Instruction Manual. Physics Laboratory MP-101. Applied Physics Department. Sardar Vallabhbhai National Institute of Technology, Surat

Instruction Manual Physics Laboratory MP-101 Applied Physics Department Sardar Vallabhbhai National Institute of Technology, Surat395007. India

Contents S.N Practical Name Page A Note to Students 3 Error Analysis 6 1 Volume of an Object by Vernier Calipers 11 2 Diameter of a wire using Screw Gauge 15 3 Radius of curvature by Spherometer 18 4 g by Simple Pendulum 22 5 Parallel axis Theorem by Bar Pendulum 24 6 Elastic Constant by Torsional Pendulum 29 7 Young s Modulus by Cantilever 32 8 Modulus of Rigidity by Barton s apparatus 35 9 Frequency of Tuning Fork by Meld s Method 38 10 Prism angle by Spectrometer 42 2

A Note to Students Introduction: The objective of Lab Experiments along with the theory classes is to understand the basic concepts clearly. The experiments are designed to illustrate important phenomena indifferent areas of Physics and to expose you to different measuring instruments and techniques. The importance of labs can hardly be overemphasized as many eminent scientists have made important discoveries in homemade laboratories. In view of this, you are advised to conduct the experiments with interest and an aptitude of learning. This manual will provide the basic theoretical backgrounds and detail procedures of various experiments that you will perform in the Physics laboratory. Before that, here are some specific instructions for you to follow while carrying out the experiments. It also outlines the approach that will be undertaken in conducting the lab. Please read carefully the followings. Specific Instructions: 1. You are expected to complete one experiment in each class. Come to the laboratory with certain initial preparation. The initial preparation will involve a prior study of the basic theory of the experiment, the procedure to perform the experiment so as to have a rough idea of what to do. In addition, it will also involve a partial preparation of the lab report (journal) in advance as mentioned later in this section. 2. You must bring with you the following materials to the laboratory: This instruction manual, journal (lab report)and graph sheets if necessary, pen, pencil, measuring scale, calculator and any other stationary items required. 3. The format of a lab report (journal) shall be as follows: (a) The first sheet (page) will contain your name, branch name and roll number, date and title of the experiment. (b) Each experiment should contain the following in order. Experiment Number, aim of experiment, apparatus needed, a brief theory with working formulae, observation tables with units, figures or diagrams whenever necessary. Write procedure in brief. 3

(c) Experimental observations: Data from experimental observations should be recorded in proper tabular format with well documented headings for the columns. The data tables should be preceded by the least counts of the instruments used to take the data and numerical value of any constant, if any, used in the table. (d) Graphs: (whenever applicable) Always label your graph properly. Be very clear to write the proper units, scale, experiment number, etc. (e) Relevant calculations: Calculations should be done neatly and carefully in proper unit. Error analyses are must to present your result. (f) Final results along with error estimates. (g) Remarks if any. 4. You must record your data directly in your lab record (Journal).Switch off any power supply etc. used and put back the components of the apparatus in their proper places. Complete the rest of the relevant calculations, error analysis, graphs (if necessary), results, and conclusion and obtain your grade of the performed experiment before leaving the laboratory from concerned Faculty (Instructor). 5. Last but not the least - please handle the instruments with care and maintain utmost discipline and decorum of the laboratory. 6. Be honest in recording your data. Never cook up the readings to get desired/ expected results. You never know that you might be heading towards an important discovery. Graphs A graph is simply a diagram illustrating the relationship between two quantities, one of which varies as the other is changed. The quantity that is changed is called independent variable, the other is called the dependent variable. The following general points should be noted: 1. Scale must not be too small loss of accuracy, scale should not be too large exaggeration of accidental errors. Scales on each axis are chosen usually the same unless one variable changes much more rapidly than the other, in which case it is plotted on a smaller scale. 2. The independent variable is placed / plotted horizontally and dependent variable placed / plotted vertically. 3. The origin need not represent the zero values of variables unless definite reference to the origin is required. 4

Temperature (deg. C) 4. Graph should be titled. It should have captions containing - a standard name of variable b its symbol, if such a thing exists, and c standard abbreviation for the unit of measure. 5. Numerals representing scale values should be placed outside the axis. Values less than unity should be written as 0.47, not.47. Use of too many ciphers should be avoided. Thus if scale numbers are 10,000; 20,000; 30,000 etc. They should be written as 1.0, 2.0, 3.0 with the caption say pressure in 10 4 N/m 2. Similarly scale numbers 0.0001; 0.0002; 0.0003 etc. Should be written as 1.0, 2.0, 3.0 with the caption say pressure in 10-4 N/m 2 6. All letterings should be easily readable from the bottom of the graph. Example: 200 180 160 140 120 100 80 60 40 20 Sample 1 Sample 2 Sample 3 0 1 2 3 4 5 Distance (cm) 5

