A block of weight `100N` is lying on a rough horizontal surface.If coefficient of friction `1/sqrt3`.The least possible force that can move the block is

Options

A. `100/sqrt3N`
B. `100sqrt3 N`
C. `50sqrt3 N`
D. `50N`

Explanation

To solve the problem of finding the least possible force that can move a block of weight 100 N on a rough horizontal surface with a coefficient of friction of \( \frac{1}{\sqrt{3}} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Weight of the block, \( W = 100 \, \text{N} \)
- Coefficient of friction, \( \mu = \frac{1}{\sqrt{3}} \)

2. **Understand the Forces Acting on the Block:**
- The weight of the block acts downwards.
- The normal force \( N \) acts upwards.
- A horizontal force \( F \) is applied at an angle \( \theta \) to reduce the frictional force.

3. **Resolve the Applied Force \( F \):**
- The horizontal component of the force is \( F \cos \theta \).
- The vertical component of the force is \( F \sin \theta \).

4. **Calculate the Normal Force:**
- The normal force \( N \) can be expressed as:
\[ N = W - F \sin \theta = 100 - F \sin \theta \]

5. **Determine the Frictional Force:**
- The frictional force \( f \) is given by:
\[ f = \mu N = \mu (100 - F \sin \theta) \]
- Substituting \( \mu \):
\[ f = \frac{1}{\sqrt{3}} (100 - F \sin \theta) \]

6. **Set Up the Equation for Motion:**
- For the block to just start moving, the applied force's horizontal component must equal the frictional force:
\[ F \cos \theta = \frac{1}{\sqrt{3}} (100 - F \sin \theta) \]

7. **Rearranging the Equation:**
- Rearranging gives:
\[ F \cos \theta + \frac{1}{\sqrt{3}} F \sin \theta = \frac{100}{\sqrt{3}} \]
- Factoring out \( F \):
\[ F \left( \cos \theta + \frac{1}{\sqrt{3}} \sin \theta \right) = \frac{100}{\sqrt{3}} \]

8. **Maximize the Denominator:**
- To minimize \( F \), we need to maximize \( \cos \theta + \frac{1}{\sqrt{3}} \sin \theta \).
- The maximum value of \( \cos \theta + \mu \sin \theta \) can be found using the formula:
\[ \sqrt{1 + \mu^2} \]
- Substituting \( \mu = \frac{1}{\sqrt{3}} \):
\[ \sqrt{1 + \left(\frac{1}{\sqrt{3}}\right)^2} = \sqrt{1 + \frac{1}{3}} = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} \]

9. **Calculate the Least Possible Force \( F \):**
- Substitute back into the equation for \( F \):
\[ F = \frac{100/\sqrt{3}}{\frac{2}{\sqrt{3}}} = \frac{100}{2} = 50 \, \text{N} \]

### Final Answer:
The least possible force that can move the block is \( F = 50 \, \text{N} \).