AB is the vertical diameter of a circle in a vertical plane. Another diameter CD makes an angle of `60^(@)` with AB, then the ratio of the time taken by a particle to slide along AB to the time taken by it to slide along CD is

Options

A. 1:1
B. `sqrt(2):1`
C. ` 1: sqrt(2)`
D. `3^(1//4) : 2^(1//4)`

Answer

To solve the problem, we need to find the ratio of the time taken by a particle to slide along the vertical diameter AB to the time taken to slide along the diameter CD, which makes an angle of 60 degrees with AB.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let AB be the vertical diameter of the circle.
- Let CD be another diameter making an angle of 60 degrees with AB.
- We need to analyze the motion of a particle sliding down these diameters under the influence of gravity.

2. **Distance Along AB**:
- The distance the particle slides down along AB can be denoted as \( h \).
- The time taken to slide down AB (T1) can be calculated using the equation of motion:
\[ h = \frac{1}{2} g T_1^2 \]
- Rearranging gives:
\[ T_1^2 = \frac{2h}{g} \]

3. **Distance Along CD**:
- The distance along CD can also be denoted as \( h \), but we need to consider the angle. The vertical component of the distance along CD can be expressed as:
\[ h = \frac{1}{2} g T_2^2 \cos(60^\circ) \]
- Since \( \cos(60^\circ) = \frac{1}{2} \), we can rewrite this as:
\[ h = \frac{1}{2} g T_2^2 \cdot \frac{1}{2} = \frac{1}{4} g T_2^2 \]
- Rearranging gives:
\[ T_2^2 = \frac{4h}{g} \]

4. **Finding the Ratio of Times**:
- Now we have expressions for \( T_1^2 \) and \( T_2^2 \):
\[ T_1^2 = \frac{2h}{g} \quad \text{and} \quad T_2^2 = \frac{4h}{g} \]
- Taking the ratio of \( T_1^2 \) to \( T_2^2 \):
\[ \frac{T_1^2}{T_2^2} = \frac{\frac{2h}{g}}{\frac{4h}{g}} = \frac{2}{4} = \frac{1}{2} \]
- Taking the square root gives:
\[ \frac{T_1}{T_2} = \frac{1}{\sqrt{2}} \]

5. **Final Result**:
- Therefore, the ratio of the time taken by the particle to slide along AB to the time taken to slide along CD is:
\[ \frac{T_1}{T_2} = \frac{1}{\sqrt{2}} \]