AN INTRODUCTION TO ERROR ANALYSIS Suppose the length of an object is measured with a meter scale and the result is given as 11.3 cm. Does it mean that the length is exactly 11.3 cm? The chances are that the length is slightly more, or slightly less, than the recorded value but as the least count of the scale is 1 mm (it cannot read fraction of a mm) the observer rounds off the result to the nearer full mm. Thus, any length greater than 11.25 cm and less than 11.35 cm we can only conclude that the actual length is anywhere between 11.25 and 11.35 cm. The maximum uncertainty (on either side) or the maximum possible error, δ1, is 0.05 cm which is half the less count of the scale. Let the object under consideration be a glass plate. To obtain the volume of the plate, suppose we measure the width b with slide callipers and the thickness t with a screw gauge whose least counts are respectively 0.1 mm and 0.01 mm. Let the result obtained, after averaging over many measurements, be b = 2.75 cm t = 2.52 mm = 0.252 cm and 1 = 11.3 cm as measured by a meter scale with one end at zero exactly! We note that the coincidences noted in the vernier scale on the head scale of the screw gauge might not have been exact and represent only the nearest exact reading. Hence these measurements also include the corresponding uncertainties each equal to half the least count. So we have 1 = 11.3 ± 0.05cm B= 2.75 ± 0.005 cm t = 0.252 ± 0.0005 cm Note that ± 0.05 cm, ± 0.005 cm, ± 0.0005 cm are actually instrumental errors. Personal errors like reading 11.3 as 11.2 or 11.4 are not taken into account. To avoid personal errors average values of many readings has to be used. The volume calculated from the recorded values of 1, b and t is V = (11.3 2.75 0.252) = 7.8309 cm 3 Take care to avoid writing cm as mm, mm as cm etc. This is also personal error but a careless one at that. However, since each observation is subject to an uncertainty, there should be an uncertainty in the result V too. How can the cumulative effect of the individual uncertanities on the final result be estimated? Let the maximum error in V due to δ1, δb, and δt be δv. Then, (V ± δv) = (1±δ1)(b ± δb)(t ± δt) 6

V + δv will corresponds to maximum positive values of δ1, δb, δt, (V + δv) = (1+ δ1)(b +δb)(t + δt) Or V(1 + δv/v) =1bt (1+δ1/1)(1 +δb/b)(1 + δt/t) Cancelling V = 1bt from both sides and using the approximation We obtain δv/v = δ1/1 + δb/b + δt/t (1 + x) (1 + y) (1 + z) = 1 + x + y + z as x<<1, y<<1, z<<1, The relative error in the product of a number of quantities is the sum of the relative errors of the individual quantities. δ 1 /1 = 0.05 / 11.3 =0.0044 δb / b = 0.005 / 2.75 = 0.0018 δt / t = 0.0005/ 0.252 = 0.002 δv / V = 0.0082 From the value V= 7.8309, we have δv = 7.8309 0.0082 = 0.064213 cm 3 (rounded off to one significant digit). The result of the measurements is therefore V = 7.8309 ± 0.06 cm 3 An important point to be noted is that writing the volume as 7.8309 cm 3 would convey the idea that the result is measured accurate to 0.0001 cm 3. We know from the calculated error that this is not the case and error is in the second decimal place itself. We are not certain that the second decimal is 3 but it may be 3 + 6. The volume may be anywhere in the range 7.77 to 7.89 cm 3. As the second decimal place is subject to such an uncertainty, it is meaningless to specify the subsequent digits. This result should therefore be recorded only up to the second decimal place. [The error could be much larger if the least counts themselves are taken into account]. Thus, V = (7.83 ± 0.06) cm 3 It is the calculation of the maximum error in the result, based on the least counts of the different instruments used that can indicate the number of significant digits to which the final result is accurate. Suppose we now measure the mass of a plate correct to a milligram and the result is 7

m = (18.34 ± 0.005)gm The density d can be calculated from m and V. d = m/ v = 18.34/ 7.83 = 2.3423 gm cm -3 To estimate the uncertainty in d, we write (d + δd) = m + δm/ - δν As the maximum density will correspond to the greatest mass and least volume. d(1 + δd/d) = 1 + δd/d = -1 As and are very much less than 1, δd/d = + The relative error in the quotient of two quantities is (also equal to the sum of the individual relative errors). = 0.0085 2.3423 = 0.02 gm/ cm -3 Therefore d= (2.3423 ± 0.02) or (2. 34 ± 0.02) gm / cm -3 [The error in measurements may be many times the least count if the instrument is not properly designed. Least count may often signify readability/ resolution and not the accuracy. Repeated measurements falling outside the least counts are indicative of this] Other situations: 1. Suppose x is the difference of two quantities a and b, whose measurements have maximum possible errors as δa and δb. What is δx? X =a b (x ± δx) = (a ± δa) (b ± δb) The maximum value of the difference x corresponds to maximum a and minimum b (x + δx) = (a + δa) (b - δb) = (a - b) + (δa + δb) 8

