Two identical blocks of mass m are attached to two separate ideal springs having force constants 16k and 9k . The blocks are displaced from their mean positions and released simultaneously in the same phase. What is the minimum number of oscillations completed by the block attached to the 16k spring when both blocks are in the same phase again at their mean position?

Step-by-step solution

The time period of a spring-mass system is given by T = 2 m K . For the two systems, the masses are identical, so the time period depends only on the force constant as T 1 K . Let T₁ be the time period of the block on the 16k spring and T₂ be the time period of the block on the 9k spring. T₁ T₂ = 9k 16k = 3 4 This implies 4T₁ = 3T₂ . Both blocks will be in the same phase again at their mean position when the time elapsed is a common multiple of their time periods. The minimum time for this to happen is t = 4T₁ = 3T