solve this---**Question:**
Q5: A cylindrical block of wood (density = 650 kg m$^{-3}$), of base area 30 cm$^2$ and height 54 cm, floats in a liquid of density 900 kg m$^{-3}$. The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly):
**Options:**
1. 52 cm
2. 65 cm
3. 39 cm
4. 26 cm
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We have a floating wooden block problem. A cylindrical block with density 650 kilograms per cubic meter floats in a liquid with density 900 kilograms per cubic meter. The block has a base area of 30 square centimeters and height of 54 centimeters. We need to find the equivalent simple pendulum length for its oscillations.
Let's calculate the mass and find the equilibrium condition. First, we calculate the mass using density times volume. The volume is base area times height, which gives us 1.62 times 10 to the minus 3 cubic meters. The mass is 650 times this volume, equals 1.053 kilograms. At equilibrium, the weight equals the buoyant force, giving us the equilibrium submerged depth h zero.
Now let's derive the oscillation equation. When the block is displaced downward by distance x from equilibrium, the net force becomes the weight minus the increased buoyant force. This simplifies to negative rho l A g times x, which is a restoring force. Applying Newton's second law gives us the equation for simple harmonic motion with angular frequency omega.
Now we find the equivalent pendulum length. The time period of the block oscillation is 2 pi times the square root of m over rho l A g. For a simple pendulum, the period is 2 pi times the square root of L over g. Setting these equal and simplifying, we get L equals m over rho l A. Substituting our values: L equals 1.053 over 2.7, which equals 0.39 meters or 39 centimeters. This is our final answer.
To summarize what we've learned: floating block oscillations follow simple harmonic motion due to buoyancy changes. The restoring force comes from the varying buoyant force as the block moves up and down. The time period depends on the block's mass, liquid density, and base area. Most importantly, the equivalent pendulum length equals the equilibrium submerged depth, giving us our final answer of 39 centimeters.