroid. c) Draw a best fit solid straight line. The line must pass through the centroid and as many data points as possible as shown in Figure 2.1. It may not necessarily pass through every data point and the origin. 1.3 Determination of gradient a) The gradient of the graph is determined by choosing two points (x1, y1) and (x2, y2) which lie on the line (NOT THE DATA POINTS) and could be read accurately. Draw a right-angle triangle using these two points. b) The value of the gradient is calculated as: m= y2 - y1 2.3 x2 - x1 Updated: 19/1/2022 7 DP014 Physics Lab Manual y ´ ´ (x2, y2) ´ Ä ´ ´ c (x´1, y1) 0 x Figure 2.1 Note: The size of the triangle is such that (x2 – x1) is greater than 50% of the x-axis data range (refer Figure 2.1). 2. Transformation to linear graph The relationship between physical quantities may not be linear. However, the relationship often can be transformed into a linear form and hence a linear graph may be plotted. For example, the relationship between the period, T of oscillation of a simple pendulum of length ! is given as T = 2p ! 2.4 g which can be expressed in linear form as T 2 = 4p 2 ! 2.5 g Thus, a plot of T2 against ! is a straight line passing through the origin with gradient given by m = 4p 2 2.6 g Updated: 19/1/2022 8 DP014 Physics Lab Manual Exercise 1. Table 2.1 shows data taken in a free fall experiment. Measurements were made of the distance of fall (y) at each precisely measured times by using digital stopwatch. Table 2.1 Distance, y Time, t (± 0.001 s) t (s) t2 (s2) (± 0.1 cm) t1 t2 t3 10.0 20.0 0.143 0.139 0.140 30.0 40.0 0.202 0.199 0.200 50.0 60.0 0.247 0.245 0.249 70.0 0.286 0.286 0.280 0.319 0.320 0.317 0.350 0.346 0.352 0.378 0.380 0.376 a) Complete the table. Use proper number of significant figures in the table entries. b) The equation of motion for an object in free fall starting from rest is y= 1 gt 2 , where g is the acceleration due to 2 gravity. This is a parabolic equation, which has the general form y = ax2 . i. Convert the curve to a straight line by plotting a graph of y against t2. ii. Determine the gradient of the line and calculate the experimental value of g. Updated: 19/1/2022 9 DP014 Physics Lab Manual 2. A physical pendulum which is made of a rod has its axis of oscillation at a distance, h from its centre of mass as shown in Figure 2.2. The period of oscillation, T is tabulated in Table 2.2. h Pin (axis of H oscillation) Centre of mass q Figure 2.2 Table 2.2 h 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 (± 0.1 cm) T (± 0.01 s) 2.65 2.54 2.39 2.20 2.00 1.83 1.55 1.26 T2 (s2) Updated: 19/1/2022 10 DP014 Physics Lab Manual The period of oscillation is given as T = 2p H -h 2.7 g where H is a constant and g is the acceleration due to gravity. a) Complete Table 2.2 b) Express equation 2.7 in linear form as equation 2.1. c) Plot a linear graph to determine H and g. Updated: 19/1/2022 11 DP014 Physics Lab Manual EXPERIMENT 3: FREE FALL MOTION Objective: To determine the acceleration due to gravity, g using free fall motion Theory: When a body of mass, m falls freely from a certain height, h above the ground, it experiences a linear motion. The body will obey the equation of motion, s = ut + 1 at 2 3.1 2 By substituting, s = –h = downward displacement of the body from the falling point to the ground u = 0 = the initial velocity of the body a = –g = the acceleration due to gravity t = time taken for the body to reach the ground we obtain the displacement of the body, h as h = 1 gt 2 3.2 2 Apparatus: A retort stand with a clamp A timer A metre rule A free fall adaptor A horizontal table A steel ball Updated: 19/1/2022 12 Procedure: DP014 Physics Lab Manual electromagnet clamp steel ball h timer retort stand trap door Figure 3.1 electromagnet N1 P : +ve N : -ve steel ball P2 retort 12 V N1 stand N P1 2 clamp timer container h P2 N3 P3 hinged trap door N 2 P3 P1 N 3 Updated: 19/1/2022 Figure 3.2 13 DP014 Physics Lab Manual 1. Set up the apparatus as in Figure 3.1 or Figure 3.2 (Set up for free fall apparatus with separate power supply to electromagnet). 2. Switch on the circuit and attach the steel ball onto the upper contact. 3. Adjust the height of the electromagnet above the point of impact. 4. Begin with a small value of h. 5. Switch off the circuit and let the ball fall. Record the value of h and t. 6. Take six sets of reading at different values of h and t. 7. Plot a graph of h against t2. 8. Calculate the gradient of the graph and determine the value of acceleration due to gravity, g. 9. Determine the uncertainty of acceleration due to gravity, Δg. 10. Compare the acceleration due to gravity, g obtained from this experiment with the standard value. Write the comments. Updated: 19/1/2022 14 DP014 Physics Lab Manual EXPERIMENT 4: LINEAR MOMENTUM Objective: To verify the principle of conservation of linear momentum. Theory: Linear momentum, p is a vector quantity defined as the product of mass agnivdenvealsocpi!ty=.mFv!or. a body of mass, m and velocity, v; its momentum is In the absence of external force, the total momentum of the system is conserved. This is known as the Principle of Conservation of Linear Momentum. Applying the principle of conservation of linear momentum in collision, the total momentum of the colliding bodies remains the same before and after the collision. Let m1 and mFi2gbuereth4e.1maansdsesu!1o, fu!t2w, ov!1c,oallniddinv!2g bodies (i.e: body 1 and body 2) as in are the velocities before and after the collision of body 1 and body 2 respectively and written as m1u!1 + m2u!2 = m1v!1 + m2v!2 4.1 m1 m2 m1 m2 u1 u2 v1 v2 Collision in 1D Figure 4.1 Updated: 19/1/2022 15 DP014 Physics Lab Manual In this experiment, we shall study the collision between two ball bearings on a curved gtraaicnks.mTohme feinrstut mbalml 1bu!e1abrienfgories released from the top of the track so that it it hits the second ball bearing which is stationary at the horizontal end of the track. A ball bearing 1 A ball bearing 1 curved track horizontal end ball bearing 2 string table h pendulum bob drawing paper carbon paper R= xo Figure 4.2b Figure 4.2a Consider the collision in x-axis only. Assume that the velocity of the ball bearing is directly proportional to its horizontal displacement, R. m1x0 = m1x1 + m2 x2 4.2 Therefore, if equation 4.2 practically satisfied, the principle of conservation of linear momentum is thus verified. Updated: 19/1/2022 16 DP014 Physics Lab Manual curved track carbon paper pendulum bob landing point of ball bearing 1 without collision x m1 m2 x1 x–axis x2 direction table (top view) xo Figure 4.3 Apparatus: A level table A curved track. (Important: The lower end of the track must be horizontal) 2 ball bearings A piece of string A pendulum bob A metre rule A piece of drawing paper (A3) A piece of carbon paper (A4) A retort stand Updated: 19/1/2022 17 DP014 Physics Lab Manual Procedure: 1. Weigh the mass of the two ball bearings, m1 and m2. 