Two simple pendulums, P and Q , are allowed to oscillate at the same location. It is observed that pendulum P completes 20 oscillations in the exact same time that pendulum Q completes 30 oscillations. What is the ratio of the length of pendulum P to the length of pendulum Q ( L_P : L_Q )?

Step-by-step solution

Let the time taken for the given number of oscillations be t . The time period of pendulum P is T_P = t 20 . The time period of pendulum Q is T_Q = t 30 . The ratio of their time periods is: T_P T_Q = t 20 t 30 = 30 20 = 3 2 The time period of a simple pendulum is given by T = 2 L g . Since T L , we have L T^2 . Therefore, the ratio of their lengths is: L_P L_Q = ( T_P T_Q )^2 = ( 3 2 )^2 = 9 4 Thus, the ratio L_P : L_Q is 9 : 4 . Answer: 9 : 4