A wooden block of mass 0.5 kg and density 800 kg m ⁻³ is attached to the free end of a vertical spring having a spring constant of 50 N m ⁻¹ , which is fixed at the bottom. The entire system is completely submerged in water. Determine the time-period of vertical oscillations of the block if it is slightly depressed and released.

Step-by-step solution

Since the block is completely submerged in water, the volume of water displaced remains constant. Therefore, the buoyant force acting on the block is constant throughout its motion. The constant buoyant force and the weight of the block only determine the equilibrium position of the spring. The restoring force for the simple harmonic motion is provided solely by the spring. The effective spring constant of the system is k_ eff = k = 50 N m ⁻¹ . The time period of vertical oscillations of a spring-mass system is giv