A particle is placed at rest inside a hollow hemisphere of radius `R`. The coefficient of friction between the particle and the hemisphere is `mu = (1)/sqrt(3)`. The maximum height up to which the particle can remain stationary is

Options

A. `(R )/(2)`
B. `(1 - sqrt(3)/(2)) R`
C. `sqrt(3)/(2) R`
D. `(3 R)/(8)`

Explanation

To solve the problem, we need to find the maximum height \( H \) up to which a particle can remain stationary inside a hollow hemisphere of radius \( R \) with a coefficient of friction \( \mu = \frac{1}{\sqrt{3}} \).

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- The particle is placed inside a hollow hemisphere of radius \( R \).
- The maximum height \( H \) is the vertical distance from the base of the hemisphere to the point where the particle can remain stationary.

2. **Identifying the Angle of Friction**:
- The coefficient of friction \( \mu \) is given as \( \frac{1}{\sqrt{3}} \).
- The angle of friction \( \theta \) can be calculated using the relation:
\[ \tan \theta = \mu = \frac{1}{\sqrt{3}} \]
- This gives us:
\[ \theta = 30^\circ \]

3. **Setting Up the Geometry**:
- When the particle is at height \( H \), it makes an angle \( \theta \) with the vertical.
- The angle between the radius at the point of contact and the vertical line is \( 30^\circ \).

4. **Using Trigonometry**:
- We can denote the horizontal distance from the center of the hemisphere to the particle as \( a \).
- From the geometry of the triangle formed, we can express the relationship between \( H \), \( a \), and \( R \):
\[ R = H + a \]
- The angle at the point of contact with the hemisphere is \( 60^\circ \) (since \( 90^\circ - 30^\circ = 60^\circ \)).

5. **Finding the Relationship Using Sine**:
- In the triangle formed, we can use the sine function:
\[ \sin(60^\circ) = \frac{a}{R} \]
- Therefore, we can express \( a \) as:
\[ a = R \sin(60^\circ) = R \cdot \frac{\sqrt{3}}{2} \]

6. **Substituting Back to Find Height**:
- Now substitute \( a \) back into the equation \( R = H + a \):
\[ R = H + R \cdot \frac{\sqrt{3}}{2} \]
- Rearranging gives us:
\[ H = R - R \cdot \frac{\sqrt{3}}{2} = R \left(1 - \frac{\sqrt{3}}{2}\right) \]

7. **Final Expression for Maximum Height**:
- Thus, the maximum height \( H \) up to which the particle can remain stationary is:
\[ H = R \left(1 - \frac{\sqrt{3}}{2}\right) \]