Mechanics:- Simple and Compound Pendulum

a compound pendulum is a physical system with a more complex structure than a simple pendulum, incorporating its mass distribution and dimensions into its oscillatory motion around a fixed axis. Understanding its dynamics involves principles of rotational mechanics and the interplay between gravitational potential energy and kinetic energy. Compound pendulums are used in various scientific and engineering applications, such as seismology for measuring earthquakes, in clocks to maintain accurate timekeeping, and in mechanical systems to study oscillatory motion dynamics.

1. Compound Pendulum # Topics a) Time period of compound pendulum b) Maximum and Minimum Time periods of a compound pendulum

2. Simple Pendulum A pendulum is suspended body oscillating under the influence of the gravitational force acting on it. This is the most accurate method for determining of acceleration due to gravity. The simple pendulum whose period of oscillation is, where, g = acceleration due to gravity l = is the length of pendulum

3. Compound Pendulum The real (or rigid body) pendulum which can be of any shape object or a rigid body capable of oscillating in a vertical plane freely about a horizontal axis passing through any point of it is called physical pendulum or compound pendulum. The moment (or torque) of a force about a turning point is the force multiplied by the perpendicular distance to the force from the turning point. Moments are measured in newton metres (Nm). Moment = F d F is the force in Newton d is the perpendicular distance Example:   A 10N force acts at a perpendicular distance of 0.50m from the turning point. What is the moment of the force? Moment = F . d = 10 x 0.50 = 5.0 Nm

4. A couple is two equal forces which act in opposite directs on an object but not through the same point so they produce a turning effect. The moment (or torque) of a couple is calculated by multiplying the size of  one  of the force (F) by the perpendicular distance between the two forces (s). E.g. a steering wheel in a car; Moment of Couple = F . s

5. ---------------------(1) ---------------------(2)

6. Equate equation (1) and (2), we get This is the differential equation of angular SHM showing that the motion of the compound pendulum is simple harmonic. The time period of oscillation is given by, ---------------------(3) ---------------------(4)

7. Let Ig be the moment of inertia of the rigid body about an axis parallel to the axis of suspension but passing through its centre of gravity G. Then from the parallel axis theorem, we can write, I = Ig + ml 2 If k be the radius of gyration of the rigid body about an axis passing through the centre of gravity G, then Ig = mk 2 . Thus we have, I = mk 2 + ml 2 = m(k 2 + l 2 ) Hence, the time period becomes, ---------------------(5) The moment of inertia about any axis parallel to that axis through the center of mass the moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass.

8. Length of Equivalent Simple Pendulum The time period of compound pendulum is, Comparing this time period of a compound pendulum with the time period of an ideal simple pendulum of length L if we use above equation, then two time periods would be the same. This length L of a compound pendulum is known as the length of equivalent simple pendulum. If the whole mass of the compound pendulum is concentrated at a point C at a distance from the point of suspension, as shown in Fig. This pendulum behave like a simple pendulum of same period.  The point C is then the centre of oscillation.

9. Maximum and Minimum Time Periods of a Compound Pendulum The time period of compound pendulum is, Squaring on both sides, we get, Differentiating the above eqn. with respect to l, we get, (a) If l = 0, the R.H.S. of eqn. (2) becomes infinite. Hence, ------------(1) ------------(2) The time period of the pendulum gradually increases as the point of suspension is shifted slowly towards the centre of mass. The time period of the pendulum is maximum when the axis of suspension passes through the centre of gravity of the pendulum. Hence the time period of the pendulum is minimum , when the distance between the centre of suspension and centre of gravity ( c.g .) becomes equal the radius of gyration of the pendulum about the horizontal axis passing through c.g ., we then have, (b) If l = k, the R.H.S. of eqn. (2) is zero.

10. Centre of suspension & centre of oscillation are mutually interchangeable The point O through which the horizontal axis of suspension passes is known as the centre of suspension. The length of equivalent simple pendulum is, If the line OG is extended up to a point C, then the point C is called the centre of oscillation. Thus, OC = OG + GC Let the distance of the c.g . from the centre of oscillation C = l′. Then we have, If the pendulum is suspended about the point O, then the time period becomes, Again, when the pendulum is inverted and suspended about the point C, then the time period T′ is,     -------(1) -------(2) Eqn. (1) & (2)

11. Thus, the centre of suspension and the centre of oscillation are interchangeable. If the distance between the centre of suspension and the centre of oscillation is L and knowing the time period about either of them, we have the value of g as follows, ------------------------------------------(3)

12. Reversible Compound Pendulum A compound pendulum with two knife edges (i.e., the point of suspension and point of oscillation) so placed that the periods of oscillation when suspended from either is the same is called a reversible compound pendulum. The time period of the compound pendulum from one side is, squaring on both sides, and the time period of the compound pendulum from other side is, ------------(1) ------------(2) squaring on both sides, ------------(3) ------------(4) Let us put T′ = T, and subtract eqn. (4) from eqn. (2), we get,

13. Continued…… If l ≠ l′, then or ------------(5) The distance between any two points on opposite sides of G and at unequal distances from G, the time periods about them being exactly equal, then the distance ( l + l’) is equal to the length of the equivalent simple pendulum and is given by L = l + l′ Experimentally, we can measure the values of l, l′ and the time period T = T′. Hence, the acceleration due to gravity can be found using eqn. (5).

