PY001/051 Compound Pendulum and Helical Springs Experiment 4 Physics 001/051 The Compound Pendulum Experiment 4 and Helical Springs Prelab 1 Read the following background/setup and ensure you are familiar with the concepts and theory required for the experiment. Background/Setup Today s laboratory session consists of two distinct experiments: (i) Use of the compound pendulum to measure the acceleration due to gravity and the radius of gyration of the pendulum and (ii) Use of a helical spring to determine Young s modulus and the modules of torsion for steel. Experiment 1: The Compound Pendulum A simple pendulum consists of a string (of negligible mass) attached at a pivot point and a mass at the other end. The motion of the mass can be described as being simple harmonic in nature. The simple pendulum can be viewed as a point mass and is a resonant system with a single resonant frequency. The period of motion for a simple pendulum is given by: l T = π, (1) g where l is the distance from the pivot point to the centre of the suspended mass. A compound pendulum is a more realistic physical system, in which the mass is distributed over the entire rigid body. In the compound pendulum a hole is drilled into the body and it is free to pivot around a rod inserted through this hole. The motion is also simple harmonic in nature and use is made of the centre-of-mass of the body and the moment of inertia in order to describe the motion. In this experiment we will study the properties of a compound pendulum. A metre stick with holes drilled through it at various positions along its length will be used. The expression for the period of a compound pendulum is given by I T = π, () mgh where I is the moment of inertia of the pendulum about the axis of suspension, m is the mass of the pendulum and h is the distance from the point of suspension to the centre-of-mass. Figure illustrates a metre stick as a compound pendulum with a rotation axis S. h 4.1
PY001/051 Compound Pendulum and Helical Springs Experiment 4 S h θ mg Figure : Compound pendulum We can express I in terms of the moment of inertia of the body about an axis through its centre-of-mass, C: I CM = mk, (3) where k is the radius of gyration about an axis through C. For any regular body, k can be readily calculated and for an irregular body k must be determined experimentally. The moment of inertia about any axis parallel to the one through C is given by the parallel axis theorem such that I = I + CM mh, (4) where h is the distance between the two axis. Substituting equations (3) and (4) into equation () we get that k + h T = π. (5) gh This function expresses the period of oscillation in terms of the geometry of the rigid body. The period is independent of mass but depends on the mass distribution (through k) and the location of the pivot point (through h). Since k is a constant for any rigid body, the period of the compound pendulum is a function of h only. We can determine the minimum value of T with respect to h by setting: T min dt k = 0 T min = π. (6) dh g Hence, by determining one can determine a value for the radius of gyration, k. By comparing equation (6) to equation (5) it can be shown that T min is obtained when h = k. 4.
PY001/051 Compound Pendulum and Helical Springs Experiment 4 If we equate the expressions for the simple pendulum in equation (1) and the compound pendulum in equation (5) and solving for h, we can find two values for h such that the compound pendulum has the same period as a simple pendulum of length l: 4 l ± l k h =. (7) There are two solutions for h, which we will label h1 and h, for which the simple pendulum and compound pendulum yield the same period. We can therefore rewrite equation (1) as h1 + h T = π. (8) g Prelab Read the rest of this lab description and set-up tables for your data collection. You should have identified the goals of the experiment and the measurements that you must make before coming into the lab. Experimental Procedure 1. Determine the theoretical value of k for the rigid body by measuring the length of the stick l and the width w and using k = ( l + w ) 1.. Find the centre of gravity of the rigid body by balancing the stick on a sharp edge. Note its position carefully. 3. Position the pivot rod through one of the holes in the stick and record h, the distance from the centre of the hole to the centre of gravity as determined in Step. 4. Measure the period of oscillation T for small amplitude oscillations (using 10 or more oscillations). Repeat the measurement at least three times in order to determine an average value for T plus its estimated error for a particular value of h (determined from Step3). 5. Repeat Step3 and Step4 above for all the holes on the stick. Note: In this method we have assumed sin θ θ (angular displacement in radians). Using your calculator investigate the maximum amplitude displacement that can be used to ensure this is true. 6. Plot a graph of your recorded values of T versus h. The graph should show that there are two values of h for which identical values of T are obtained as predicted by equation (7). 7. From your plot find the minimum value of T, Tmin with respect to h and, hence, determine the radius of gyration, k (from equation 6). Compare this value to the theoretical value obtained in Step1 above using the dimensions of the metre stick. Which method is more accurate? 8. Using the theoretical value of k from Step1, plot the theoretically predicted variation of the period with h on the same graph where you displayed your experimental results. Compare and comment. 4.3
