A wooden block is placed on an inclined plane. The block just begins to slide down when the angle of the inclination is increased to 45°. What is the coefficient of the friction?

Options

A. 0.25
B. 0.75
C. 1
D. 0.5

Answer

To solve the problem of finding the coefficient of friction for a wooden block on an inclined plane that begins to slide at an angle of 45°, we can follow these steps:

### Step 1: Understand the Forces Acting on the Block
When the block is on an inclined plane, two main forces act on it:
- The gravitational force (weight) acting downward, which is \( mg \).
- The normal force acting perpendicular to the inclined surface.

### Step 2: Resolve the Gravitational Force
The gravitational force can be resolved into two components:
- The component parallel to the incline: \( F_{\parallel} = mg \sin(\theta) \)
- The component perpendicular to the incline: \( F_{\perpendicular} = mg \cos(\theta) \)

For our case, where \( \theta = 45° \):
- \( F_{\parallel} = mg \sin(45°) = mg \frac{\sqrt{2}}{2} \)
- \( F_{\perpendicular} = mg \cos(45°) = mg \frac{\sqrt{2}}{2} \)

### Step 3: Write the Expression for Normal Force
The normal force \( N \) is equal to the perpendicular component of the weight:
\[ N = mg \cos(45°) = mg \frac{\sqrt{2}}{2} \]

### Step 4: Write the Expression for Frictional Force
The frictional force \( F_f \) that opposes the motion is given by:
\[ F_f = \mu N \]
Where \( \mu \) is the coefficient of friction.

### Step 5: Set Up the Equation for Motion
At the point when the block just begins to slide, the frictional force is equal to the component of the gravitational force acting down the incline:
\[ F_f = F_{\parallel} \]
Thus, we have:
\[ \mu N = mg \sin(45°) \]

### Step 6: Substitute the Values
Substituting the expressions for \( N \) and \( F_{\parallel} \):
\[ \mu \left(mg \frac{\sqrt{2}}{2}\right) = mg \frac{\sqrt{2}}{2} \]

### Step 7: Simplify the Equation
We can cancel \( mg \frac{\sqrt{2}}{2} \) from both sides (assuming \( mg \frac{\sqrt{2}}{2} \neq 0 \)):
\[ \mu = 1 \]

### Conclusion
The coefficient of friction \( \mu \) is equal to 1.