A pendulum is suspended in a ligt and its period of oscillation when the lift is stationary is `T_(a)`. What must be the accleraion of the lift for the period of oscillation of the pendulum to be `T_(0)//2`?

Options

A. `2g` downward
B. `2g` upward
C. `3g` downward
D. `3g` upward

Explanation

To solve the problem, we need to analyze the pendulum's behavior in a lift that is accelerating. Here are the steps to derive the required acceleration of the lift for the period of oscillation of the pendulum to be \( \frac{T_0}{2} \).

### Step-by-Step Solution:

1. **Understanding the Period of a Pendulum**:
The period of a simple pendulum when it is stationary (not accelerating) is given by the formula:
\[ T_0 = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

2. **Finding the New Period**:
We want the new period \( T \) to be \( \frac{T_0}{2} \):
\[ T = \frac{T_0}{2} = 2\pi \sqrt{\frac{L}{g'}} \]
where \( g' \) is the effective acceleration due to gravity when the lift is accelerating.

3. **Setting the Periods Equal**:
Since we have two expressions for the period:
\[ \frac{T_0}{2} = 2\pi \sqrt{\frac{L}{g'}} \]
Substituting \( T_0 \):
\[ \frac{1}{2} \cdot 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{L}{g'}} \]

4. **Simplifying the Equation**:
Cancel \( 2\pi \) from both sides:
\[ \frac{1}{2} \sqrt{\frac{L}{g}} = \sqrt{\frac{L}{g'}} \]

5. **Squaring Both Sides**:
To eliminate the square roots, square both sides:
\[ \left(\frac{1}{2}\right)^2 \frac{L}{g} = \frac{L}{g'} \]
This simplifies to:
\[ \frac{L}{4g} = \frac{L}{g'} \]

6. **Rearranging for \( g' \)**:
Cross-multiplying gives:
\[ g' = 4g \]

7. **Understanding the Acceleration of the Lift**:
The effective acceleration \( g' \) is the result of the gravitational force and the lift's acceleration \( a \):
\[ g' = g + a \]
Substituting \( g' \) from our previous result:
\[ 4g = g + a \]

8. **Solving for the Lift's Acceleration**:
Rearranging gives:
\[ a = 4g - g = 3g \]

### Final Answer:
The acceleration of the lift must be \( 3g \) upward.