A cuboidal piece of wood has dimensions a, b and c. its relatively density is d. it is floating in a large body of water such that side a is vertical. It is pushed down a bit and released. The time period of SHM executed by it is

Options

A. `2pisqrt(abc)/g`
B. `2pisqrt(g/(da)`
C. `2pisqrt((bc)/(dg))`
D. `2pisqrt((da)/g)`

Answer

To find the time period of the simple harmonic motion (SHM) executed by the cuboidal piece of wood when it is pushed down and released, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the System**:
- We have a cuboidal block of wood with dimensions \( a \), \( b \), and \( c \).
- The relative density of the wood is \( d \).
- The block is floating in water with side \( a \) vertical.

2. **Establish the Equilibrium Condition**:
- When the block is floating, the weight of the block is balanced by the buoyant force.
- The weight of the block is given by \( mg \), where \( m \) is the mass of the block.
- The buoyant force is given by \( \text{Buoyant Force} = \rho_l \cdot V_d \cdot g \), where \( \rho_l \) is the density of the liquid (water), \( V_d \) is the volume of water displaced, and \( g \) is the acceleration due to gravity.

3. **Volume and Weight Relationships**:
- The volume of the block \( V \) is \( V = a \cdot b \cdot c \).
- The mass of the block can be expressed as \( m = \rho_s \cdot V \), where \( \rho_s \) is the density of the solid wood.

4. **Displacement and Buoyant Force**:
- When the block is pushed down by a distance \( x \) and released, the new displaced volume of water is \( V_d = b \cdot c \cdot x \).
- The buoyant force when displaced is \( \text{Buoyant Force} = \rho_l \cdot (b \cdot c \cdot x) \cdot g \).

5. **Setting Up the Equation**:
- At equilibrium, the weight of the block equals the buoyant force:
\[ \rho_s \cdot (a \cdot b \cdot c) \cdot g = \rho_l \cdot (b \cdot c \cdot h) \cdot g \]
- This simplifies to:
\[ \rho_s \cdot a = \rho_l \cdot h \]
- Here, \( h \) is the height of the submerged part of the block.

6. **Finding the Time Period**:
- The restoring force when displaced by \( x \) is proportional to the displacement \( x \):
\[ F = -\rho_l \cdot (b \cdot c) \cdot g \cdot x \]
- The effective spring constant \( k \) can be identified as:
\[ k = \rho_l \cdot (b \cdot c) \cdot g \]
- The mass \( m \) of the block is:
\[ m = \rho_s \cdot (a \cdot b \cdot c) \]
- The time period \( T \) of SHM is given by:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
- Substituting the values:
\[ T = 2\pi \sqrt{\frac{\rho_s \cdot (a \cdot b \cdot c)}{\rho_l \cdot (b \cdot c) \cdot g}} \]
- Simplifying gives:
\[ T = 2\pi \sqrt{\frac{d \cdot a}{g}} \]
- where \( d = \frac{\rho_s}{\rho_l} \) is the relative density.

### Final Answer:
The time period of SHM executed by the cuboidal piece of wood is:
\[ T = 2\pi \sqrt{\frac{d \cdot a}{g}} \]