The force required to just move a body up the inclined plane is double the force required to just prevent the body from sliding down the plane. The coefficient of friction is `mu`. If `theta` is the angle of inclination of the plane than `tan theta` is equal to

Options

A. `(a)mu`
B. `(b)3 mu`
C. `(c)2 mu`
D. `(d)0.5 mu`

Explanation

To solve the problem, we need to analyze the forces acting on a body on an inclined plane and derive the relationship between the angle of inclination (θ) and the coefficient of friction (μ). Here’s the step-by-step solution:

### Step 1: Understand the Forces Acting on the Body
When a body is placed on an inclined plane, the forces acting on it include:
- The weight of the body (mg), acting downwards.
- The normal force (N), acting perpendicular to the inclined surface.
- The frictional force (f), which can act either up or down the incline depending on the situation.

### Step 2: Identify the Two Scenarios
1. **Force to Move the Body Upwards (F1)**: This force must overcome both the component of gravitational force acting down the incline (mg sin θ) and the frictional force acting downwards.
2. **Force to Prevent Sliding Down (F2)**: This force must balance the component of gravitational force acting down the incline (mg sin θ) and the frictional force acting upwards.

### Step 3: Establish Relationships
From the problem, we know:
- \( F_1 = 2 F_2 \)

#### For F1 (Force to move up):
The equation can be written as:
\[ F_1 = f + mg \sin \theta \]
Where the frictional force \( f = \mu N \) and \( N = mg \cos \theta \). Thus:
\[ f = \mu mg \cos \theta \]
So,
\[ F_1 = \mu mg \cos \theta + mg \sin \theta \]

#### For F2 (Force to prevent sliding):
The equation can be written as:
\[ F_2 = mg \sin \theta - f \]
Substituting for f:
\[ F_2 = mg \sin \theta - \mu mg \cos \theta \]

### Step 4: Substitute F2 into the F1 Equation
Now, substituting \( F_2 \) into the equation \( F_1 = 2 F_2 \):
\[ \mu mg \cos \theta + mg \sin \theta = 2(mg \sin \theta - \mu mg \cos \theta) \]

### Step 5: Simplify the Equation
Expanding and simplifying:
\[ \mu mg \cos \theta + mg \sin \theta = 2mg \sin \theta - 2\mu mg \cos \theta \]
Rearranging gives:
\[ \mu mg \cos \theta + 2\mu mg \cos \theta = 2mg \sin \theta - mg \sin \theta \]
This simplifies to:
\[ 3\mu mg \cos \theta = mg \sin \theta \]

### Step 6: Divide by mg
Dividing through by mg (assuming mg ≠ 0):
\[ 3\mu \cos \theta = \sin \theta \]

### Step 7: Rearranging to Find tan θ
Dividing both sides by \( \cos \theta \):
\[ 3\mu = \tan \theta \]

### Conclusion
Thus, we find:
\[ \tan \theta = 3\mu \]

### Final Answer
The value of \( \tan \theta \) is equal to \( 3\mu \).

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