2. Set up the apparatus as in Figure 4.2a. 3. Mark point A on the curved track as in Figure 4.2a. 4. Release ball bearing 1 from point A. 5. Observe the landing point of ball bearing 1 on the floor in order to place the carbon paper on top of a drawing paper so as to mark accurately the point where the ball bearing will land on the floor. 6. Release ball bearing 1 from point A and measure the distance from the bottom end of the pendulum bob to the landing point of ball bearing 1, xo. Repeat this step to obtain at least five readings and calculate the average value of xo. 7. Place ball bearing 2 on the end of the horizontal track, the path of ball bearing 1 so that one dimension (1D) collision occurs. Note: Make sure the landing point of both ball bearings are along the x-axis. 8. Release ball bearing 1 from point A (Refer to Figure 4.2b). These two ball bearings will subsequently collide and fall freely in a projectile motion. 9. Measure the position of landing points for each ball bearing. x1 for ball bearing 1 and x2 for ball bearing 2. (Refer to Figure 4.3) 10. Repeat step (7) to (9) for at least another four sets. 11. Does your result satisfy equations 4.2? Give comments. Updated: 19/1/2022 18 DP014 Physics Lab Manual EXPERIMENT 5: FRICTION Objective: To determine the coefficients of static friction and kinetic friction. Theory: F Static friction (at the onset of motion): R f mg Figure 5.1 By referring to Figure 5.1, if force F is increased, frictional force f also increases accordingly and the object still remain at rest. However, for a certain value of F, the object starts to move. At this stage, the frictional force is known as the limiting static frictional force fs which is the maximum value of f. Hence, fs = µsR 5.1 fs = µs mg 5.2 where, µs = coefficient of static friction fs = static frictional force R = normal reaction force Updated: 19/1/2022 19 DP014 Physics Lab Manual Kinetic friction (in motion): Now, if the object is in motion, the frictional force is known as kinetic friction fk. The kinetic frictional force is less than the static frictional force. That explains why it is difficult to move an object which is initially at rest, but once it is in motion, less force is needed to maintain the motion. fk = µkR 5.3 where µk = coefficient of kinetic friction fk = kinetic frictional force R = normal reaction force Since fk < fs, therefore µs > µk. Apparatus: A piece of plywood A wooden block Two sets of slotted mass (2 g, 5 g, 10 g, 20 g) Sand granules or small metal pallets (each about 2 g) Weighing pan or a tin can A set of pulley with clamp A piece of string Plasticine An electronic balance Updated: 19/1/2022 20 DP014 Physics Lab Manual Procedure: resting slotted mass, mr pulley wooden block mb plywood string sand granules plasticine table weighing pan or e tin can slotted mass Figure 5.2 1. Weigh the mass of the wooden block, mb. 2. Set up the apparatus as in Figure 5.2. Make sure that the string from the block is tied up horizontally to the pulley. Mark the initial position of the wooden block. 3. Add the slotted mass into the weighing pan or tin can gradually until the wooden block begins to slip. If the wooden block still remains static, begin adding the sand granules or the metal pallets gradually until the block starts to move. Note: Add the mass gently to avoid impulsive force. 4. Weigh and record the total mass, mh required to move the wooden block. Repeat this step three times to get the average value of mh. 5. Add different masses of mr onto the wooden block and repeat step (3) to (5). 6. Repeat step (5) for at least five different values of mr. Updated: 19/1/2022 21 DP014 Physics Lab Manual 7. Plot a graph of fs against R where fs = mh g and R = (mr + mb ) g. 8. Calculate the gradient of the graph and determine µs, the coefficient of static friction from the graph. 9. Repeat step (3) but exert a little push (using tic-tac pen) to the wooden block every time each mass is added. Weigh and record the total mass, mh when the block moves slowly and steadily along the plywood. 10. Add different masses of mr onto the wooden block and repeat step (9). 11. Repeat step (10) for at least five different values of mr. 12. Plot a graph of fk against R where fk = mh g and R = (mr + mb ) g. 13. Calculate the gradient of the graph and determine µk, the coefficient of kinetic friction from the graph. 14. Determine the uncertainty of both µs and µk. 15. Compare the value of µs and µk. Does this confirm to the theory? Updated: 19/1/2022 22 DP014 Physics Lab Manual EXPERIMENT 6: THERMAL CONDUCTION Objective: To determine the thermal conductivity of glass. Theory: Thermal conductivity k could be expressed in terms of the rate of flow of heat, dQ = -kA dT 6.1 dt dx where dQ = rate of heat flow dt A = tangential surface area for heat flow /Cross-sectional area dT = temperature gradient dx Relationship between temperature T and time t for this experiment is given by log T0 - logT = kt 6.2 Brx or, log T = - kt + log T0 6.3 Brx where T = temperature (in Kelvin) T0 = 293 K t = time B = 4.84x 106 J m-3 K-1 (specific heat capacity of water) r = average radius of the boiling tube x = thickness of the wall of the boiling tube Updated: 19/1/2022 23 Apparatus: DP014 Physics Lab Manual A boiling tube digital A digital thermometer thermometer A mercury thermometer A 1000 cm3 beaker stirrer Two stirrers boiling tube A cork ice-water mixture A stopwatch warm water A vernier calipers beaker A retort stand and clamp Ice cubes Hot water Procedure: retort stand stirrer 1 cm mercury thermometer Figure 6.1 Updated: 19/1/2022 24 DP014 Physics Lab Manual 1. Measure the internal and external diameters of a boiling tube and calculate the average radius r and the thickness x of the wall of the boiling tube. 2. Fill up a beaker with water and ice. 3. By referring to Figure 6.1, clamp the boiling tube on a retort stand and lower the boiling tube into the beaker until the whole of the boiling tube almost submerge in the ice and water mixture. The temperature of the ice-water mixture inside the beaker should be 0 °C. 4. Pour hot water into the boiling tube until the water level inside the tube reaches about 1 cm below the ice-water level in the beaker. 5. Insert the stirrer and thermometer through the cork. 6. Record at regular time t and the corresponding temperature T starting from 30 ⁰C until it drops to 3 ⁰C. The ice-water mixture in the beaker and the warm water in the boiling tube should be constantly stirred throughout the experiment. 7. Tabulate t, T and log T. 8. Plot a graph of log T against t. 9. Calculate the gradient of the graph and determine the thermal conductivity of glass, k. 10. Determine the uncertainty of k. 11. Compare the result with the standard value k = 0.8 W m-1 K-1 for glass. Give comments. Updated: 19/1/2022 25 SEMESTER II

