A body of mass M is kept on a rough horizontal surface (friction cofficient = ? ). A person is trying to pull the body by applying a horizontal force but the body is not moving. The force by the surface on A is F where

When a body of mass M is placed on a rough horizontal surface and a person applies a horizontal force to pull it, but the body does not move, we can analyze the situation using the concepts of friction and forces. The force exerted by the surface on the body is known as the normal force, and it plays a crucial role in determining the frictional force that opposes the applied force.

Understanding Forces at Play

In this scenario, we have two primary forces acting on the body:

• Applied Force (F_a): This is the horizontal force exerted by the person trying to pull the body.
• Frictional Force (F_f): This is the force that opposes the motion of the body due to the rough surface. It is determined by the coefficient of friction (μ) and the normal force (N).

Normal Force and Friction

The normal force (N) is the force exerted by the surface perpendicular to the body. For a horizontal surface, the normal force is equal to the weight of the body, which can be expressed as:

N = M * g

where g is the acceleration due to gravity (approximately 9.81 m/s²). The frictional force can be calculated using the formula:

F_f = μ * N

Substituting the expression for the normal force, we get:

F_f = μ * (M * g)

Condition for No Motion

For the body to remain stationary, the applied force must be less than or equal to the maximum static frictional force. This can be expressed as:

F_a ≤ F_f

Substituting the expression for the frictional force, we have:

F_a ≤ μ * (M * g)

Determining the Coefficient of Friction

If we know the applied force (F_a) and the mass of the body (M), we can rearrange the inequality to find the coefficient of friction:

μ ≥ F_a / (M * g)

This means that the coefficient of friction must be at least equal to the ratio of the applied force to the product of the mass and gravitational acceleration for the body to remain stationary.

Example Calculation

Let's say we have a body with a mass of 10 kg, and a person is applying a force of 20 N. We can calculate the minimum coefficient of friction required to prevent motion:

First, calculate the weight of the body:

N = M * g = 10 kg * 9.81 m/s² = 98.1 N

Now, using the applied force:

μ ≥ F_a / (M * g) = 20 N / 98.1 N ≈ 0.204

This means that the coefficient of friction must be at least 0.204 for the body to remain stationary under the applied force of 20 N.

In summary, the interaction between the applied force, normal force, and frictional force determines whether the body will move or stay at rest. Understanding these relationships is key to solving problems involving forces and motion on surfaces.