A wooden rod of a uniform cross section and of length `120 cm` is hinged at the bottom of the tank which is filled with water to a height of `40 cm`. In the equilibrium position, the rod makes an angle of `60^@` with the vertical. The centre of buoyancy is located on the rod at a distance (from the hinge) of

Options

A. `20cm`
B. `40cm`
C. `60cm`
D. `75cm`

Explanation

To solve the problem, we need to find the distance from the hinge to the center of buoyancy of the wooden rod submerged in water. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a wooden rod of length 120 cm hinged at the bottom of a tank filled with water to a height of 40 cm. The rod makes an angle of 60° with the vertical in its equilibrium position. We need to find the distance from the hinge to the center of buoyancy of the submerged part of the rod.

### Step 2: Determine the Submerged Length of the Rod
Since the rod is at an angle of 60° with the vertical, we can use trigonometry to find out how much of the rod is submerged in water.

- The vertical height of the rod submerged in water can be calculated using:
\[ h = L \cdot \cos(\theta) \]
where \(L\) is the length of the rod (120 cm) and \(\theta\) is the angle with the vertical (60°).

### Step 3: Calculate the Submerged Length
Using the formula:
\[ h = 120 \cdot \cos(60°) \]
Since \(\cos(60°) = \frac{1}{2}\):
\[ h = 120 \cdot \frac{1}{2} = 60 \text{ cm} \]
This means that the rod extends 60 cm vertically downwards from the hinge.

### Step 4: Determine the Effective Submerged Length
However, the water level is only 40 cm high. Therefore, the actual submerged length of the rod is limited to the water height:
- The submerged length of the rod is 40 cm.

### Step 5: Find the Center of Buoyancy
The center of buoyancy is located at the center of the submerged part of the rod. Since the submerged part is 40 cm long, the center of buoyancy will be at half of this length:
\[ \text{Center of Buoyancy} = \frac{40 \text{ cm}}{2} = 20 \text{ cm} \]

### Step 6: Calculate the Distance from the Hinge
Since the hinge is at the bottom of the rod, the distance from the hinge to the center of buoyancy is:
\[ \text{Distance from hinge} = 20 \text{ cm} \]

### Final Answer
Thus, the center of buoyancy is located at a distance of **20 cm** from the hinge.

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