A block starts moving up a fixed inclined plane of inclination `60^(@)C` with a velocityof `20 ms^(-1)` and stops after 2s. The approximate value of the coefficient of friction is `[g = 10 ms^(-2)`]

Options

A. 3
B. 3.3
C. 0.27
D. 0.33

Answer

To solve the problem, we need to analyze the motion of the block on the inclined plane and apply the equations of motion along with the forces acting on the block. Here’s a step-by-step solution:

### Step 1: Identify the given data
- Inclination angle, θ = 60°
- Initial velocity, u = 20 m/s
- Final velocity, v = 0 m/s (the block stops)
- Time, t = 2 s
- Acceleration due to gravity, g = 10 m/s²

### Step 2: Calculate the acceleration of the block
Using the equation of motion:
\[ v = u + at \]
Substituting the known values:
\[ 0 = 20 + a \cdot 2 \]
Solving for acceleration (a):
\[ a = \frac{-20}{2} = -10 \, \text{m/s}^2 \]

### Step 3: Analyze the forces acting on the block
The forces acting on the block include:
1. Gravitational force (mg)
- Component parallel to the incline: \( mg \sin(60°) = mg \cdot \frac{\sqrt{3}}{2} \)
- Component perpendicular to the incline: \( mg \cos(60°) = mg \cdot \frac{1}{2} \)
2. Normal force (N) acting perpendicular to the incline.
3. Frictional force (f) acting down the incline, which can be expressed as:
\[ f = \mu N = \mu mg \cos(60°) = \mu mg \cdot \frac{1}{2} \]

### Step 4: Write the equation of motion along the incline
The net acceleration of the block can be expressed as:
\[ a = -g \sin(60°) - \frac{\mu mg \cos(60°)}{m} \]
Substituting the values:
\[ -10 = -10 \cdot \frac{\sqrt{3}}{2} - \mu \cdot 10 \cdot \frac{1}{2} \]

### Step 5: Simplify and solve for the coefficient of friction (μ)
Rearranging the equation:
\[ -10 = -5\sqrt{3} - 5\mu \]
\[ 5\mu = -10 + 5\sqrt{3} \]
\[ \mu = \frac{-10 + 5\sqrt{3}}{5} \]
\[ \mu = -2 + \sqrt{3} \]

### Step 6: Calculate the approximate value of μ
Using the approximate value of \(\sqrt{3} \approx 1.732\):
\[ \mu \approx -2 + 1.732 \]
\[ \mu \approx -0.268 \]

Since we are looking for the approximate value, we can round this to:
\[ \mu \approx 0.27 \]

### Conclusion
The approximate value of the coefficient of friction is:
\[ \mu \approx 0.27 \]

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