Cancelling x = a b, δx =δa + δb In a sum or difference of two quantities, the uncertainty in the result is the sum of the actual uncertainties in the quantities (Not the relative uncertainties). 2. If p = First δ(1 + m) = δ1 + δm Y 2 can be dealt with as a product of y and y. Questions: 1. Suppose x = (a+b)/ (c-d). To minimize the uncertainty in x, which of the four quantities must be measured to greatest accuracy, if all four quantities a, b, c, d are of the same order of magnitude? 2. The period of a simple pendulum is measured with a stop watch of accuracy 0.1 second. In one trial 4 oscillations are found to take 6.4 secs, in another 50 oscillations take 81 secs. In this relative uncertainty depends only on the least count of the instrument in this case the stop watch? How can the relative uncertainty in the period be minimized? 3. The refractive index of a glass slab may be determined using a vernier microscope as follows. The microscope is focused on a marking on an object placed on a platform and the reading, a, on the vertical scale is noted. The glass slab is placed over the object. The object appears raised. The microscope is raised to get the image is focus and the position on the scale, b, is again noted. The last reading, c, is found raising the microscope to focus on a tiny marking on the top surface of the slab. The least count of the vernier scale is 0.01mm. The readings a, b, c are respectively 6.128 cm, 6.497 cm, and 6.128 cm. Calculate the refractive index and the percentage error in the result. Express the result to the accuracy possible in the experiment, along with the range of error. 9

Note: In the above case cited we have used our judgment i.e. the ability to estimate the reading to ONE HALF the least count of the instrument. If we take that the actual error is ONE least count on either side of the measured quantity all the errors calculated in the above cases would be doubled. References 1. Practical physics by G.L.Squires, Cambridge University Press,4 th edition, 2001. 2. A text book of Practical Physics by M.N. Srinivasan, S. Balasubramanian and R. Ranganathan, Sultan Chand and Sons, First edition, 1990. 10

EXPERIMENT-1 VOLUME OF AN OBJECT BY VERNIER CALIPERS Aim: To measure the length and diameter of the given solid cylinder using vernier calipers and to calculate the volume of the solid cylinder. Apparatus: Vernier calipers and solid cylinder. Theory: The vernier calipers consists of main scale (MS) graduated in cm along the lower edge of the frame and inches along the upper edge of the frame. To one end of the frame, two fixed jaws are provided. There are two movable jaws, which slide over the main scale, and it can be fixed at any point using a screw S. The movable jaw attachment carries a vernier scale (VS). Procedure i) To find the Least Count Least Count is the minimum measurement that can be made with the given instrument. The value of 1 main scale division (MSD) and number of divisions (n) on the vernier scale are determined. The least count of the instrument is calculated using the formula Least Count (LC) = ii) To find the Zero Error (Z.E) and Zero Correction(Z.C) The instrument is checked to find initial error. If there is any initial error, suitable correction is to be done. When the two jaws are in contact, the graduations are made such that the zero of the MS coincides with the zero of the VS. In this case, the instrument has no error. If the zero of the VS is on the right side of the zero of the MS, it has positive error. Then the zero correction is negative. If the zero of the VS is on the left side of the zero of the MS, it has negative error. Then zero correction is positive. In each case, note the division of the vernier scale(vs) which is coinciding with any one of the main scale divisions. This is vernier scale coincidence (VSC). Applying this in the given formula, the value of zero error is determined. Hence zero correction (ZC) is also determined. 11

iii) To find the length and diameter of the solid cylinder The given cylinder is gently placed in between the two lower jaws of the vernier calipers such that the length of the cylinder is parallel to the scale. The completed main scale reading (MSR) is taken by noting the position of zero of the vernier scale (VS). The vernier scale coincidence (VSC) is then noted. VSC is that particular division which coincides with any one of the main scale divisions, making a straight line. The MSR and VSC are noted in the tabular column I for different settings of the cylinder. Then the Vernier Scale Reading (VSR), Observed Reading (OR) and Correct Reading (CR) are calculated using the formula given. The average value of CR is determined, which gives the length of the given cylinder. Similarly the readings for diameter of the solid cylinder are taken by placing the cylinder suitably in between the lower fixed jaws and the readings are noted in the tabular column II. No error Positive error Negative error No correction Negative correction Positive correction VSC = 3 div VSC = 7 div ZE = + (VSC) x LC ZE = -(10-VSC) x LC = + 3 x 0.01 cm = - (10-7) x 0.01cm = + 0.03 cm = - 0.03 cm ZC = - 0.03 cm ZC = + 0.03 cm Zero error and zero correction of the given vernier callipers Formula: Volume of the solid cylinder = πr 2 l 12

Where l = length of the cylinder r = radius of the cylinder Length / diameter of the solid cylinder = Observed reading +Zero correction = {MSR + (VSC x LC)} + ZC Where MSR = Main Scale Reading VSC = Vernier Scale Coincidence LC = Least Count ZC = Zero Correction Vernier Scale Reading (VSR) = Vernier Scale coincidence (VSC) x Least Count (LC) Observed Reading (OR) = Main Scale Reading (MSR) + Vernier Scale Reading (VSR) Correct Reading (CR) = Observed Reading (OR) + Zero Correction (ZC) Observations (i) Least count: Value of 10 MSD = cm Value of 1 MSD = cm No. of division on the vernier scale (n) = (ii) The three possible initial error and initial correction: Least Count (LC) = = =.cm Observation Table:I To find the length of the solid cylinder LC = cm ZE = cm ZC = cm Sr. No MSR VSC VSR = VSC x LC OR = MSR + VSR Length CR = OR + ZC Unit cm div cm cm cm 1 2 3 4 5 13

