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1 Sandeep Bala 1st Year B.Tech. Indian Institute of Technology Mumbai , India. Torsion Pendulum Experiment at the International Physics Olympiad Introduction The International Physics Olympiad is held every year to test the theoretical and experimental skills of the best physics students from high schools all around the world. The theory part of the examination has three long problems. to be solved in five hours. The experimental part of the examination is given 40% weightage and may consist of two short experiments or one long experiment to be carried out in five hours. This year the 30th Physics Olympiad was held at Padua in Italy and 291 students from 62 countries participated. Here, a long experiment on torsion pendulum was chosen to test the skills of the students and the examination was conducted in two batches. By choosing this problem the organisers had ensured that the experimental set-up was simple, inexpensive and could be quickly and easily manufactured with uniform quality in large numbers ( ~ 150) so that every student faced the same level of difficulty in conducting the experiment. There were no delicate components and sufficient length of wire was provided for replacement in case a careless student broke it while assembling the torsion pendulum. The problem chosen was based on secondary and higher secondary school level physics. It tested the skills of the students (a) to assemble the experimental set-up, (b) to make measurements using only the given instruments, (c) to present the collected data appropriately in the form of tables, (d) to derive the necessary equations for estimating physical quantities from experimentally determined data, (e) in choosing the best way of obtaining the results with least errors and (f) in the estimation of errors to say something about the reliability of the conclusions. (In fact more weightage was given to the estimation of errors than to the actual obtained value.) ~ R-E-S-O-N-A-N-C-E--I-J-un-e

2 V N t i I I J I I./ j'f. Figure 1. Sketch of apparatus with horizonfdl torsion axis, C, C 2 w-wir4 ov vertical. PositivesIde, ~~ e Apparatus Figure 1 shows the experimental set-up. The pendulum has an outer cylinder A of mass MI. A is not longitudinally uniform (part of it is hollow) and has hole 0 at one end. Through the hole a steel wire passes. After clamping the steel wire at C l' the handle H is used to pull and keep the wire under tension before clamping at C 2" A is locked to the wire using the 'sunk' screw S 1. Through the other end of A a uniform threaded inner rod B of length 1 and mass M z can be screwed in and out along the cylinder axis, to vary the length of the pendulum x (measured with respect to 0) between Xl and x z. The inner rod cannot be removed from the outer rod nor can it be completely screwed inside. Asmall hex nut (hn) is used to lock the position of the inner rod every time one moves the threaded rod. (The mass of the hex nut is included in M 1 and is considered to be a part of the outer cylinder.) The direction along the steel wire (C - 1 C z ) is the ~ RESONANCE I June 2000 T1

3 torsion axis and O-X is along the axis of the cylinders. When O X is along O-N the 'elastic restoring torque' is zero. The direction of the gravitational force is of course the vertical. Theory The experimental problem required one to study the torsional oscillations of the pendulum (i) with the torsion axis of the pendulum along the vertical direction (not shown in the figure) and (ii) with the torsion axis in the horizontal plane (as shown in Figure 1). The first case is quite simple and is a standard simple harmonic motion problem. Here, a displacement of the pendulum in the horizontal plane will twist the wire resulting in an increase in the potential energy U e associated with the 'shear elasticity' of the wire. The elastic torque tries to restore the pendulum to its original position. For small angular twists the elastic restoring torque Te' is proportional to )h' the angle between the cylinder axis O-X and the direction O-N. where K is the torsional constant of elasticity. If / is the moment of inertia of the 'pendulum', the equation of motion is similar to that for a simple harmonic motion and the expression for the period (= 'l') of torsional oscillations is (1) r (x) = 2n (/ / K)0.s. (2) The second case is more complex. With the apparatus set-up as shown in Figure 1, if the pendulum is displaced in a plane perpendicular to the wire in the plane of the paper, not only is there an increase in the elastic potential energy but also there is a change in the gravitational potential energy. The expressions for the restoring torque T and the potential energy U in this case can be written as follows. T = -K ( )- )0) + (M l + M) g R(x) Sin ), (3) n ~ r-e-s-o-n-a-n-c-e--i-j-u-ne

