A solid of density P and radius R is floating in liquid of density sigma with half of its volume submerged when sphere is pressed down slightly and released . It executes SHM of period T . Then T will be?

To determine the period \( T \) of the simple harmonic motion (SHM) for a solid sphere floating in a liquid, we can start by applying principles from fluid mechanics and dynamics. When the sphere is pressed down slightly and released, it experiences a restoring force due to the buoyancy acting on it. This force is what causes the oscillation, and we can derive the period of this oscillation using the properties of the sphere and the liquid.

Understanding the Forces at Play

When the sphere is floating, it displaces a volume of liquid equal to the weight of the sphere. Given that half of the sphere's volume is submerged, we can express the buoyant force \( F_b \) acting on the sphere as:

• Volume of the sphere, \( V = \frac{4}{3} \pi R^3 \)
• Weight of the sphere, \( W = P \cdot V = P \cdot \frac{4}{3} \pi R^3 \)
• Buoyant force, \( F_b = \sigma \cdot V_{submerged} = \sigma \cdot \frac{1}{2} V = \sigma \cdot \frac{1}{2} \cdot \frac{4}{3} \pi R^3 \)

At equilibrium, the weight of the sphere equals the buoyant force:

\( P \cdot \frac{4}{3} \pi R^3 = \sigma \cdot \frac{1}{2} \cdot \frac{4}{3} \pi R^3 \)

Restoring Force and SHM

When the sphere is displaced a small distance \( x \) downward, the volume of liquid displaced changes, which alters the buoyant force. The new buoyant force can be expressed as:

\( F_b' = \sigma \cdot \left( \frac{1}{2} V + A \cdot x \right) \)

where \( A \) is the cross-sectional area of the sphere. The change in buoyant force \( \Delta F \) can be calculated as:

\( \Delta F = F_b' - F_b = \sigma A x \)

This change in buoyant force acts as the restoring force, which is proportional to the displacement \( x \). According to Hooke's law, we can relate this to SHM:

\( F = -k x \)

where \( k \) is the effective spring constant. In our case, \( k = \sigma A \).

Finding the Period of SHM

The period \( T \) of a mass-spring system is given by the formula:

\( T = 2\pi \sqrt{\frac{m}{k}} \end{equation}

Here, \( m \) is the mass of the sphere, which can be expressed as:

\( m = P \cdot V = P \cdot \frac{4}{3} \pi R^3 \)

Substituting \( k \) and \( m \) into the period formula, we have:

\( T = 2\pi \sqrt{\frac{P \cdot \frac{4}{3} \pi R^3}{\sigma A}} \end{equation}

Final Expression for the Period

To find the area \( A \) of the sphere's cross-section, we use:

\( A = \pi R^2 \end{equation}

Substituting \( A \) into the period equation gives:

\( T = 2\pi \sqrt{\frac{P \cdot \frac{4}{3} \pi R^3}{\sigma \cdot \pi R^2}} \end{equation}

After simplifying, we arrive at:

\( T = 2\pi \sqrt{\frac{4P R}{3\sigma}} \end{equation}

This formula provides the period of the simple harmonic motion for the sphere floating in the liquid. The key takeaway is that the period depends on the density of the sphere, the density of the liquid, and the radius of the sphere, illustrating the interplay between buoyancy and oscillatory motion.