A block has been placed on an inclined plane . The slope angle of `theta` of the plane is such that the block slides down the plane at a constant speed . The coefficient of kinetic friction is equal to :

Options

A. ` sin theta`
B. ` cos theta`
C. zero
D. ` tan theta`

Explanation

To solve the problem of finding the coefficient of kinetic friction (μ) when a block slides down an inclined plane at a constant speed, we can follow these steps:

### Step 1: Understand the Forces Acting on the Block
When the block is on the inclined plane, two main forces act on it:
- The gravitational force acting downwards, which can be divided into two components:
- Parallel to the incline: \( mg \sin \theta \)
- Perpendicular to the incline: \( mg \cos \theta \)
- The normal force (N) acting perpendicular to the surface of the incline.
- The frictional force (f) acting opposite to the direction of motion, which is upward along the incline.

### Step 2: Set Up the Equation for Constant Speed
Since the block is sliding down at a constant speed, the net force acting on it along the incline is zero. Therefore, the frictional force must balance the component of gravitational force acting down the incline:
\[ f = mg \sin \theta \]

### Step 3: Express the Frictional Force
The frictional force can also be expressed in terms of the coefficient of kinetic friction (μ) and the normal force (N):
\[ f = \mu N \]

### Step 4: Determine the Normal Force
The normal force (N) acting on the block is equal to the perpendicular component of the gravitational force:
\[ N = mg \cos \theta \]

### Step 5: Substitute the Normal Force into the Friction Equation
Now, we can substitute the expression for the normal force into the friction equation:
\[ \mu (mg \cos \theta) = mg \sin \theta \]

### Step 6: Simplify the Equation
We can cancel \( mg \) from both sides of the equation (assuming \( m \neq 0 \)):
\[ \mu \cos \theta = \sin \theta \]

### Step 7: Solve for the Coefficient of Kinetic Friction
Now, we can isolate μ:
\[ \mu = \frac{\sin \theta}{\cos \theta} \]

### Step 8: Recognize the Trigonometric Identity
The expression \( \frac{\sin \theta}{\cos \theta} \) is equal to \( \tan \theta \):
\[ \mu = \tan \theta \]

### Conclusion
Thus, the coefficient of kinetic friction (μ) is equal to \( \tan \theta \).

### Final Answer
The coefficient of kinetic friction is \( \mu = \tan \theta \).

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