PY001/051 Compound Pendulum and Helical Springs Experiment 4 9. By rewriting equation () as 4π k T = + h g h it is possible to replot your data as a linear relationship of T versus h. Fit a curve to this plot (using software such as CurveExpert) and extract the values for g and k by fitting a curve of the form: T 39.478 b = b + h. a h How do these extracted values of g and k compare to predicted values for g and the value for k determined above? Experiment : Elastic Constants for Steel Wire In the laboratory session two weeks ago we investigated Young s modulus, Y, and the modulus of rigidity, η, for a brass rod through sound wave propagation. In this experiment, we will measure these constants for a steel helical spring with a mass suspended on the end. Experiment.1: Modulus of Rigidity for Steel 1 The period of horizontal oscillation of a torsion pendulum is given by T hor I = π, (9) k where I is the moment of inertia of the suspended mass about its axis of rotation and k is the torsional constant. The moment of inertia for a thin rod suspended at its centre of mass is given by where m is the mass of the rod and l is its length. ml I = (10) 1 The torsional constant k is related to the modulus of rigidity, η. For a wire of crosssectional diameter d and total length L it can be shown that π d 4 η k =. (11) 3L We can eliminate k and I from equation (9) and rewrite the period of oscillation in terms of readily measurable quantities: π 3IL ml L T hor = = π 3. (1) 4 π d η 1 π d 4 η 1 A value of 8. x 10 10 Nm - is expected 4.4
PY001/051 Compound Pendulum and Helical Springs Experiment 4 Experimental Procedure 1. Measure the parameters m, l for the suspended rod and L and d for the steel wire. Note that L is related to the diameter of a single spiral of the spring, D, and the total number of turns, N, by L = π DN. Include appropriate errors on all your values.. Determine the period of horizontal oscillations, T hor, for the system by turning the system and timing at least 10 complete oscillations. Repeat the measurement at least three times in order to get the average period of oscillation (and include appropriate errors). 3. From the average experimental value of and using equation (1), determine the modulus of rigidity, η, for steel and compare to tabulated values (these can be obtained from any standard physics text book). Comment on your results. T hor Experiment : Young s Modulus for Steel It can be shown that the motion of a mass m suspended on the end of a spring is simple harmonic in nature and that the period of vertical oscillations is given by T ver where k is the spring constant (stiffness) of the spring. T ver m = π, (13) k For an elastic material it is also known that the force applied to the material is related to Young s modulus, Y, for that material (c.f. Experiments with Sound two weeks ago) and it is also related to the spring constant, k. It can be shown that for a helical spring the relationship between k and Y is given by 4 Yd k =, (14) 3 8ND where d is the cross-sectional wire diameter, N is the number of turns in the spring and D is the average diameter of a single spiral of the spring. Hence, we can rewrite equation (13) in terms of readily measurable quantities and find that the period of oscillation is given by T ver 3 8mND = π. (15) 4 Yd A value of.1 x 10 11 Nm - is expected 4.5
PY001/051 Compound Pendulum and Helical Springs Experiment 4 Experimental Procedure 1. Measure the mass m of the suspended rod and N, D and d for the steel wire. Include appropriate errors on all your values. (Assume the mass of the spring is negligible). Determine the period of vertical oscillations, T ver, for the system by forcing vertical oscillations on the spring and timing at least 10 complete oscillations. Repeat the measurement at least three times in order to get the average period of oscillation (and include appropriate errors). 3. From the average experimental value of T ver and using equation (15), determine Young s modulus, Y, for steel and compare to tabulated values (these can be obtained from any standard physics text book). Comment on your results. 4.6