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• Published: 21 June 2024

Wave-momentum shaping for moving objects in heterogeneous and dynamic media

• Bakhtiyar Orazbayev   ORCID: orcid.org/0000-0002-3597-1873 1 , 2 ,
• Matthieu Malléjac   ORCID: orcid.org/0000-0003-0896-638X 2 ,
• Nicolas Bachelard   ORCID: orcid.org/0000-0001-6011-4905 3 , 4 ,
• Stefan Rotter   ORCID: orcid.org/0000-0002-4123-1417 4 &
• Romain Fleury   ORCID: orcid.org/0000-0002-9486-6854 2  

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• Electronics, photonics and device physics

Light and sound waves can move objects through the transfer of linear or angular momentum, which has led to the development of optical and acoustic tweezers, with applications ranging from biomedical engineering to quantum optics. Although impressive manipulation results have been achieved, the stringent requirement for a highly controlled, low-reverberant and static environment still hinders the applicability of these techniques in many scenarios. Here we overcome this challenge and demonstrate the manipulation of objects in disordered and dynamic media by optimally tailoring the momentum of sound waves iteratively in the far field. The method does not require information about the object’s physical properties or the spatial structure of the surrounding medium but relies only on a real-time scattering matrix measurement and a positional guide-star. Our experiment demonstrates the possibility of optimally moving and rotating objects to extend the reach of wave-based object manipulation to complex and dynamic scattering media. We envision new opportunities for biomedical applications, sensing and manufacturing.

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Ever since the emergence of optical tweezers 1 , 2 , the non-contact manipulation of objects using electromagnetic 3 , 4 and acoustic waves 5 , 6 , 7 has become a central paradigm in quite diverse fields ranging from optomechanics to bioacoustics. Sound waves, in particular, offer distinct advantages, as they are biocompatible and harmless and their short wavelengths can penetrate a wide range of heterogeneous, opaque and absorbing media. Another key feature of acoustics is its wide frequency range, spanning from hertz to gigahertz, which facilitates the manipulation of particles varying in size from a few centimetres to a few micrometres. In this way, not only Mie 8 , 9 , 10 and Rayleigh particles can be addressed, but also complex objects including individual biological cells 11 , 12 , 13 .

Although various strategies have already been developed to collectively or selectively manipulate objects and particles, these techniques always rely on controlled and static environments. Collective dynamic positioning of particles trapped in the potential wells of a pressure field has been achieved in one 14 , two 15 , 16 and three 17 , 18 , 19 , 20 dimensions. Typically, by generating appropriate standing waves 21 , particles or objects are trapped in the vicinity of the pressure nodes or antinodes, depending on their contrast ratio with the surrounding fluid 13 . More advanced strategies have also been developed to address the selectivity problem of standing-wave-based trapping involving acoustic vortices 22 or the use of additional systems such as lenses 23 , metasurfaces 24 or holograms 18 , 25 , 26 , 27 , 28 . Considerable attention has also been paid to the development of on-chip acoustofluidic and acoustophoretic devices 12 , 29 , 30 , 31 and wave-controlled microrobots 32 , 33 , 34 , 35 , 36 , 37 , 38 for lab-on-a-chip and biomedical applications. However, the requirements for a precisely controlled static environment and proximity to the target substantially restrict the applicability of these various techniques in many real-world scenarios. Practical cases involve disordered or dynamic environments where manipulation must occur at a considerable distance from the object that needs to be manipulated.

