A simple pendulum consists of a small mass of 72 g suspended from a light inelastic cord. During oscillations with 50 amplitude, this pendulum completes 60 oscillations every 1 minute. Calculate:

Question

A simple pendulum consists of a small mass of 72 g suspended from a light inelastic cord. During oscillations with 50 amplitude, this pendulum completes 60 oscillations every 1 minute. Calculate:

3.1. The length of the cord and the amplitude. (4)

3.2. The maximum linear velocity and the maximum linear acceleration of the mass. (5)

3.3. The maximum force and the minimum force in the light cord. (6)

Answer 1

I have to assume that "50" is an angle in degrees. I report answers to two sig figs, while carrying at least 3 places in calculations.

3.1) For the length L use ω = 2πf = √(g/L) where f = 60 cycles/min = 1 Hz, so L = 9.8/(2π)² = 0.248 ≅ 0.25 m.

Sketch the pendulum at max displacement of 50° and draw a horizontal line to its vertical position. The pendulum is driven by its potential energy mgh, where h is the difference in height between the two positions. With 50° as the top angle of the triangle formed, cos 50° = (L - h)/L =.6428, and L - h = 0.160. Amplitude A = h = 0.248 - 0.160 = 0.088 m.

3.2) Max speed for SHM is Aω = h•2π = 0.553 ≅ 0.55 m/s. Max acceleration is Aω² = 3.47 ≅ 3.5 m/s².

3.3) Max tension is at bottom of swing, T = mg + mv_max²/r = 72 [9.8 + (0.553)² /0.248] = 790 N.

Min T is at 50° angle, T = mg cos 50° = 450 N.