The average length of the solid cylinder (L)= cm =..x 10-2 m Observation Table: II To find the diameter of the solid cylinder LC =.. cm ZE = cm ZC =. cm Sr. No MSR VSC VSR = VSC x LC OR = MSR + VSR Diameter CR = OR + ZC Unit cm div cm cm cm 1 2 3 4 5 The average diameter of the solid cylinder (D) = cm = x 10-2 m Calculation 1. The average radius of the solid cylinder r =. x 10-2 m 2. Volume of the solid cylinder = π r 2 l =..m 3 Result 1. The length of the solid cylinder l = -------x 10-2 m 2. The diameter of the solid cylinder d = x 10-2 m 3. The volume of the solid cylinder v = x 10-6 m 3 14

EXPERIMENT-2 DIAMETER OF A WIRE USING SCREW GAUGE Aim : i. To measure the diameter of a wire by using micrometer screw gauge. ii. To measure the thickness of a metal glass plate by using micrometer screw gauge. Apparatus: Micrometer screw gauge, wire, metal plate or glass plate etc. 15

Procedure: 1. Find out the distance (X) through which screw travels on the main scale in five rotations of the circular scale. Hence distance travelled by the circular scale in one rotation is called the pitch of the screw. 2. Note down the number of divisions on the circular scale. Hence knowing pitch, L.C. (Least count) of the screw can be determined (See diagram - 1). 3. Find out zero error of micrometer screw. This can be found out as follows: When the two jaws of micrometer screw gauge are brought in contact with each other without applying unduer pressure, the zero on the circular scale should coincide with the reference line on main scale. If this happens then there is no zero error (See diagram - 2). If this is not observed then the micrometer screw has zero error. There are two possibilities Zero of circular scale is above the reference line: When zero of the circular scale is above the reference line as shown in diagram - 3, then zero error is negative. Count the divisions from zero of circular scale coinciding with reference line (m). Negative sign is for negative error. The value of zero error can be found out by multiplying it by least count. [(m) L.C.] Zero of circular scale is below the reference line: When zero of the circular scale is below the reference line as shown in diagram - 4, then zero error is positive. The division of circular scale coinciding with reference line is noted (+m). Positive sign is for positive error. The value of zero error can be found out by multiplying it by least count. [+(m) L.C.] Note that any micrometer screw gauge apparatus can have one and only one possibility out of the three i.e. no zero error, negative zero error and positive zero error. Also zero error is related to the apparatus. 4. Hold the wire, between the two jaws of micrometer screw gauge, whose diameter is to be measured without applying undue pressure. Note down the reading of the main scale (M.S.R.) or pitch scale (P.S.R.). This is a. 5. Note down the circular scale reading coinciding with reference line. This is C.S.D. (n). Multiply this by L.C. to obtain circular scale reading (C.S.R.). 16

6. The total reading (T) can be found out by adding main scale reading and circular scale reading i.e. T = a + b. 7. To obtain correct reading, zero error which is found out in procedure (3) is subtracted from total reading. Thus correct reading of the diameter is obtained. 8. Five readings of the diameter of the wire are taken by using procedure (4) to (7). Hence mean corrected diameter of the wire can be measured. 9. To find the thickness of the metal plate (or glass plate) instead of the wire, glass or metal plate is held between the jaws of the micrometer screw gauge. The procedure (4) to (8) is repeated, resulting in mean thickness of the plate (metal or glass). Observations: (i) Distance (X) travelled by the circular scale in five rotations =... cm (ii) Pitch of the screw = =... (iii) Number of divisions on the circular scale =... (iv) L.C. = Pitch No. of divisions on circular scale =... (v) Zero error, Z = ±(m) (L.C.) =... Observation table: Object Obs. M.S.R. a (cm) Wire 1 2 3 4 5 Plate 1 2 3 4 5 C.S.D. n (div) C.S.R. b=n x (L.C.) cm Total Reading cm Correct Reading= T-Z (cm) Result: 1. Diameter of wire =... cm 2. Thickness of the plate (metal or glass) =... cm 17