4 v (0) = - ftdo = 0.5 K (0- ( 0 )2 + (M l + M) g R(x) CosO, after choosing the constant of integration as the reference point of zero energy with respect to which the potential energies are given. 0 is the angle made by the pendulum axis O-X with the vertical, 0 0 is the angle between O-N and the vertical and R(x) is the distance of the centre of mass of the pendulum with respect to O. Depending on the extent of insertion of the inner rod into the outer cylinder, x and R(x) vary. The expression for V(O) shows some interesting features: for -n/2 < 0 < n/2, the second term of V( 0) decreases with increase in I 0 I (i.e. on either side of the vertical), whereas the first term decreases until 0 = 0 0 and increases on further increasing 0 on the positive side of the vertical or always increases with its increase on the negative side of the vertical. Thus, on the positive side of the vertical we will always get a 0 = 0 e for which V( 0) is a minimum. But on the negative side of the vertical we may get a minimum only if the second term is sufficiently large. Figure 2 shows plots of U*(O) (= V(O) / {(M l )gr(x)}) for two choices of f3 (f3 = K/{2(M l )gr(x)}). The doubling of the potential energy minimum is. seen for a sufficiently small (4) = ' 2 "c :l 1.2 >- ~ cti * ::> 0.8 ~ ----L --'- -' ' -2-1 o 2 8 in radians Figure ~~ RESONANCE I June

5 value of f3 and is termed 'bifurcation'. The 'bifurcation' can be observe~ only if the 'symmetry of the system is broken' by having a non zero 8 o ' This sort of bifurcation is seen in particle physics and statistical mechanics with various kinds of symmetry breaking. We note that for the set-up of Figure 1, the 'minimum' on the positive side of the vertical is more stable, located farther from the vertical than the 'minimum' on the negative side. With the set-up shown in Figure 1, we see the existence of two equilibrium positions ('bifurcation') by loading the free end of the threaded rod with a heavy nut so that both Ml + M2 and R(x) increase. At the equilibrium position corresponding to 8 = 8 e' the net restoring torque is zero and hence using (3) we can write, The equation of motion of the pendulum can be written as follows: (5) Since the equilibrium position does not change with time d 2 8 e /dt 2 = 0 and (6) can be rewritten as follows using 8 = 8 e + 8 d J(x)(d 2 8 d /dt 2 )=-K (8 e + 8 d - 80) + (M l + M)g R(x) Sin(8 e + 8 d ) = - K (8 e - ( 0 ) + (M l + M 1 ) g R(x) Sin8 e Cos8 d -K 8 d + (M l )g R(x) Cos8 e Sin8 d. For small deviations 8 d from the equilibrium position, Cos8 d ::::: 1 and Sin8 d ::::: 8 d and we can use (5) to cancel the first two terms. Equation 7 is also similar to that for simple harmonic motion. Interestingly the period of oscillation r depends on the equilibrium angle 8. e r=2n[l(x)/{k-(m 1 +M)gR(x)Cos8)]0.s. (8) Since x can be varied, we must also know how R(x) and lex) vary with x. Both the cylinders constituting the pendulum are ~ RESONANCE I June 2000

