Question

A block of wood is floating in water; it is depressed slightly and then released to oscillate up and down. Assume that the top and bottom of the block are parallel planes which remain horizontal during the oscillations and that the sides of the block are vertical. Show that the period of the motion (neglecting friction) is \(2 \pi \sqrt{h / g}\), where \(h\) is the vertical height of the part of the block under water when it is floating at rest. Hint: Recall that the buoyant force is equal to the weight of displaced water.

Short answer

The period of the motion is \( 2\pi \sqrt{h/g} \).

Step-by-step solution

- Identify the forces acting on the block

When the block is at rest in the water, the buoyant force equals the gravitational force on the block. Let the block have a mass of m, and the height of the submerged part of the block be h. The downward gravitational force is given by \( F_g = mg \) and the buoyant force is given by \( F_b = \rho_{water} Ah g \), where \( \rho_{water} \) is the density of water, A is the cross-sectional area of the block.

- Write the equilibrium condition

At equilibrium, the gravitational force is balanced by the buoyant force: \( mg = \rho_{water} Ah g \). This implies that \( m = \rho_{water} Ah \).

- Analyze the restoring force

Consider the block displaced a small distance y from its equilibrium position. The volume of the displaced water changes by \( \text{Volume} = Ay \), so the change in buoyant force is \( \rho_{water} Ayg \). This extra buoyant force acts as a restoring force, which is proportional to the displacement y but in the opposite direction: \( F = -\rho_{water} Ayg \).

- Apply Hooke's Law

This restoring force can be compared to Hooke's law for simple harmonic motion, \( F = -ky \), where k is the effective spring constant: \( k = \rho_{water} Ag \).

- Determine the mass-spring system's period

The period T for a mass-spring system can be given by \( T = 2\pi \sqrt{m/k} \). Substitute \( k = \rho_{water} Ag \) and \( m = \rho_{water} Ah \) into this equation: \( T = 2\pi \sqrt{\frac{\rho_{water} Ah}{\rho_{water} Ag}} = 2\pi \sqrt{\frac{h}{g}} \).

Key concepts

Buoyant Force

The buoyant force is a fundamental concept in fluid mechanics that helps explain why objects float or sink in a fluid. It is the upward force exerted by a fluid that opposes the weight of an object immersed in it. This force is equal to the weight of the displaced fluid.
To understand buoyant force in the context of our exercise, recall that the block of wood floating in water experiences two main forces: the gravitational force pulling it down and the buoyant force pushing it up. When these are equal, the block floats at equilibrium. The buoyant force can be mathematically described as:

\( F_b = \rho_{water} Ah g \),

where \( \rho_{water} \) is the density of water, \( A \) is the cross-sectional area of the block, \( h \) is the height of the submerged part of the block, and \( g \) is the gravitational acceleration.
By understanding buoyant force, we see how displacing a fluid results in an upward force that allows objects to float and oscillate if disturbed.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic motion where an object oscillates back and forth around an equilibrium position. A key feature of SHM is that the restoring force is directly proportional to the displacement.
In the case of the block of wood oscillating in water, the SHM is seen as the block moves up and down through the water's surface. When the block is displaced from its equilibrium position by a small distance \( y \), the restoring force acts to return it to equilibrium. This force is given by:

\( F = -\rho_{water} A y g \).

This relation shows us that the restoring force is proportional to the displacement \( y \) and acts in the opposite direction, making it a perfect example of SHM. The negative sign indicates that the force is in the opposite direction of the displacement, which is a defining characteristic of SHM.

Restoring Force

The restoring force is the force that brings an object back to its equilibrium position when displaced. In simple harmonic motion, this force is proportional to the displacement and acts in the opposite direction.
For the block of wood in water, when the block is pushed down (or pulled up), it displaces more (or less) water. This changes the buoyant force. If the block is displaced a small distance \( y \) from equilibrium, the extra buoyant force (restoring force) can be given by:

\( \text{Extra Buoyant Force} = \rho_{water} A y g \).

Since this force acts in the direction opposite to the displacement, we write it as:

\( F = -\rho_{water} A y g \).

This relationship outlines the nature of the restoring force in SHM and shows us how the block's motion is influenced by both the initial displacement and the properties of the water and block.

Mass-Spring System Period

The period of a mass-spring system is the time it takes for one complete cycle of oscillation. For the block of wood in water, we can compare its oscillatory motion to a simple mass-spring system.
The period \( T \) of a simple mass-spring system is given by the formula:

\( T = 2\pi \sqrt{\frac{m}{k}} \).

In our exercise, \( m = \rho_{water} Ah \) (the mass of the block), and the spring constant \( k = \rho_{water} A g \). Substituting these into the period formula, we get:

\( T = 2\pi \sqrt{\frac{\rho_{water} Ah}{\rho_{water} A g}} \).

Simplifying this, we find:

\( T = 2\pi \sqrt{\frac{h}{g}} \).

This final expression gives us the period of the harmonic oscillatory motion of the block in water, showing that it depends only on the height of the submerged part of the block \( h \) and the acceleration due to gravity \( g \). This connects the properties of the block and the fluid to the characteristics of its motion.