Here, we propose and experimentally demonstrate a wave-momentum shaping approach that requires only far-field information and allows us to move and rotate objects even in disordered or dynamic environments. Instead of relying on potential wells to trap the object, we continuously find and send the optimal mode mixture that transfers an optimal amount of momentum to the object. This mode mixture is updated during the motion as the scattering changes. The method is experimentally demonstrated in a macroscopic two-dimensional acoustic cavity containing a movable object and a collection of scatterers. Far-field scattering matrix measurements allow us to determine the optimal wavefronts for shifting or rotating the object at each moment in time. Remarkably, the method neither requires knowledge or modelling of acoustic forces nor any prior information on the physical properties of the object or disorder. Only a guide-star measurement of the object’s position or rotation angle is needed, which is here provided by a camera. The remarkable robustness of the method is emphasized by implementing it in a dynamic scenario, where the scatterers composing the environment move randomly. The method may be transposed to other platforms and scales, such as ultrasound or light for the motion of microscopic bodies.

Principles of wave-momentum shaping

The idea of wave-momentum shaping was inspired by recent developments in adaptive optics and disordered photonics, for which wavefront shaping techniques have been significantly advanced to focus light in disordered media or to compensate for aberrations and multiple-scattering for various purposes 39 , 40 , 41 , 42 , 43 . In the most straightforward implementation, a feedback mechanism allows a quantity of interest, such as the optical power focused at a given point, to be iteratively optimized by tuning the incident wavefronts 39 . On the other hand, more advanced approaches, such as Wigner–Smith operators derived from a system’s scattering matrix, can focus light in disorder optimally 44 , exert a maximal electromagnetic force or torque on static objects 45 , 46 , or potentially even cool an ensemble of levitated particles 47 , 48 . Wave-momentum shaping applies these ideas to the manipulation of moving objects. It combines the optimal character of Wigner–Smith approaches with iterative guide-star techniques, necessitated by the dependence of the S matrix on the object position, which influences the complex scattering process and constantly modifies the field speckle.

Consider the experimental set-up illustrated in Fig. 1a . It consists of an acoustic multimode waveguide supporting ten modes ( N  = 10) at the operational frequency f 0  = 1,590 Hz (audible sound) corresponding to an acoustic wavelength of λ 0 ≈ 0.22 m. In the central part, we introduced a movable object (ping-pong ball) with a radius of 20 mm (~0.1 λ 0 ), which floats on the surface of a water tank. Within this tank, several static cylindrical scatterers (depicted as black cylinders) protrude above the water level, thus creating a complex scattering landscape. Two arrays of ten speakers are placed on both sides, labelled 1 and 2, which allow us to control the incident acoustic mode mixtures \(\left\vert {{{{\boldsymbol{\Psi }}}}}_\mathrm{in}^{(1,2)}\right\rangle\) . These incident waves are linearly scattered into outgoing mode mixtures \(\left\vert {{{{\boldsymbol{\Psi }}}}}_\mathrm{out}^{(1,2)}\right\rangle\) , which can be measured using microphones placed in the waveguide’s asymptotic regions ( Methods and Supplementary Information Sections 1 and 2 ). From such measurements, it is possible to deduce the scattering matrix S ( t ), which evolves with time because, as the target object moves, it modifies the scattering in the central region. This dynamic scattering matrix obeys the relation \(\left\vert {{{{\boldsymbol{\Psi}}}}}_\mathrm{out}(t)\right\rangle ={{S}}(t)\left\vert {{{{\boldsymbol{\Psi}}}}}_\mathrm{in}(t)\right\rangle\) , where we gathered the states related to both sides into single column vectors.

a , We consider a parallel-plate acoustic waveguide supporting ten modes at the working frequency. It contains cylindrical rigid scatterers (in black). The bottom surface of this waveguide is formed by the water in this container, so that the spherical object (orange ball) can float and move freely. Wave-momentum shaping consists of finding and sending, at each time t m , the optimal mode mixture to push the ball along an arbitrarily chosen path (orange line). We achieve this by real-time far-field measurements, which allow us to track the evolution of the scattering matrix S as the object moves. We deduce the wavefronts to be injected by the external speaker arrays such that they optimally deliver the target momentum. b , Example of a scattering matrix measured at a given time t m in our experiment. c , Difference between the scattering matrix at t m and the one measured at t m −1 , showing the influence of the translation of a small object on scattering. We use the information collected at three consecutive time steps to derive the mode mixture that optimally pushes the ball in the desired direction. The static scatterers are later replaced with dynamic ones.

Source data

Figure 1b shows an example of the measured scattering matrix, of dimension 2 N  × 2 N , which is composed of four N  ×  N sub-blocks, the reflection and transmission matrices r (1),(2) and t (1),(2) , describing how each of the ten modes scatter on each side. The figure encodes the amplitude of the matrix coefficients in the transparency of the squares and the phase in their colour. Clearly, mode mixing occurs due to the complex scattering. Quite intuitively, to make object manipulation possible, such a scattering matrix must depend on the position of the movable object. This dependence is evidenced by repeating the scattering matrix measurement after slightly moving the object (by a distance equal to a quarter of its radius, l  = 5 mm). The difference is plotted in Fig. 1c . We observe that although the changes are small in magnitude, consistent with the fact that only one scatterer is moved, some information about the object’s motion seems to be embedded in these changes.

