An ice cube is kept on an inclined plane of angle `30^(@)`. The coefficient to kinetic friction between the block and incline plane is the `1//sqrt(3)`. What is the acceleration of the block ?

Options

A. zero
B. `2 ms^(-2)`
C. `1.5 ms^(-2)`
D. `5 ms^(-2)`

Explanation

To find the acceleration of the ice cube on the inclined plane, we can follow these steps:

### Step 1: Identify the Forces Acting on the Ice Cube
The forces acting on the ice cube are:
1. The gravitational force (weight) acting downwards: \( F_g = mg \)
2. The normal force \( N \) acting perpendicular to the inclined plane.
3. The frictional force \( F_f \) acting up the incline.

### Step 2: Resolve the Gravitational Force
The weight of the ice cube can be resolved into two components:
- Perpendicular to the incline: \( F_{g\perp} = mg \cos \theta \)
- Parallel to the incline: \( F_{g\parallel} = mg \sin \theta \)

For \( \theta = 30^\circ \):
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 30^\circ = \frac{1}{2} \)

Thus,
- \( F_{g\perp} = mg \cdot \frac{\sqrt{3}}{2} \)
- \( F_{g\parallel} = mg \cdot \frac{1}{2} \)

### Step 3: Calculate the Normal Force
The normal force \( N \) is equal to the perpendicular component of the weight:
\[ N = mg \cos \theta = mg \cdot \frac{\sqrt{3}}{2} \]

### Step 4: Calculate the Frictional Force
The frictional force \( F_f \) can be calculated using the coefficient of kinetic friction \( \mu \):
\[ F_f = \mu N = \frac{1}{\sqrt{3}} \cdot \left( mg \cdot \frac{\sqrt{3}}{2} \right) = \frac{mg}{2} \]

### Step 5: Write the Equation of Motion
The net force acting on the ice cube along the incline can be expressed as:
\[ F_{\text{net}} = F_{g\parallel} - F_f \]
Substituting the values:
\[ F_{\text{net}} = mg \cdot \frac{1}{2} - \frac{mg}{2} = 0 \]

### Step 6: Apply Newton's Second Law
According to Newton's second law:
\[ F_{\text{net}} = ma \]
Since \( F_{\text{net}} = 0 \):
\[ 0 = ma \]
This implies that:
\[ a = 0 \]

### Conclusion
The acceleration of the ice cube on the inclined plane is \( 0 \, \text{m/s}^2 \). This means the ice cube moves with constant velocity.

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