A block of mass m is connected to a light rod of length l . One end of the rod is pivoted to a point in such a way that it can be rotated in a vertical plane . Initially rod is vertical with the block a its lowest position . If velocity given to the block is `sqrt(ngl)` then it is found to complete the circle successfully . Find the value of n .

Explanation

To solve the problem step-by-step, we will analyze the energy conservation between the lowest point (point A) and the highest point (point B) of the circular motion of the block attached to the rod.

### Step 1: Understand the System
We have a block of mass \( m \) attached to a light rod of length \( l \) that can rotate in a vertical plane. Initially, the rod is vertical, and the block is at the lowest position. The block is given an initial velocity of \( \sqrt{n g l} \).

### Step 2: Identify Points A and B
- **Point A**: The lowest point where the block is given the initial velocity.
- **Point B**: The highest point in the circular motion.

### Step 3: Write the Energy Conservation Equation
According to the principle of conservation of mechanical energy:
\[ \text{Total Energy at A} = \text{Total Energy at B} \]
This can be expressed as:
\[ KE_A + PE_A = KE_B + PE_B \]

### Step 4: Calculate Energies at Point A
At point A:
- Kinetic Energy \( KE_A = \frac{1}{2} m v^2 = \frac{1}{2} m (\sqrt{n g l})^2 = \frac{1}{2} m n g l \)
- Potential Energy \( PE_A = 0 \) (taking the lowest point as the reference level)

Thus, the total energy at point A is:
\[ E_A = \frac{1}{2} m n g l + 0 = \frac{1}{2} m n g l \]

### Step 5: Calculate Energies at Point B
At point B (the highest point):
- Kinetic Energy \( KE_B = 0 \) (for the block to just complete the circle, the velocity at the highest point should be 0)
- Potential Energy \( PE_B = m g (2l) \) (the height at point B is \( 2l \))

Thus, the total energy at point B is:
\[ E_B = 0 + m g (2l) = 2 m g l \]

### Step 6: Set the Energies Equal
Now, set the total energy at point A equal to the total energy at point B:
\[ \frac{1}{2} m n g l = 2 m g l \]

### Step 7: Simplify the Equation
We can cancel \( m g l \) from both sides (assuming \( m, g, l \neq 0 \)):
\[ \frac{1}{2} n = 2 \]

### Step 8: Solve for \( n \)
Multiply both sides by 2:
\[ n = 4 \]

### Final Answer
The value of \( n \) is \( 4 \).

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