The S matrix’s dependence on the object’s position is relevant in the context of its dynamic manipulation. If we denote by α either the x or y coordinate of the movable object, the momentum transferred to it upon scattering Δ p α can be calculated from the expectation values of the operator C α  = −i∂/∂ α for the superposition states \(\left\vert {{{{\boldsymbol{\Psi }}}}}_\mathrm{in,out}\right\rangle\) (ref. 45 ). The momentum transferred to the particle upon scattering is the difference between the momentum of the outgoing and incident mode mixtures in the vicinity of the particle. Assuming unitary scattering ( \({{{{{S}}}}}^{{\dagger} }{{{{S}}}}={\mathbb{1}}\) ), one can demonstrate the following link between this momentum transfer and the variation of S with α ( Methods ):

where the Hermitian operator

is known as a generalized Wigner–Smith (GWS) operator 44 . Equation ( 1 ) means that the momentum imparted locally onto the moving object upon scattering is related to the expectation value of Q α for the specific input state \(\left\vert {{{{\boldsymbol{\Psi }}}}}_\mathrm{in}\right\rangle\) in the far field. A direct consequence of equation ( 1 ) is that if the input state \(\left\vert {{{{\boldsymbol{\Psi }}}}}_\mathrm{in}\right\rangle\) is chosen to be an eigenvector of Q α , the momentum kick on the object will be proportional to its eigenvalue. Therefore, choosing the eigenstate with the highest eigenvalue as the input mode mixture will optimize the transfer of momentum to the object in the direction α . This is the basic physical principle behind wave-momentum shaping.

Linear-momentum transfer

We first apply wave-momentum shaping to the transfer of linear momentum and experimentally demonstrate complete control over the trajectory of a moving object in a complex scattering medium, which is static for now. We start from the set-up of Fig. 1 and apply an iterative motion algorithm that works as follows: (1) Initially, the object is at rest. We send three random wave fields Ψ m −3 , Ψ m −2 and Ψ m −1 to move it slightly but randomly and measure the S m −3 , S m −2 and S m −1 matrices at three different nearby points, whose positions ( x m −3 ,  y m −3 ),( x m −2 ,  y m −2 ) and ( x m −1 ,  y m −1 ) are measured by a camera. (2) From these measurements, we estimate the components d S /d α of the gradient of S with respect to the coordinates α  =  x ,  y , using discrete derivative approximations, equations ( 3 ) and ( 4 ). (3) We compose Q α from equation ( 2 ) and diagonalize it to obtain mode mixtures and momentum expectations (eigenvalues) in α  =  x m ,  y m . (4) We send a superposition of eigenvectors of Q x and Q y in proportion to move the object in a desired direction and measure S again once the object has moved and stabilized. (5) The process is iteratively repeated based on the last three measured S matrices until the object arrives at the desired destination. The method does not require calibration or access to the interior of the medium.

Figure 2a demonstrates the successful guiding of an object within a disordered medium using acoustic-wave-momentum shaping. Several snapshots of the moving ball are blended into one picture to illustrate better the path followed by the moving scatterer. Supplementary Video 1 (ref. 49 ) was recorded by our camera. Remarkably, the acoustic fields injected from the far field can continuously move the floating ball through a chosen S-shaped trajectory within the disordered medium (the total path length is around four λ 0 ). Note that the path is discretized into intermediate checkpoints (blue discs) arranged in a zigzag manner about the S-shaped trajectory to enable a good estimation of the S matrix gradient (Supplementary Information Section 1.6 and Supplementary Fig. 6 ). Note that the object is not trapped but moved by successive acoustic pushes, much like a hockey player guiding a puck.

a , A set of points, in blue, are chosen to define an overall S-shaped path to be followed by the moving ball, whose successive positions, as captured by a camera, are shown by orange discs. The ball successfully reaches each blue point, where the S matrix is measured. Crucially, these checkpoints are chosen to zigzag about the S-shaped path so that the last three consecutive measurements contain optimal information on the gradient of the S matrix with respect to the ball coordinates. b , Net momentum imparted to the ball at three different times on the path (black arrow), and its decomposition over the modes injected from the two sides. Note that these were not derived from the ball dynamics but rather inferred from the momentum expectation value of the injected Wigner–Smith eigenstates. Remarkable agreement with the actual direction of the ball velocity, reported in a (black arrows), is observed. See Supplementary Video 1 .

To illustrate the contribution of each mode and in which sense the input state at each time step is optimal, Fig. 2b compares at three distinct times the momentum expectation value of the input superposition (black arrow) with those of the components of its individual modes alone (coloured arrows). It is clear that each mode contributes to pushing the ball in the correct final direction, and the total push is due to the collective action of all modes. Some modes do not push the ball precisely in the desired direction. Yet, this mixture is optimal given the constraints on the wave spatial degrees of freedom imposed by the disordered medium at this specific location. We also compare the total momentum expectation (black arrows in Fig. 2b ), which is a theoretical prediction, to the actual velocity of the ball (black arrows in Fig. 2a ), which is an experimental observation. The remarkable agreement between the expected momentum push and the measured velocity directions confirms that we successfully implemented our wave-momentum shaping strategy. We conclude that the unavoidable absorption losses present in any experiment, which alter the field amplitudes more than their phases, do not significantly influence the direction of the momentum push predicted by the unitary theory. The interested reader will find other path instances in Supplementary Video 2 (ref. 49 ).