6 laterally uniform. Therefore, we can take their centres of mass to lie on the pendulum axis O-X. Also, since the inner cylinder part is uniform its centre of mass will be at its centre, i.e. at a distance R2 (= x -1/2) from O. If Rl is the distance of the centre of mass of the outer cylinder part from 0, R (= the distance of the centre of mass of the pendulum as a whole from 0) can be written as follows making use of the expression for R2 : R(x) = {R 1 Ml / (M 1 ) -I M2 / 2(M 1 )} + x M2 / (M 1 + M) (9) The moment of inertia of the pendulum I(x) is the sum of the moments of inertia of the outer cylinder part (= II) and the inner cylinder part (= I). Since the outer cylinder is fixed II does not vary with x. Assuming the lateral dimensions of the inner cylinder to be negligible we can write the following expressions for 12 (using the parallel axis theorem) and I(x) ( = II ), Experiments 12 = Mi 2 / 12 + M2 (x -l/2)2 I(x) = II + (Mi 2 / 3) - x Mi + x 2 M2 (10) The main aim of the experiment was to determine the periods of oscillation for different choices of x for the torsion pendulum assembled with its torsion axis in the horizontal plane and then to compare it with the functional form expected using the theoretical equation relevant for that specific pendulum. Only the total mass Ml + M2 (~ 40 gms) was given as data. The measurements had to be carried out using only some adhesive tape (if necessary), at-shaped rod, millimeter graph papers, a timer, a graduated ruler and a right triangular plate. No weighing scale was provided and the inner cylinder could not be separated from the outer cylinder. Since only a non-programmable scientific calculator was allowed, the slopes and intercepts of linear plots could not be determined by feeding the data to the calculator and running a least square analysis programme. The problem in the above form seems a bit difficult for high school students. Therefore, at the Olympiad the problem was -RE-S-O-N-A-N-C-E--I-J-u-ne ~ m-

7 divided into several sub-parts and the student was asked to follow the 'steps' given in the instruction sheet. Thus, the student was asked first to write an equation for R(x) as a function of x and the parameters Ml' M 2, Rl and I, as in (9). Then the student had to measure R(x) for several values of x (at least 3) (obviously before attaching the pendulum to the steel wire). After preparing suitable tabular columns, by the screw like motion of the inner threaded rod, convenient and nearly equally spaced values x could be chosen. The pendulum is balanced on a T -shaped rod. The distance of the point of balance from 0 gives R(x). Equation (9) tells us that a plot of R(x) against x is going to be linear and the slope of the straight line plot can be used to get M/(M 1 ). Since the total mass is known, the masses M 1 and M 2 can be determined making use of the slope of the plot. The third step required the student to write the equation for the total moment of inertia I(x) as a function of x and the parameters M 2, II and 1 as in (10). In the fourth step with the torsion axis in the horizontal plane the pendulum equation of motion had to be obtained as a function of 8, x, K, 8 0, Ml' M 2, I(x) and R(x) as in (6). In the fifth step the pendulum had to be assembled with its torsion axis in the horizontal plane and K had to be estimated from the equilibrium angle 8. This requires one to first write e equation (5). From this equation it is obvious if some how we set up 8 0 = 0, the equation will be satisfied for 8 e = whatever may be the value of x and we will not be able to get K by plotting the data. Therefore, it is prudent to fix the pendulum to the wire (using S 1) with a substantial value for 8 0, The instructions given also say that the equilibrium angle 8 e should substantially deviate from the vertical and should be measured for five or more values of x. As no protractor was available, 8 e had to be determined from Cos8 e which could be obtained by measuring h, the height of the tip of the pendulum from the level of 0 (cf. Figure 1) for the chosen x and using Cos8 e = h/x. h could be measured using the right triangular plate and the graduated ruler. Alternatively, -~ ~ R-E-S-O-N-A-N-C-E--I-J-un-e

