Obtain an expression for the acceleration of a body moving down a rough inclined plane.

To derive the expression for the acceleration of a body moving down a rough inclined plane, we need to consider the forces acting on the body. This involves understanding the components of gravitational force, friction, and the normal force. Let’s break this down step by step.

Forces Acting on the Body

When a body is placed on an inclined plane, several forces come into play:

• Gravitational Force (Weight): This acts vertically downward and is given by \( W = mg \), where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity.
• Normal Force (N): This acts perpendicular to the surface of the incline.
• Frictional Force (F_f): This opposes the motion of the body down the incline and is given by \( F_f = \mu N \), where \( \mu \) is the coefficient of friction.

Breaking Down the Forces

To analyze the motion, we need to resolve the gravitational force into two components:

• The component parallel to the incline: \( W_{\parallel} = mg \sin(\theta) \)
• The component perpendicular to the incline: \( W_{\perpendicular} = mg \cos(\theta) \)

Here, \( \theta \) is the angle of inclination of the plane. The normal force \( N \) is equal to the perpendicular component of the weight, so:

N = mg \cos(\theta)

Calculating the Frictional Force

Using the normal force, we can express the frictional force:

F_f = \mu N = \mu mg \cos(\theta)

Applying Newton's Second Law

According to Newton's second law, the net force acting on the body along the incline is equal to the mass of the body multiplied by its acceleration:

F_{net} = ma

The net force acting down the incline can be expressed as:

F_{net} = W_{\parallel} - F_f = mg \sin(\theta) - \mu mg \cos(\theta)

Setting Up the Equation

Now, substituting this into Newton's second law gives us:

mg \sin(\theta) - \mu mg \cos(\theta) = ma

Solving for Acceleration

We can simplify this equation by dividing through by \( m \) (assuming \( m \neq 0 \)):

g \sin(\theta) - \mu g \cos(\theta) = a

Thus, the expression for the acceleration \( a \) of the body moving down a rough inclined plane is:

a = g \sin(\theta) - \mu g \cos(\theta)

Understanding the Result

This equation shows that the acceleration of the body depends on both the angle of the incline and the coefficient of friction. As the angle increases, the gravitational component acting down the incline increases, leading to greater acceleration. Conversely, a higher coefficient of friction reduces the acceleration, as it increases the opposing force. This relationship is crucial in applications such as vehicle dynamics on slopes or designing safe ramps.