Angular-momentum transfer

An advantage of the variational principle presented above is that α is not restricted to the x or y coordinate but can be any observable target parameter influencing the scattering. A relevant example we consider in the following is the rotation angle θ of an object. This choice will allow us to create an acoustic motor and rotate objects from a distance by sending audible sound. Consider the angular momentum transferred to a rotating object constructed from three balls glued together. Its centre is the rotation axis, which is fixed within the disordered medium (Fig. 3a ). The instantaneous scattering matrix S ( t m ) is measured at consecutive time instances t m with 20° angle step to harness the angular-momentum operator Q θ and provide a way to induce the optimal transfer of torque from the field to the object. Figure 3b reports the experimentally measured value of θ as a function of time. In this experiment, we first selected eigenvectors of Q θ with positive eigenvalues, consistent with the anticlockwise rotation initially observed during the experiment (blue shaded part of the figure). Then, we abruptly switched to input states with negative eigenvalues (red shaded part). The observation of a reversal of the rotation direction, reported in Fig. 3b , is, thus, consistent with theoretical expectations (Supplementary Video 3 (ref. 49 )).

a , We use audible sound to rotate an object constructed from three balls glued together in a disordered medium. First, we move the target in the anticlockwise direction (left part of the figure) and then abruptly switch its direction of rotation (right part). At each step (20°), we extract from a far-field S matrix measurement the Wigner–Smith operator with respect to the rotation angle θ , which allows us to send the wavefront with maximal angular-momentum transfer. b , Measured angle versus time, confirming the rotation of the object in the anticlockwise then clockwise directions.

Manipulating in dynamic disorder

As our manipulation method is based on real-time measurements of the instantaneous scattering matrix, the scattering environment can change in time as well. To demonstrate this, we added other floating balls (like the moving target ball) inside the cavity. This experiment and its goal are illustrated in Fig. 4a . The ball we want to control is the orange one. The blue ones are the added balls, which are anchored with light strings to prevent them from colliding with the target ball. The blue balls carry small metallic nuts, which allows us to randomize their motion by varying in time the magnetic field inside the cavity. We wish to control the path of the orange ball and make it follow a shape that looks like a period of a sine function (dashed blue line). Figure 4b shows the successive measured positions of all balls during the experiment. Unlike the blue balls, which move unpredictably, the orange ball closely follows the predesigned sinusoidal path. The deviations of the orange ball from this target path, shown as a blue line in Fig. 4c , are tiny. For comparison, we also plot as a red line the average distance browsed by the blue balls away from their initial position, which fluctuates more strongly in magnitude and speed, underlining the extreme control that we can maintain on the trajectory of the target (details of the different timescales of the dynamic media are discussed in Supplementary Information Section 8 ). Supplementary Video 4 (ref. 49 ) shows the robustness of the manipulation in the dynamically changing random medium in Fig. 2c .

a , We let all scatterers move freely under time-dependent external perturbations and wish to control the trajectory of the orange ball, guiding it on a sinusoidal path. The blue balls have a metallic nut glued to them, allowing us to randomize their motion by applying fast magnetic perturbations with a moving external electromagnet. They are loosely anchored to the ground by strings to avoid colliding with the orange ball. b , Experimental trajectories measured by a camera, demonstrating the successful control of the ball’s trajectory even in this extreme dynamic scenario. c , Comparison between the measured deviation of the target ball centre from the intended sinusoidal path (blue), and the large fluctuations of the other scatterers from their initial positions (red).

Acoustic pressure field maps

We provide another point of view on wave-momentum shaping by probing the acoustic pressure field in the vicinity of the moving object ( Methods ). For this purpose, we exert the optimal linear-momentum push on the ball along different directions, such as + x or − y , and make it rotate in opposite directions. The acoustic field map is measured. From the displayed pressure profiles (Fig. 5a–d ), we see that the speckle field tends to create hot spots of acoustic pressure that push the object in the right direction. Conversely, experimental pressure distributions for the eigenvalues with the smallest absolute value of the corresponding Wigner–Smith operator exhibit no hot spot near the particle and tend to put it in a silent zone (Supplementary Fig. 8 ). Note that input states that are optimal for motion along x can still exhibit a non-zero expectation value in the y direction. However, they remain the most efficient at pushing along x . Therefore, combining eigenvectors to control expectation values in unwanted directions may provide a way to further refine the algorithm and the precision of the motion. Sometimes the object is in a location in the medium where it cannot be pushed in the right direction given the available degrees of freedom in the speckle. This is not a problem as these points are isolated, and the algorithm will make the object catch up with the trajectory at the next points. To conclude, it is striking to observe how the optimal pressure field can be prepared around the object without knowing anything about the wave–matter interaction involved nor about the object’s shape or environment. This is a clear advantage of the present method compared to conventional methods based on trapping.

a , b , Measured distribution of acoustic pressure amplitudes associated with the optimal transfer of linear momentum along the x direction ( a ) and the − y direction ( b ). c , d , The rotating object is illustrated for clockwise ( c ) and anticlockwise rotations ( d ). The cylindrical scatterers are represented by black discs, and the moving object is outlined with dashed circles. With our approach, we can create the optimal speckle field allowed by the scattering medium. We automatically create the best possible hot spot next to the object to achieve each prescribed momentum kick.

In this Article, we report on the experimental control of an object’s translation and rotation in a complex and dynamic scattering medium using wave-momentum shaping. An iterative manipulation protocol, based solely on knowledge about the far-field scattering matrix of the system and a position guide star, enables the optimal transfer of linear and angular momentum from an acoustic field to manipulate an object in both static and dynamic disordered media. The dynamically injected wavefronts generate the optimal field speckle near the object so that it moves, much like a hockey player guiding a puck, through successive momentum kicks. This method is free of potential traps, robust against disorder and is tolerant of changes in time to the surrounding medium throughout the manipulation. Remarkably, the method is rooted in momentum conservation and does not require any knowledge of the object being manipulated, but only a guide-star measurement of its position. In addition, it does not require any modelling of interaction forces, making the protocol very general and broadly applicable to many real-life scenarios (including different waves, scales, objects and so on). Future efforts will focus on developing methods for objects of various sizes, for example, by transposing the method to ultrasonic frequencies to manipulate smaller objects, as well as extensions to the control of several objects. For this, note that the frequency degree of freedom could also be leveraged. An extension to three-dimensional manipulation would require adequate adaptations, such as a three-dimensional cavity, extraction of three-dimensional coordinates and gradients, and an increased number of channels. Although this seems like a promising direction that would enable motion in a fluid without support for applications in microrobotics, two-dimensional manipulation is already suitable for a vast variety of micromanipulation and acoustofluidic applications in which controlling two of the object’s coordinates is sufficient, for example, tissue engineering 50 , 51 , drug delivery 52 , 53 and biological analysis under a microscope 12 , among others.

