AB is a vertical diameter of a circle, whose plane is vertical and PQ is a diameter inclined at an angle `theta` to AB. If the time of sliding down PQ be twice that of sliding down AB, then `cos theta = `

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. `(1)/(4)`
D. None

Answer

To solve the problem, we need to analyze the motion of a particle sliding down two different diameters of a circle: AB (vertical) and PQ (inclined at an angle θ to AB). The key information given is that the time taken to slide down PQ is twice that of sliding down AB.

### Step-by-Step Solution:

1. **Identify the Distances**:
- Let the length of the diameter AB be \( d \).
- The distance traveled along AB is \( d \).
- The distance traveled along PQ can be expressed as \( d \cos \theta \) since PQ is inclined at an angle \( \theta \).

2. **Use the Equation of Motion**:
- For the vertical diameter AB, the distance traveled \( d \) can be described by the equation of motion:
\[ d = \frac{1}{2} g T_1^2 \]
where \( g \) is the acceleration due to gravity and \( T_1 \) is the time taken to slide down AB.

3. **For the Inclined Diameter PQ**:
- The distance traveled along PQ can be described as:
\[ d \cos \theta = \frac{1}{2} (g \cos \theta) T_2^2 \]
where \( T_2 \) is the time taken to slide down PQ.

4. **Relate the Times**:
- Given that \( T_2 = 2 T_1 \), we can substitute \( T_2 \) in the equation:
\[ d \cos \theta = \frac{1}{2} (g \cos \theta) (2 T_1)^2 \]
- This simplifies to:
\[ d \cos \theta = \frac{1}{2} (g \cos \theta) (4 T_1^2) \]
- Further simplifying gives:
\[ d \cos \theta = 2 g \cos \theta T_1^2 \]

5. **Equate the Two Expressions**:
- From the first equation for AB, we have:
\[ d = \frac{1}{2} g T_1^2 \]
- Substitute \( d \) from this equation into the equation for PQ:
\[ \frac{1}{2} g T_1^2 \cos \theta = 2 g \cos \theta T_1^2 \]

6. **Cancel Common Terms**:
- We can cancel \( g \cos \theta T_1^2 \) from both sides (assuming \( g \cos \theta T_1^2 \neq 0 \)):
\[ \frac{1}{2} = 2 \]
- This implies:
\[ \frac{1}{2} = 2 \cos \theta \]

7. **Solve for \( \cos \theta \)**:
- Rearranging gives:
\[ \cos \theta = \frac{1}{4} \]

### Final Answer:
\[ \cos \theta = \frac{1}{4} \]