Starting from rest , a body slides down at `45^(@)` inclined plane in twice the time it takes to slide down the same distance in the absence of friction. The coefficient of friction between the body and the inclined plane is

Options

A. 1. 0.8
B. 2. 0.75
C. 3. 0.25
D. 4. 0.33

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let:
- \( d \) = distance slid down the incline
- \( t_1 \) = time taken to slide down the incline without friction
- \( t_2 = 2t_1 \) = time taken to slide down the incline with friction
- \( g \) = acceleration due to gravity
- \( \theta = 45^\circ \) = angle of the incline
- \( \mu \) = coefficient of friction

### Step 2: Write the equations for distance
In the absence of friction, the distance \( d \) can be expressed as:
\[ d = \frac{1}{2} a_1 t_1^2 \]
where \( a_1 = g \sin \theta \).

With friction, the distance can be expressed as:
\[ d = \frac{1}{2} a_2 (2t_1)^2 = 2 a_2 t_1^2 \]
where \( a_2 = g \sin \theta - \mu g \cos \theta \).

### Step 3: Set the equations equal to each other
Since both expressions represent the same distance \( d \), we can set them equal:
\[ \frac{1}{2} a_1 t_1^2 = 2 a_2 t_1^2 \]

### Step 4: Cancel \( t_1^2 \) and simplify
Cancelling \( t_1^2 \) from both sides gives:
\[ \frac{1}{2} a_1 = 2 a_2 \]
This simplifies to:
\[ a_1 = 4 a_2 \]

### Step 5: Substitute for \( a_1 \) and \( a_2 \)
Substituting the expressions for \( a_1 \) and \( a_2 \):
\[ g \sin \theta = 4 \left( g \sin \theta - \mu g \cos \theta \right) \]

### Step 6: Factor out \( g \) and simplify
Dividing through by \( g \) (assuming \( g \neq 0 \)):
\[ \sin \theta = 4 \left( \sin \theta - \mu \cos \theta \right) \]
Expanding this gives:
\[ \sin \theta = 4 \sin \theta - 4 \mu \cos \theta \]
Rearranging yields:
\[ 4 \mu \cos \theta = 3 \sin \theta \]

### Step 7: Solve for \( \mu \)
Dividing both sides by \( 4 \cos \theta \):
\[ \mu = \frac{3}{4} \tan \theta \]

### Step 8: Substitute \( \theta = 45^\circ \)
Since \( \tan 45^\circ = 1 \):
\[ \mu = \frac{3}{4} \cdot 1 = 0.75 \]

### Final Answer
The coefficient of friction \( \mu \) between the body and the inclined plane is:
\[ \mu = 0.75 \]

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