Experimental set-up

The set-up consists of a water-filled tank (Fig. 1a and Supplementary Fig. 1a ) coupled at the top to a two-dimensional air waveguide terminated at both ends by anechoic terminations. The water tank’s width, length and height are 100 × 100 × 3 cm, respectively. The two-dimensional acoustic waveguide above it has a width of 104 cm, a length of 180 cm and a height of 8 cm. Two columns of ten microphones (ICP 130F20, 1/4 inch, PCB Piezotronics) separated by 5 cm were placed between the tank and the anechoic terminations on each side to measure the distribution of the complex pressure field inside the waveguide. Two columns of ten amplified loudspeakers (Monacor MSH-115, 4 inches, with in-house amplifiers) were placed horizontally on the bottom of each end of the air waveguide to ensure the efficient excitation of all ten modes inside the acoustic structure. The incident wave states were generated with a Speedgoat Performance Real-Time Target Machine controller (I/O 135, sampling rate 10 kHz) with 40 inputs and 20 outputs. The same equipment was used to acquire the corresponding far-field scattering. To generate the proper acoustic wavefronts, the controller was programmed by Matlab/Simulink to control the voltage of the 20 loudspeakers. The voltages produced by 40 microphones corresponding to the pressure signals were then acquired. Sensor conditioners (483C05, PCB Piezotronics) were used to precondition the microphone signals. Examples of captured signals and post-processing are shown in Supplementary Fig. 2 .

The moving target scatterer was a ping-pong ball (diameter 4 cm and weight 4.17 g) floating on the water’s surface. The static disorder scatterers were plastic cylinders of various diameters (2 to 4 cm) immersed in the water tank. They extended above the surface but without reaching the upper part of the two-dimensional air waveguide. The cylinders and the ball were coated with candle wax (a hydrophobic film) to prevent the ball from sticking to the static scatterers by capillarity. The real-time position of the ball (Fig. 4 ) was captured by an ultra-wide webcam (Logitech Brio), working in full high-definition resolution (1,920 × 1,080 pixels) at a high refresh rate (60 frames per second). A small iron nut was glued to the moving target. It was placed at its initial (starting) position by using an electromagnet attached to a mechanical arm (Supplementary Fig. 1b ), which was moved in a volume (1,000 × 1,000 × 110 mm) above the water tank by three high-precision linear stages (Newport IMS stages with displacement error <0.05 mm).

The rotating object shown in Fig. 3 comprised three ping-pong balls glued together in a line. A needle was fixed to the bottom of the tank, and the centre of the middle ball was impaled on the needle to prevent linear displacement of the object and allow only rotation while limiting friction. The three balls were painted with different patterns to facilitate the detection of the instantaneous angle.

To create the dynamic scattering medium, we used ten ping-pong balls and glued small metallic nuts onto them. The scattering balls, evenly positioned around the intended paths, were attached to the bottom of the water tank by 3- to 8-cm-long nylon threads. The random fluctuations of these scatterers were exacerbated by randomly moving the mechanical arm over the balls while randomly switching the state of the attached electromagnet. The disorder scatterers were placed at a significant enough distance from the target scatterer, which was still free-floating, to avoid any collisions.

Finally, the top plates above the water tank could be replaced with carefully designed perforated plates (holes with a diameter of 1 mm and forming a square array with a period of 10 mm; Supplementary Fig. 1e ) to allow the field inside the waveguide to be scanned by a microphone placed on the robotic arm outside the waveguide.

Scattering matrix measurement

The complex scattering matrix S relates the incoming with the outgoing flux-normalized modes through a set of 2 N linearly independent equations (2 N  = 20 is the total number of propagative modes, ten from each side): ψ out  =  S   ψ in . Solving the scattering matrix S requires measuring N independent wave mode distributions excited by a combination of speakers that form an orthogonal basis. Our experiment uses an orthonormal basis, and only one speaker is excited at a time with the 1,590 Hz harmonic signal. For each excitation, the data collected by the microphone arrays on both sides can be used to determine the incident and outgoing modes ψ in,out . With the hardware we used, this takes about 80 ms. Therefore, after 2 N orthogonal excitations (1.6 s), the scattering matrix is solved for that particular scattering configuration. This type of raw scattering matrix is neither perfectly symmetric nor unitary and is subsequently regularized by discarding its very small antisymmetric part and rescaling its subunitary eigenvalues while keeping their phases (Supplementary Fig. 3 ).

Construction of GWS operators

The construction of the GWS operator Q α is based on gradient approximations, which require successive measurements of the scattering matrix S ( t ) at three different positions (time).

To derive the translation GWS operators Q x and Q y for the ball at position ( x m ,  y m ) and time instance t m , we need, in addition to the scattering matrix S m measured at the actual position, the scattering matrices S m −1 and S m −2 measured at the two previous time instances t m −1 and t m −2 , when the ball was at coordinates ( x m −1 ,  y m −1 ) and ( x m −2 ,  y m −2 ), respectively.