EXPERIMENT-3 RADIUS OF CURVATURE BY SPHEROMETER Aim To find the radius of curvature of a convex lens using spherometer. Apparatus A spherometer, a plane glass slab, a convex lens of appropriate diameter (about 8cm to 10 cm), a vernier calipers and wooden blocks for supporting the lens Theory It is a device to measure the thickness of a thin plate and the radius of curvature of any spherical surface (concave /convex mirror or a lens). It carries a small vertical scale usually divided into millimetres. The body of the instrument is supported on three legs whose lower tips form an equilateral triangle and lie in one plane. A screw which carries a circular scale (having 100 or 50 divisions) at its top is so supported that the tip of this screw is at circum- centre of the triangle formed by the tips of the legs. The distance through which the screw advances along the vertical scale in one full rotation is called pitch of the spherometer. It is usually 1 mm or 0.5 mm. If the pitch is 1 mm and the circular scale has 100 divisions, then this means that when the circular scale is rotated by 100 divisions, the screw moves through distance 1mm. Therefore when rotated through one division, it moves through 0.01 mm. This is the least count of the instrument.. 18

Zero Error- If the instrument is correct, then the zero of the circular scale should coincide with the zero of the vertical scale when the tip of the screw is in the plane of the tips of the legs. This is seldom so and hence the instrument has initial error called zero error. The correction whether positive or negative depends upon the final reading whether taken by moving the screw downward or upward. It is not quite necessary to find the zero error if the number of divisions through which the circular scale is rotated is measured correctly and carefully. To determine the radius of curvature of a spherical surface- The formula for calculating R can be easily derived. Fig 1 Fig 2 From the geometry of the circle (Fig 1), we have AD 2 = ED. DF or r 2 = h(2r-h) or 2 R h = r 2 + h 2 19

or R = r 2 /2h + h/2 If l is the length of each side of the equilateral triangle ABC formed by joining the tips of the three outer legs, then as shown in Fig 2, I/2 = r cos = r Substituting r =, we have Procedure 1. Find the least count of the spherometer. 2. Raise the screw by tuning its head so that it may be above the plane containing the tips of the three legs. Place the spherometer on the page of your notebook on which experiment is to be done. Press the spherometer to get the impression of the tips of three legs. Mark the position of each of the three points by drawing a small circle around each point. Measure the length of each side of the triangle formed by joining these points and let the mean value. Let the mean length be l cm. 3. Set the convex lens firmly on a horizontal surface (table).place the spherometer on the surface of convex lens after raising its central leg through four or five complete rotations of the circular scale. Turn the screw so that the central leg while moving downwards touches the surface of the lens. Note the circular scale reading against the edge of the vertical scale. Repeat this step three times 4. Place the spherometer on the surface of the plane glass slab or a table. Turn the screw in the same directions as in the previous step till tip of the central leg just touches the plane surface. Count the number of complete rotations and the additional number of circular divisions moved. 5. Repeat the step three times by moving the central leg every time in the upward Observations: direction through a sufficient large distance as compared to the value of h. (i) Distance moved by the screw in 4 complete rotations = (ii) Pitch of the screw = (iii)total number of divisions on circular scale (n) = (iv) Least count of the spherometer = Pitch/ n = (v) Mean distance between the legs l = 20

Observation Table: Sr.No. No. Complete rotation (m) No. of additional circular division moved (X) h= (m pitch) + (x L.C.) Mean h 1 2 3 4 5 Calculation: Formula- Radius of curvature of the lens Results: Radius of curvature of the lens R=... cm 21

Experiment-4 g BY SIMPLE PENDULUM Aim: Determine the gravitational acceleration g using the simple pendulum s period Apparatus: Meter stick, simple pendulum, timer. Theory: A pendulum consists of a "bob" attached to a string that is fastened such that the pendulum assembly can swing or oscillate in a plane. For an ideal pendulum, all the mass is considered to be concentrated at a point in the center of the bob. Some of the parameters of a simple pendulum include the length (L), the mass of the bob (m), the angular displacement θ through which the pendulum swings, and the period T of the pendulum, which is the time it takes the pendulum to swing through one complete oscillation. Fig.1. When the angular displacement is minimal (θ < 10 0 ) the period of a pendulum can be determined with the following equation. Notice that the period of a pendulum is independent of the mass of the bob. Also, for small displacement angles, the period is independent of θ. Thus, when the period T and length L of a pendulum can be accurately ascertained, it is possible to accurately determine the acceleration of gravity. 22

where L/T 2 represents the slope of data collected and graphed. Observation Table Length of string L (cm) Time taken for 20 oscillation t (sec) t 1 t 2 t =(t 1 +t 2 )/2 Period of oscillation T=t/20 (Sec) T 2 (Sec 2 ) L/T 2 (cm/sec 2 ) 90 80 70 60 50 40 From The Calculation From the Graph(L T 2 ) =-------------- cm/sec 2 ( =------------ cm/sec 2 Result: 1. Gravitational acceleration g =... cm/sec 2 (from the calculation) 2. Gravitational acceleration g =... cm/sec 2 (from the calculation) 23

