Two simple pendulums, A and B, undergo small oscillations. A graph plotting the magnitude of their angular acceleration ( | | ) against angular displacement ( ) yields two straight lines passing through the origin. If the line for pendulum A makes an angle of 60^ with the -axis, and the line for pendulum B makes an angle of 30^ with the -axis, what is the ratio of the length of pendulum A to the length of pendulum B

Step-by-step solution

For a simple pendulum undergoing small oscillations, the restoring torque is = -mgl . The moment of inertia is I = ml^2 . The magnitude of angular acceleration is | | = | | I = g l . Thus, the slope of the | | versus graph is m = g l = , where is the angle of inclination with the -axis. This gives l = g . For pendulum A: l_A = g 60^ = g 3 . For pendulum B: l_B = g 30^ = g 3 . The ratio is l_A l_B = g 3 g 3 = 1 3 . Answer: 1 : 3