A pendulum bob is rotated in a vertical circle with one end of string of length l fixed at a position . At a certain instant bob is at its lowest position and is moving with speed u . Calculate the magnitude of change in velocity when string becomes horizontal for a moment .

Options

A. `sqrt((u^(2)-gl))`
B. `sqrt(2(u^(2)-gl))`
C. `sqrt(3(u^(2)-gl))`
D. `2sqrt((u^(2)-gl))`

Explanation

To solve the problem of calculating the magnitude of change in velocity of a pendulum bob when it moves from its lowest position to a horizontal position, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- At the lowest position, the pendulum bob has a speed \( u \).
- The height of the bob at the lowest position is \( h = 0 \).

2. **Determine the Height at Horizontal Position**:
- When the bob is at the horizontal position, it is raised by a height equal to the length of the string \( l \) (since the string is fixed at one end).
- Therefore, the height \( h \) at the horizontal position is \( h = l \).

3. **Use Conservation of Energy**:
- The mechanical energy of the system is conserved. Thus, the change in kinetic energy equals the change in potential energy.
- At the lowest point, the kinetic energy \( KE_1 \) is:
\[ KE_1 = \frac{1}{2} m u^2 \]
- The potential energy \( PE_1 \) at the lowest point is:
\[ PE_1 = 0 \]
- At the horizontal position, the potential energy \( PE_2 \) is:
\[ PE_2 = mgh = mg l \]
- Let \( v \) be the speed of the bob at the horizontal position. The kinetic energy \( KE_2 \) at this position is:
\[ KE_2 = \frac{1}{2} m v^2 \]

4. **Set Up the Energy Conservation Equation**:
- According to the conservation of energy:
\[ KE_1 + PE_1 = KE_2 + PE_2 \]
- Substituting the values:
\[ \frac{1}{2} m u^2 + 0 = \frac{1}{2} m v^2 + mg l \]

5. **Simplify the Equation**:
- Cancel \( m \) from all terms (assuming \( m \neq 0 \)):
\[ \frac{1}{2} u^2 = \frac{1}{2} v^2 + g l \]
- Rearranging gives:
\[ \frac{1}{2} v^2 = \frac{1}{2} u^2 - g l \]
- Multiplying through by 2:
\[ v^2 = u^2 - 2g l \]

6. **Calculate the Change in Velocity**:
- The change in velocity \( \Delta v \) when moving from the lowest to the horizontal position can be calculated using the Pythagorean theorem:
\[ \Delta v = \sqrt{(v^2 + u^2)} \]
- Substituting \( v^2 \):
\[ \Delta v = \sqrt{(u^2 - 2g l + u^2)} = \sqrt{(2u^2 - 2g l)} = \sqrt{2(u^2 - g l)} \]

### Final Result:
The magnitude of change in velocity when the string becomes horizontal is:
\[ \Delta v = \sqrt{2(u^2 - g l)} \]