EXPERIMENT-5 ACCELERATION DUE TO GRAVITY USING BAR PENDULUM (Parallel axis Theorem by Bar Pendulum) Aim: To determine acceleration due to gravity using bar pendulum. Apparatus: Bar pendulum, wedge, stop watch, meter rod and graph paper Theory: The purpose of this experiment is to use the angular oscillations of a rigid body in the form of a long bar for determining the acceleration due to gravity. The particular form of the body is chosen for the sake of simplicity in performing the experiment. The bar is hung from a knifeedge through one of the many holes along the length, as shown in Fig. 1. It is free to oscillate about the knife-edge as the axis. Any displacement θ, from the vertical position of the equilibrium would give rise to an oscillatory motion just as in the case of a simple pendulum. The difference is that since it is a rotating rigid body we have to consider the torque of the gravitational force which give rise to the angular acceleration. Hence the period of oscllation is given by where I is the moment of inertia of the bar about the axis of rotation through O, M is its mass and d the distance between O and the centre of mass of the bar C (Fig. 1). (1) Fig. 1. Experimental set-up of bar pendulum. 24

The parallel axis theorem gives I0 being the moment of inertia of the bar about the parallel axis through its centre of mass; so that equation (1) becomes (2) (3) Note that T becomes very large as d 0 as well as d. You can check that T as a function of d undergoes through a minimum. The plot of T as a function of d has a shape shown in Fig. 2. From equation (3), Fig. 2. Plot between time period T, and distance d. For a given T,it is a quadratic equation in d. So there are two values d1 and d2, the sum of which is or, 25

(4) Procedure: 1. Find the centre of mass of the bar by balancing it on a knife-edge and put a mark there. This is necessary to be able to measure the distance of the various points of suspension from the centre of mass, i.e., ds. 2. The bar has a number of holes in it. A Knife-edge can be passed through a hole and secured to the bar by means of two nuts at the end of the edge. The bar is then suspended, with edge resting on the glass plates of a bracket fixed to the wall. Check that it is the edge, not rough top, of the knife-edge that is resting and that the bar is free of the bracket. 3. Decide on how many oscillations to time by using error estimates in all the measurements. Set the bar oscillating with a small amplitude and time the oscillations Decide on where do you start the count. Take three readings for each hole, i.e., the same value of d. Start at one end of the bar and systematically take readings for determining T for each available value of d till you reach closest to the centre of mass. 4. Plot a graph of T against d as in Fig.(2). Draw a smooth curve through the points. Draw a line parallel to the d-axis cutting the curve at points A and B. Then QA = d1 and QB = d2. Also T = CQ. Get g from equation (4). Do it for at least two values of T. Observation (i) (ii) (iii) Weight of bar M=.gm Length of bar L=.. cm Moment of inertia of a bar w.r.t. the axix passing through center of gravity (C.G.), I cg =Ml 2 /12=. gm.cm 2 26

Observation Table:I Sr. No. d (cm) No of Oscillation 1 2 3 4 5 Time of Oscillation (sec) 1 2 3 Mean Time t (sec) Time period (sec) Calculation: = cm/sec 2 Error Calculation: Observation Table: II Sr. No. Distance from C.G. to suspension point d (cm) d 2 cm 2 Time for 20 oscillations sec t 1 t 2 Aveg. t sec Time period T=t/20 sec I=Mg/4Π 2 [T 2 d] gm.cm 2 Md 2 gm.cm 2 2 I=I cg+md gm.cm 2 Calculation I=Mg/4Π 2 [T 2 d] (compare with I=I cg +Md 2 ) From Graph (I d 2 ) Slope=AB/BC=Weight of Bar=...gm 27

Intercept=I cg =... gm.cm 2 Results: 1. Gravitational acceleration g =... cm/sec 2 (from observation Table I) 2. The Parallel axis theorem in verified and obtained results from the observation table 2. (i) Weight of Bar M=... (From Graph) Weight of Bar M=... (From Calculation) (ii) Moment of Inertia I cg =...(From Graph) Moment of Inertia I cg =... (From Calculation) 28

EXPERIMENT-6 TORSIONAL PEDULUM RIGIDITY MODULUS Aim: To determine the rigidity Modulus of the given wire by dynamical method. Apparatus: Torsional pendulum, stop watch, screw guage, vernier calipers, scale. Theory: A heavy cylindrical disc suspended from one end of a fine wire whose upper end is fixed constitutes a Torsional pendulum. The disc is turned in its old plane to twist the wire, so that on being released, it executes torsional vibrations about the wire as axis. Let ϴ be the angle through which the wire is twisted. Then the restoring couple set up in it is equal to (Πna 4 θ)/2l=cθ Where (Πna 4 )/2l=c (1) is the twisting couple per unit (radian) twist of the wire. This produces an angular acceleration (dw/dt) in the disc therefore if I is the moment of inertia of the disc about the wire we have Idw/dt=-C. θ dw/dt= -(C/I). θ i.e the angular acceleration (dw/dt) of the angular displacement(θ) and therefore its motion is simple harmonic hence time period is given by (2) 29

