Question

A cubic block of side \(l\) and mass density \(\rho\) per unit volume is floating in a fluid of mass density \(\rho_{0}\) per unit volume, where \(\rho_{0}>\rho .\) If the block is slightly depressed and then released, it oscillates in the vertical direction. Assuming that the viscous damping of the fluid and air can be neglected, derive the differential equation of motion and determine the period of the motion. Hint Use archimedes' principle: An object that is completely or partially submerged in a fluid is acted on by an upward (bouyant) equal to the weight of the displaced fluid.

Short answer

Answer: The period of the oscillatory motion is 2π.

Step-by-step solution

Find the volume submerged when the block is in equilibrium

When the block is in equilibrium, the buoyant force acting on the object will be equal to its weight. Let \(V_s\) be the volume of the block submerged in the fluid. Using Archimedes' principle, \[\rho_{0}gV_s = \rho gl^3\] Solving for \(V_s\), \[V_s = \frac{\rho}{\rho_0}l^3\]

Determine the additional submerged volume when the block is depressed

Let the block be depressed by a distance \(y\) further into the fluid. Then, the additional volume \(V_{add}\) submerged is given by, \[V_{add} = l^2y\]

Establish the relationship between buoyant force and new submerged volume

Due to the additional submerged volume, the buoyant force will increase. Using Archimedes' principle again for the total submerged volume \((V_s + V_{add})\), we have \[\rho_{0}g(V_s + V_{add}) = (\rho_{0}gV_s) + (\rho_{0}gl^2y)\]

Derive the differential equation of motion for the block

We can now use Newton's second law to find the equation of motion. The net force acting on the block is given by the difference between the buoyant force and the weight of the block. The acceleration of the block is given by \(\frac{d^2y}{dt^2}\). \[\rho_{0}gl^2\frac{d^2y}{dt^2} = (\rho_{0}gV_s) + (\rho_{0}gl^2y) - (\rho gl^3)\] Plugging the expression for \(V_s\) from step 1, \[\rho_{0}gl^2\frac{d^2y}{dt^2} = (\rho_{0}g\frac{\rho}{\rho_0}l^3) + (\rho_{0}gl^2y) - (\rho gl^3)\] Simplifying gives, \[\rho_{0}l^2\frac{d^2y}{dt^2} = \rho_{0}l^2y\] Dividing by \(\rho_{0}l^2\), \[\frac{d^2y}{dt^2} = y\] This is the differential equation of motion for the block.

Find the period of oscillatory motion

The equation we derived in step 4 is a simple harmonic oscillator equation, with the general form \[\frac{d^2y}{dt^2} = -\omega^2y\] where \(\omega\) denotes the angular frequency. Comparing with our previous equation, we have \[\omega^2 = 1\] Thus, \(\omega = 1\) (we can disregard the negative solution here, as frequency must be positive). The period (T) of the oscillatory motion is given by \[T = \frac{2\pi}{\omega}\] Plugging in our calculated value for \(\omega\), \[T = 2\pi\] Hence, the period of the oscillatory motion is \(2\pi\).

Key concepts

Archimedes' principle

Archimedes' principle is a fundamental law in fluid dynamics. It states that a body submerged in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the body. This principle helps us understand why objects float or sink. In our exercise, a cubic block floats because the buoyant force balances its weight. When the block is slightly pushed down, it continues to move due to the buoyant force change. This creates an oscillating effect.

To apply Archimedes' principle, consider the block at equilibrium. The weight of the displaced fluid equals the weight of the block. Here's how it works:

• The fluid density is \( \rho_0 \).
• The block’s density is \( \rho \).
• The submerged volume provides the necessary buoyant force.

The principle's real-world applications include ship design, submarines, and even hot air balloons. Knowing this, you can calculate how deep an object will float in any fluid.

Harmonic Oscillation

Harmonic oscillation occurs when an object oscillates back and forth. It's driven by restoring forces proportional to the displacement from its equilibrium position. In the case of our exercise, the block exhibits harmonic oscillation as it moves vertically after being disturbed. This type of motion is characterized by sinusoidal waves and is common in various systems, from springs to pendulums.

The differential equation \( \frac{d^2y}{dt^2} = y \) deduced in the step-by-step solution reflects this. It is a classic simple harmonic motion equation. Such models are described by:

• Displacement \( y \) from equilibrium
• Angular frequency \( \omega = 1 \)
• Amplitude determined by initial conditions

These concepts provide insight into predicting the object's motion and calculating its period, which in this exercise is \( 2\pi \). Harmonic oscillation is essential for understanding vibrations, wave motion, and even electrical circuits.

Newton's Second Law

Newton's Second Law is a cornerstone of physics. It states that the force acting on an object equals its mass times its acceleration: \( F = ma \). This law helps derive the motion equation for the block in our exercise. When displaced, the net force results from the combination of buoyant force and the block's weight.

Using Newton's Second Law, we can set up the problem as:

• Calculate net buoyant force when the block is displaced.
• Consider only forces in the direction of motion (vertical).
• Establish the relationship using densities and volumes involved.

The resulting equation \( \rho_{0}l^2 \frac{d^2y}{dt^2} = \rho_{0}l^2y \) came from applying Newton's principle alongside Archimedes'. This allowed us to determine the block's period, showcasing how forces dictate motion.

Buoyant Force

Buoyant force is a phenomenon that occurs when an object is immersed in a fluid. This force acts upward, counterbalancing the object's weight. According to Archimedes' principle, it's equal to the weight of the displaced fluid. In our exercise, the buoyant force allows the block to float partially submerged in the fluid, and when further depressed, it increases proportionally.

Key concepts to remember about buoyant force include:

• Depends on fluid density \( \rho_0 \)
• Relies on volume of fluid displaced
• Critical for maintaining equilibrium or initiating motion

The role of buoyant force in oscillations is pivotal; it acts as a restoring force when the block moves from its equilibrium position. This force is why objects submerged in water feel lighter and why they can float, forming the basis for designing various structures and devices. It's a practical and palpable concept, resonating through many areas in science and engineering.