Question

The acceleration of a body sliding down the inclined plane, having coefficient of friction \( \mu \), is

• A. a= \( g(\sin\theta + \mu \cos\theta) \)
• B. a=\( g(\sin\theta - \mu \cos\theta) \)
• C. a= \( g(\cos\theta - \mu \sin\theta) \)
• D. a=\( g(\cos\theta + \mu \sin\theta) \)

Hint:

In motion on an inclined plane, the component of gravitational force along the plane is \( mg \sin\theta \), and frictional force is \( \mu mg \cos\theta \). The net acceleration is obtained by subtracting the frictional force from the gravitational force component along the incline.

Answer 1

The Correct Option is B

Step 1: Understanding Forces Acting on the Body
- The forces acting on the body on an inclined plane of angle \( \theta \) are:
- Gravity \( mg \) acts downward.
- Normal reaction \( N \) acts perpendicular to the plane.
- Friction force \( f \) acts opposite to motion along the plane.
Step 2: Resolving Forces Along the Inclined Plane
- The gravitational force along the inclined plane: \[ F_{\text{gravity}} = mg \sin\theta \] - The normal force perpendicular to the plane: \[ N = mg \cos\theta \] - The frictional force opposing the motion: \[ f = \mu N = \mu mg \cos\theta \] Step 3: Applying Newton’s Second Law
- The net force along the inclined plane:
\[ F_{\text{net}} = mg \sin\theta - \mu mg \cos\theta \] - The acceleration \( a \) is given by: \[ a = \frac{F_{\text{net}}}{m} = g (\sin\theta - \mu \cos\theta) \] Step 4: Conclusion
Thus, the acceleration of the body is: \[ \mathbf{g(\sin\theta - \mu \cos\theta)}. \]