From 1 and 2 In case of a circular disc whose geometric axes coincide with the axis of rotation. The moment of inertia I is given by where M is the mass of disc and R is the radius of the disc. Procedure: 1. The radius of the suspension wire is measured using a screw gauge. 2. The length of the suspension wire is adjusted to suitable values like 40 cm, 60 cm, 80 cm, 100 cm. etc. 3. The disc is set in oscillation. Find the time for 20 oscillations twice and determine the mean period of oscillation ' T '. 4. Calculate moment of inertia of the disc using the expression, I = (1/2)MR 2. 5. Determine the rigidity modulus from the given mathematical expression. Observation: (i) Radius of the wires ( ) =... cm (ii) Radius of the disc (R)=... cm (iii) Mass of the disc (M) =... gm Observation Table: 40 60 80 100 120 Length of string L (cm) Time taken for 20 oscillation t (sec) t 1 t 2 t =(t 1 +t 2 )/2 Period of oscillation T=t/20 (sec) T 2 (sec 2 ) L/T 2 (cm.sec -2 ) Mean L/T 2 (cm.sec -2 ) 30

Calculation: From Calculation: =-------------------- dynes/cm 2 From Graph: Plot a curve for l Vs T 2 and calculate the slope. =------------------- dynes/cm 2 Result: The rigidity modulus of the given wire using dynamical method η= ------------------dynes/cm 2 (From Calculation) η = ------------------dynes/cm 2 (From Graph) 31

EXPERIMENT-7 YOUNG S MODULUS BY CANTILEVER METHOD Aim: To determine Young s modulus by cantilever method Apparatus: Meter rule, half-meter rule, G-clamp, retort stand and clamp, Thread, 50 g slotted mass hanger, set of 50 g slotted masses, wooden block, pair of vernier calipers, micrometer screw gauge, scissors Theory: A cantilever is a beam supported on only one end. The beam carries the load to the support where it is resisted by moment and shear stress. Cantilever construction allows for overhanging structures without external bracing. In most applications, objects are assumed to be rigid for the purpose of simplification. When supposedly rigid materials are subject to great forces, there is a permanent deformation. When subject to a particular stress, or force per unit area, materials will respond with a particular strain, or deformation. If the stress is small enough, the material will return to its original shape after the stress is removed, exhibiting its elasticity. If the stress is greater, the material may be incapable of returning to its original shape, causing it to be permanently deformed. At some even greater value of stress, the material will break. There are different types of stress: tension or tensile stress, compression or compressive stress, shear stress, and hydraulic stress. The quantity for all types of stress, however, can be defined as follows: where F is the force applied and A is the cross-sectional area of the material. The standard units of stress are [N/m 2 ]. The quantity strain can be defined as follows: where L is the original length of the material, and L is the change in length that results after the stress is applied. 32

The Young s modulus E of wood of the meter rule is given by where M = mass of the slotted mass, d = deflection of the end of the ruler. s = gradient of graph d against M Fig. 1. Procedure: 1. The average thickness of the meter rule is measured by using a vernier caliper and the reading is recorded. 2. By using a micrometer screw gauge, the average width of the meter rule is measured and the reading is recorded. 3. The apparatus is set up as shown in Fig. 1. 4. A 10 g of slotted mass is hung and the length of the deflection is observed. 33

5. The reading obtained is tabulated in a table. 6. Step 4 and 5 is repeated by using a slotted mass of 20 g, 30 g, 40 g, 50 g and 60 g. 7. A graph of deflection of the end of the ruler, d against mass, M is plotted. 8. The Young s modulus, E of wood of the meter rule is calculated. Observation: (i) Reading of the thickness and width of the meter rule (1) t 1 =. cm (2) t 2 =. cm (3) t 3 = cm (4) t= (t 1 +t 2 +t 3 )/3= cm (2) b 1 =. cm (2) b 2 =. cm (3) b 3 = cm (4) b= (b 1 +b 2 +b 3 )/3= cm Observation Table: Sr.No. Mass (g) Distance (d) cm Average Loading (d 1 ) Unloading (d 2 ) d= (d 1 +d 2 )/2 1 0 2 10 3 20 4 30 5 40 6 50 7 60 8 70 9 80 10 90 11 100 Calculation: From Calculation: = N/m 2 From Graph:(d M) =.. N/m 2 s = gradient of graph d against M Results: 34

EXPERIMENT-8 MODULUS OF RIGIDITY BY BARTON S APPARATUS Aim: To determine the modulus of rigidity of the material of a given rod by static method. Apparatus: Torsion apparatus, Slotted weights with hanger, Thread, Meter scale and Screw gauge. Theory: Working Formula: C= 2ΠR Where η is the modulus of the rigidity of the materials of a given rod. M is the mass suspended g is the acceleration due to gravity (g =980 cm/sec 2 ) L is the length of the rod between the two pointers 35

C is the circumferences of the pulley R is the radius of the pulley. r is the radius of the rod. ϴ 1 is the angle of twist produced at pointer No.1 ϴ 2 is the angle of twist produced at pointer No.2 Observations: (i) Least count of screw gauge LC = (ii) Circumference of the pulley C = cm. (iii)radius of the pulley R = cm. (iv) Length of the rod between the two pointers = L= Observation Table: Sr.No. Mass Suspended M (g) 1 0 2 500 3 1000 4 1500 5 2000 6 2500 Reading on Scale(ϴ) ϴ= ϴ 1 + ϴ 1 ϴ 2 ϴ 2 loading unloading M/ ϴ Average M/ ϴ Calculation: Actual Value, η= 4.55 10 11 dynes/cm 2 Percentage of error = From Graph (M ) Slope=M/ ϴ Slope 36