8 one could determine Sin B (= p/x) by measuring p, the horizone tal distance between 0 and the topmost point of the pendulum (cf. Figure 1). P could be measured by fixing a graph paper on the table using adhesive tape and taking projections of 0 and the tip of the pendulum on this paper with the help of the right triangular plate and the ruler. The next step is to use the data collected (as described above) to determine K with the help of (5). It is important to note that the analysis is easier if the data can be represented by a linear equation. Equation (5) can be considered to be a linear equation of the type a y = b z + c, where y (= R(x)SinB) and z (= Be) are variables, a (= (M l + M)g), b (= K) and c (= - KBo) are constants. Using the experimentally determined Be for n choices of x we can set up n such equations. R(x) needed for this purpose, if not already determined could be calculated with the help of the slope and intercept of the earlier plot corresponding to (9). By solving these simultaneous equations we get b/a and cia using which K and Bo can be determined. The solutions of the simultaneous equations can be obtained algebraically by taking two equations at a time. Choosing different pairs of such equations we can get several estimates of K and Bo and also the average of these estimates. However, such a method is not only time consuming but may also lead to greater errors in the quantities determined. A graphical method is therefore preferred. The simultaneous equations can be solved graphically by plotting y against z. Since a, band c are constants, ideally this plot should be a straight line. Therefore, the slope and intercept of the straight line which deviates least from the experimental points of this plot give estimates of b/a and cia, respectively from which K and 8 0 can be determined. Nearly 42.5% of the marks were allotted for the fifth and sixth steps indicating the importance of careful choice of x, careful measurements and the plot in these steps. In the seventh step the torsional pendulum is set up with its torsion axis along the vertical. With such an arrangement]l and 1 had to be estimated from the periods of torsional oscillations r -R-ES-O-N-A-N-C-E--I-J-u-ne ~~ ~-

9 determined for several values of x (at least 5). For this situation the expression for r is given by (2). Equation 2 can also be made to look like a linear equation by using (10) and rewriting it as Note that it is not physically possible to have x = 0 or I. After rearrangement, (11) also can be recognised to be a linear equation, y = b z + c, with a new set of variables: y = (K,.,;z / 4n2 - X Z M 1 ), z = x and constants: b = -Mzl andc = (11 + M1lz /3). Again the linear simultaneous equations set up using the experimental data can be solved algebraically or graphically. In the graphical method, the slope of the straight line which deviates least from the experimental plots of y against z give -M z I. This can be used to get I since M z is already determined as described above. Equating the intercept of the straight line plot to (11 + M z [2/3) we can then determine 11 also. At the end of the seventh step all the system parameters needed to predict the period of torsional oscillations as ~ function of x, with the torsion axis in the horizontal plane would have been determined, see (8). The eighth and final step is to determine r as a function of x in this configuration of the pendulum and discuss whether it is a decreasing or increasing function of x. This can be compared with the r predicted using (8). It was also necessary to discuss the reliability of the results obtained. The obvious thing to do is to obtain error estimates from the plots and use them in the discussion. It would be interesting to discuss the extent of errors which might have been introduced by the assumptions used in the analysis. One such assumption pertains to the tension in the steel wire since we do not have precise control over this factor. Is it possible that K depends on the tension to which the steel wire is subjected either during the experiment or at the time of setting up the apparatus? Ifwe had stretched the wire close to the elastic limit or beyond the elastic limit, can K and e be affected? To what extent do these parameters reflect the 'history' of the wire and ~ ~~ R-E-S-O-N-A-N-C-E--I-J-un-e

10 reflect the 'fatigue'? Tom Morfett of England pondered about such questions and got the special prize for the 'most original experimental solution' even though he did not get 100% marks. It was not impossible to score 100% in this examination. Konstantin Kravstov of Russia showed this by scoring 100% marks. Finally, one may also ponder over other practical applications of such a set-up. Since the time period of the pendulum depends on the inclination of torsion axis, by measuring 'f accurately using modern electronic timers one can determine the direction of the 'vertical' and the magnitude of g. Such measurements may be of use in aircraft and artificial gravity experiments. Ingenhouszian Motion? What we call Brownian motion was first observed by a Dutch pl~ysician, Jan Ingenhousz (in 1785), by looking at trajectories of finely divided charcoal on the surface of alcohol. He was also apparently the court physician of Maria Theresa, for which he received the princely sum of 5000 gold gulden, thus giving him time off to study the motion of charcoal on alcohol. Incidentally, he is probably better known in the biological community as the person who first noted that sunlight was essential for the process by which plants converted carbon dioxide to oxygen, or photosynthesis. He also vaccinated the family of George III of England against smallpox. A reference is 'Dictionary of Scientific Biology, ed. C C Gillespie, Scribners, NY, (1973), p.11. Perhaps Ingenhousz did not publish in a refereed journal. Gautam Menon Institute for Mathematical Sciences, Chennai ~~ RESONANCE I June