14. In actual practice, it is extremely difficult to find the positions of the axes for the time period T and T′ to be exactly equal. Continued…… According to Friedrich Wilhelm Bessel, it is found that, if the two periods are nearly equal is also sufficient. Then the time periods of the compound pendulum of both sides (after squaring) using eqns. (1) and (3) can be written as, ------------(1) ------------(3) On subtracting, above equations, we get, ------------(6) ------------(7) ------------(8)

15. Continued…… It can be solved by using partial fractions as follow, where A and B are undetermined constants. Comparing the coefficient of l and l′ of eqns. (8) and (10), we get, ------------(9) ------------(10) ------(8) A + B = T 2 and A – B = T′ 2 Substituting the values of A and B in eqn. (9), we get, ------------(11)

16. Continued…… Equation (11) is called Bessel’s formula which is used to determine the acceleration due to gravity ‘g’ using reversible compound pendulum. ------------(11) As T = T′, (T 2 – T′ 2 ) → 0 but ( l – l′ ) is appreciable. As the second term in the denominator of eqn. (11) is very small as compared to the first term, hence ignoring the second term. It is enough to determine the position of centre of gravity by balancing the pendulum on a horizontal knife edge and measuring l and l′. Thus, ------------(12)

17. Kater’s Reversible Pendulum is the compound pendulum, and this is used to give an accurate value of g. T he principle is based on reversibility of centre of suspension and centre of oscillation. It is therefore called Kater’s reversible pendulum . I t consists of a metal rod of uniform cross section, a heavy cylindrical weight W 1 can be F ixed near its one end so that the centre of mass of the rod is shifted towards this end. T wo weights W 1 and W 2 , one heavy of metal and others light of wood can be made to slide along the length of the bar and clamped at any position. T wo knife edges k 1 and k 2 facing each other may be adjusted near the two edges as shown in Fig. Kater’s Reversible Pendulum Kater’s Pendulum

18. Kater’s Reversible Pendulum Continued…… The pendulum is suspended with a knife edge k 1 on the horizontal rigid support and time period T is measured. The pendulum is reversed and allowed to oscillate about the knife edge k 2 . The time period T′ are again measured. The mass W 1 is adjusted up or downwards to make the time periods from the two knife edges nearly equal. Fix up this mass W 1 , the mass W 2 is moved by means of a screw attached with it to make the final adjustment to equality of the time periods. The rod is then balanced on a sharp knife edge horizontally and its centre of mass is marked. Distance of two knife edges from centre of mass give l and l′ . Knowing l, l′, T and T′ we can calculate the value of g using Bessel’s formula.

19. Advantages of a Compound Pendulum Over a Simple Pendulum In compound pendulum the length being the distance between the knife edges can be accurately measured. In the simple pendulum since the point of suspension and the centre of mass of the bob are both indefinite, thus the effective length of the pendulum cannot be measured acurately . In compound pendulum, being of large mass, can oscillate for a very long time before coming to rest and hence the period of oscillation can be determined with great accuracy of measuring the time for 100 to 200 oscillations. In a simple pendulum the oscillation die out very soon on account of small mass of the bob and the accuracy is limited.

20. Moment of inertia (  I  ) is defined as the sum of the products of the mass of each particle of the body and square of its perpendicular distance from the axis. It is also known as rotational inertia. The moment of inertia reflects the mass distribution of a body or a system of rotating particles, with respect to an axis of rotation. The moment of inertia only depends on the geometry of the body and the position of the axis of rotation, but it does not depend on the forces involved in the movement. 1. A thin uniform bar, one meter long is allowed to oscillate under the influence of gravity about a horizontal axis passing through its one end. Calculate: (a) the length of equivalent simple pendulum, (b) the period of oscillation of the bar and (c) its angular velocity. Solution:

21. M.I. of uniform bar about the axis passing through its one end and perpendicular to its length is given by, (a) The length of equivalent simple pendulum is, In this case, the distance between centre of suspension and centre of gravity (b) The period of oscillation of the bar, (c) The angular velocity of the bar is,

22. 2. A uniform circular disc of radius R oscillates in a vertical plane about a horizontal axis. Find the distance of the axis of rotation from the centre for which the period is minimum. Also evaluate the value of this period. Assume circular disc acts as a compound pendulum, the time period is given by, Solution: where, l is the distance between the centre of suspension and the centre of mass of the disc, and k is the radius of gyration. The period of a compound pendulum is minimum when l = k For the disc, the momentum of inertia about an axis parallel to the axis of suspension and passing through its centre of mass, i.e., about its own axis is given by, For example, consider a solid disk with radius =  R R and total mass =  m m.

23. Thus we have,

24. Q.1. Define compound pendulum. Show that a compound pendulum executives SHM. Find its periodic time. Q.1.Explain the concept of length of equivalent simple pendulum. Q.2.Derive the condition for maximum and minimum time period of the compound pendulum. Q.3. Define centre of suspension and centre of oscillation. Show that in compound pendulum they are interchangeable. Q.2.Show that in reversible compound pendulum the Bessel’s formula to calculate the acceleration due to gravity g is given by, Q.4. Explain Kater’s reversible pendulum to measure the accurate value of acceleration due to gravity g. Long answer questions Short answer questions