A wooden block of volume 5.24 ×10^-4 m^3 floats in water,...

A wooden block of volume $5.24 \times 10^{-4} \mathrm{~m}^{3}$ floats in water, and a small steel object of mass $m$ is placed on top of the block. When $m=0.310 \mathrm{~kg}$, the system is in equilibrium and the top of the wooden block is at the level of the water. (a) What is the density of the wood? (b) What happens to the block when the steel object is replaced by an object whose mass is less than $0.310 \mathrm{~kg}$ ? (c) What happens to the block when the steel object is replaced by an object whose mass is greater than $0.310 \mathrm{~kg}$ ?

Key Concepts

Archimedes' Principle

This principle states that any object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. It is fundamental for understanding why objects float or sink based on the relationship between the object's weight and the buoyant force provided by the displaced fluid.

Buoyant Force

The buoyant force is the upward force exerted by a fluid that opposes the weight of an immersed object. It depends on the density of the fluid and the volume of fluid displaced by the object, and it is crucial for determining the floating or sinking behavior of objects.

Density

Density is defined as mass per unit volume and is an intrinsic property of materials. In the context of floating, comparing the density of an object (or material) with the density of the fluid helps determine whether the object will float, sink, or remain neutrally buoyant.

Equilibrium Conditions for Floating Bodies

For an object to float in equilibrium, the total weight of the object must be equal to the buoyant force (i.e., the weight of the displaced fluid). This concept is applied when additional mass is added to or removed from a floating object, affecting its submerged volume and equilibrium level.

Mass-Displacement Relationship

When extra mass is added atop a floating object, the total weight increases, causing the object to displace more fluid until the buoyant force balances the increased weight. Conversely, reducing the mass decreases the displaced volume. This relationship explains the adjustments in the float level when different masses are applied to a floating body.

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