Question

P4. A clock keeps time using the periodic motion of a simple pendulum. The pendulum consists of a string of length L and a bob of mass m-5.00 kg attached to the end of the string. The pendulum has a period T-1.00 s. The initial angle (0) at 0 is equal to 0.175 rad. The bob is released from rest (i.e. -0) at -0. The angle between the string and the vertical is given by the equation: e-a cos (or + φ). Determine the following: a. (3) The angular frequency ω of the pendulum. b, (8) The phase angle ф and the amplitude c. (3) The length L of the pendulum. d. (5) The initial mechanical energy E of the system. Now damping is introduced and the angle θ is given by the equation: θ-9"e" cos (or + φ) . It is observed that after one hour the maximum angle of the pendulum has reduced to 0.122 rad due to the damping force e. (6) Calculate the coefficient α WWE

Answer 1

Р4 (a) The expression for the angular frequency of the pendulum is 27T Т The period of the pendulum is, T 1.00s Calculate the angular frequency of the pendulum as follows: 27T T 27T 1.00s 27T (or 6.28 rad/s (b) The expression for the angle which is between the string and the vertical is, 0=e cos ( +) ot m Differentiate the above equation with t as follows: d ө. c d )) (0)= os(a t + _ dt dt de (-osin dt =-ae sin (at + The bob is released from rest, then the initial angular speed of the pendulum is zero at t 0

de 0 dt Then calculate the angle ø as follows: de -we sin (at dt m 0 - - ов, sin (co(0) + ф) m sin 0 sin sin 0 0 The initial angle at time t is, 0 (t) — е сos (ot +ф) +(0) _ m ө cos(at + Calculate the amplitude () at t 0 as follows: ө(t) — ө cos ar e(0) e cos (0) 0.175 rad (cos 0) m m e=0.175 rad - m (c) The expression for the period of the pendulum is L T 27T g

Then the expression for the length of the pendulum is T g L 472 Calculate the length of the pendulum as follows: T2g L - 2 47 (1.00s) (9.8m/s? 4т? 0.25m (d) The expression for the mechanical energy of the pendulum is, 1 E -Ia 2 max The expression for the maximum angular frequency of the pendulum is max m Since, the pendulum is a rod, the n the moment of inertia of the pendulum is, I =mL2 Then the expression for the mechanical energy of the pendulum is

1 E 2 - max (mL)(00) The angular frequency for one complete cycle is, Calculate the mechanical energy of the pendulum as follows: 1 (mL )(0,0)* mL2(e E 2 |(5.00kg)(0.25 m) ((0.175 rad) (27) 2 0.1889 J - 0.2J