Question

A compound pendulum is made up of a rod of length L, with mass M and a solid sphere of radius r, with mass m (see figure below). The pendulum is pivoted about one end and released from rest from and angle of 0, (angle with the vertical). (a) Find the distance, dom, of center of mass of this pendulum from its pivot. (b) Draw a free body diagram and write down Newton's 2nd Law (for rotation) for the pendulum at time t. (c) What is the equation of motion of the pendulum? (d) Write down the solution to the equation of motion expressing the angular displacement as a function of time (@-t equation) and sketch this solution. (e) Write an expression for angular frequency o, time period T, and frequency f of motion of this pendulum. [Moment of inertia of rod about one end = 1/3 ML?, moment of inertia of solid sphere about its center of mass (center of square) = 2/5 mr? ] [12] L CMI ----------------------- 2r CM2

Answer 1

Diagram for the compound pendulum is as shown below along with the values, 3 MRod (ms) =) a> using the formula - v Msphere (M)=1 - Mehm, the rema Metms B 10 cm) 24 - dicm = S (см. - distance of em, day from pivot Anover - 2 mtm 112 mg - 2 1920 Anawre Mo tm (ots) b- F. B.D. weiting torque equ about. point oi To = 12 imgesinoting (late) sino mg -(mgle + mg (Late)o Toeque is dt² restoring + (² m3 + m (6+)?) Above ear is an eqr of sit. m., and its oot is 0 = 0 Sin (wt + x) Egeaph ;) where, mgomg (uts) M 6² +(²m5 tm B + m (Last 3 and &=I = 1 /2 Aps at t=0, 0 = 0. in the sol of o Ro where, to = me? 3 wa t 10 = 8. cap wt littles of motions so