With these three matrices S m , S m −1 and S m −2 , the gradient of S can be derived with the following approximation formula:

With the gradient estimated, the construction of the GWS operators Q x and Q y is direct:

The error in the gradient approximation and, therefore, in the operators Q x and Q y , depends on the shape of the triangle formed by the three measurement points, with the best results obtained for an equilateral triangle and worse for a flat scalene one (Supplementary Fig. 6 ). A detailed analysis of the triangle’s influence on the derivation of the GWS operator is provided in Supplementary Information Section 6 . Therefore, for the best manipulation of the object position, the path is drawn with a zigzag line to minimize the error in the GWS operators.

Similarly, the rotation GWS operator Q θ requires the measurement of S m , S m −1 and S m −2 for three consecutive vane angles θ m , θ m −1 and θ m −2 taken at times t m , t m −1 and t m −2 .

The gradient approximation is, in that case, derived with a backward three-point derivative

where δ θ is the angle difference between the time instances t m and t m −1 .

The GWS operator constructed to control the rotation of the vanes, therefore, reads as follows

Injection of optimal input mode mixtures

As explained in the text, finding the optimal mode mixture to be injected to give the optimal momentum push to the object follows from equation ( 1 ). We, therefore, provide a short proof for this important equation.

For a particle in free space, the change of momentum transferred to it upon scattering Δ p α can be calculated from the expectation values of the operator C α  = −i∂/∂ α for the superposition states \(\left\vert {{{{\boldsymbol{\Psi }}}}}_\mathrm{in,out}\right\rangle\)

We can then write:

Using S † S  = 1, we can write 〈 Ψ in ∣ d/d α ∣ Ψ in 〉 = 〈 S Ψ in ∣ S  d/d α ∣ Ψ in 〉 and obtain

The term in parentheses is nothing but \(\frac{\mathrm{d}S}{\mathrm{d}\alpha}\left\vert\mathbf{\Psi}_{{{{\mathrm{in}}}}}\right\rangle\) , and we directly get

In refs. 44 , 45 , it was shown that equation ( 1 ) continues to hold, even when the target particle is embedded in a scattering environment. In this way, the momentum push expected for a given far-field input is expressed as the expectation value of the GWS operator Q α , which is Hermitian. Therefore, the eigenvector of Q α provides the optimal momentum push with the highest eigenvalue.

Having measured the Wigner–Smith operators, we diagonalize them and find the eigenvector with the highest eigenvalue. We use the eigenvalues to calculate the optimal mode mixture to be injected to give the optimal momentum push to the particle. For example, if we want to move the object by Δ x and Δ y : (1) We diagonalize Q x and Q y . (2) We obtain their eigenvectors with highest eigenvalues Ψ x , y , with eigenvalues calculated as δ x and δ y . (3) We construct the optimal input state \(\frac{\Delta\, x}{\delta x}{\mathbf{\Psi}}_{x}+\frac{\Delta\, y}{\delta y}{\mathbf{\Psi}}_{y}\) . This input state is multiplied by the coupling coefficient matrix of the speakers M to give the required voltage amplitudes and phases required on each speaker (Supplementary Figs. 2 and 5 ). In practice, to determine the direction we want to go, we measure the position of the ball ( x ,  y ) at a given time (Supplementary Fig. 4 ) and compare it with the position of the next checkpoint on the trajectory, which we try to reach up to a certain threshold distance before moving on to the next checkpoint.

Data availability

Source data are provided with this paper. These data are also available via Zenodo at https://doi.org/10.5281/zenodo.10207638 (ref. 49 ). All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The codes that support the findings of this study are available from the authors upon reasonable request.

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Acknowledgements

This work was supported by the Swiss National Science Foundation (Grant No. SPARK CRSK-2_190728 to R.F. and B.O.), by internal EPFL funding from the Swiss Federal Institute of Technology in Lausanne (to R.F. and M.M.) and by Nazarbayev University under the faculty-development competitive research grants programme for 2022–2024 (Grant No. 11022021FD2901 to B.O.) and for 2024–2026 (Grant No. 201223FD2606 to B.O).

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Bakhtiyar Orazbayev

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Université de Bordeaux, CNRS, LOMA, UMR5798, Talence, France

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Contributions

N.B., S.R. and R.F. proposed the idea, raised the funding and supervised the project. B.O. built the experimental set-up, devised the methods and wrote the codes. B.O. performed the wave-momentum transfer experiments with M.M. B.O. and M.M. wrote the methods and the supplementary information. B.O., M.M. and R.F. wrote the initial draft of the manuscript. All authors contributed to the writing of the final draft and critically discussed the results.

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Correspondence to Romain Fleury .

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Supplementary information

Supplementary information.

Supplementary Text and Figs. 1–9.

Supplementary Video 1

Experimental video demonstrating object manipulation in disordered medium along the long path.

Supplementary Video 2

Experimental video demonstrating object manipulation in disordered medium along the different trajectories.

Supplementary Video 3

Experimental video demonstrating object rotation in a disordered medium.

Supplementary Video 4

Experimental video demonstrating the ability to manipulate the object in a dynamic disordered medium.

Source Data Fig. 1

Source data for S matrices.

Source Data Fig. 2

Source data for the path points and vectors.

Source Data Fig. 3

Source data for rotation angles and scatterer positions.

Source Data Fig. 4

Source data for ball deviation and fluctuations of medium.

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Orazbayev, B., Malléjac, M., Bachelard, N. et al. Wave-momentum shaping for moving objects in heterogeneous and dynamic media. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02538-5

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DOI : https://doi.org/10.1038/s41567-024-02538-5

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