Results: = ---------------------( calculation) dynes/cm 2 Percentage error = dynes/cm 2 = ---------------------( graph) dynes/cm 2 Percentage error = dynes/cm 2 37

EXPERIMENT-9 FREQUENCY OF TUNING FORK BY MELD S METHOD Aim: To determine the frequency of AC mains by Melde s experiment Apparatus: Electrically maintained tuning fork, A stand with clamp and pulley, A light weight pan, A weight box, Analytical Balance, A battery with eliminator and connecting wires etc. Theory: STANDING WAVES IN STRINGS AND NORMAL MODES OF VIBRATION: When a string under tension is set into vibrations, transverse harmonic waves propagate along its length. When the length of string is fixed, reflected waves will also exist. The incident and reflected waves will superimpose to produce transverse stationary waves in the string. The string will vibrate in such a way that the clamped points of the string are nodes and the point of plucking is the antinode. Fig. 1. The Envelope of a standing waves A string can be set into vibrations by means of an electrically maintained tuning fork, thereby producing stationary waves due to reflection of waves at the pulley. The loops are formed from the end of the pulley where it touches the pulley to the position where it is fixed to the prong of tuning fork. Fig. 2 38

(i) For the transverse arrangement, the frequency is given by where L is the length of string in fundamental modes of vibrations, T is the tension applied to the string and m is the mass per unit length of string. If p loops are formed in the length L of the string, then (ii) For the longitudinal arrangement, when p loops are formed, the frequency is given by Procedure: Constant length 1. Set experiments apparatus in longitudinal position and put some weight in pan and adjust length of string in such a way that you find loops. 2. Rotate apertures by 90 0 to set a B position for same value of weight in pan and same length repeat same process. The loops formed is twice at that in (1) 3. Note readings and repeat process. Constant Mass 4. Set experiments apparatus in transverse position and put some weight in pan and adjust length of string in such a way that you find loops. 5. Change length of string for same value of weight in pan as the no. of produced loops should increases or decreases. 39

Observations: (i) Mass of the pan, m 1 = gm (ii) Mass of unit length of string, m = gm/cm Part A: Constant length Sr. No. No of loops (p) M=m 1 +m 2 (g) T=Mg p 2 T Average p 2 T Calculation f=... Hz Part B: Constant Mass Sr. No. Mass Suspended on string M=m 1 +m 2 (g) T=Mg No of loops (p) Length one loop L=l/p of Average 40

Calculation: f=... Hz Results: 1. Frequency of string = f=..hz 2. Frequency of tunning fork= N= 2f= Hz 41

EXPERIMENT-10 PRISM ANGLE BY SPECTROMETER Aim: To determine the angle of the given prism Apparatus: A Sodium lamp, Spectrometer, a spectrometer prism, a spirit level, magnifying glass, a table lamp etc. Theory: When a beam of light strikes on the surface of transparent material (Glass, water, quartz crystal etc.), a portion of the light is transmitted and the other portion is reflected. When a beam of light strikes on a plane surface, the angle of reflection will be the same as angle of incidence. If the angle between two reflected ray is measured as θ, then the angle of the prism is Procedure: A= θ/2 1. Cross wires of the telescope are adjusted properly and then the telescope is focused on a distant object. A clear and sharp image is got in telescope with the proper adjustment of the telescope screws. This will make the telescope, set for parallel rays. 2. The telescope is now turned to a position where the axis of telescope and collimator are in the same line. The slit of collimator is adjustment so that a fine and sharp image of the 42

sodium lamp that illuminates the slit is obtained in the telescope. Now slit is at focus of telescope wire and is adjusted for parallel rays. 3. Determination of angle of prism. To find the angle of prism, the prism is paced at centre of prism table with its base perpendicular to the axis of collimator and the edge towards collimator. Parallel rays from collimator will be incident on refracting surface AB and AC are reflected by reach face. 4. The telescope is now turned to receive the light reflected from face AB and its position is so adjusted that the image of slit is at the centre of vertical cross-wires of telescope. The position of both venires of telescope is recorded. 5. The telescope is then turned and it receives the light reflected from face AC. Then it is adjusted till the image of both slit of collimator coincides with the centre of vertical cross wires of telescope. The position of both the venires is again recorded. 6. The mean difference of these two reading gives the angle 2A, and half of this angle of prism A. Observations: (i) Vernier constant of spectrometer = 43

Observations Table: Sr.No. Venier V 1 Position of Telescope Venier V 2 Position of Telescope 1 2 3 Face AB Face AC Difference Face AB Face AC Difference Result: 1. Mean value of = 2. Angle of prism A = /2 =. 44