Pushing Force making an angle θ to the horizontol is applied on a block of weight W placed on a horizontol table. If the angle of friction is Φ, the magnitude of force required to move the body is ?

To determine the magnitude of the force required to move a block on a horizontal table when a pushing force is applied at an angle, we need to analyze the forces acting on the block. This involves understanding the concepts of friction, weight, and the components of the applied force.

Understanding the Forces Involved

When a force is applied at an angle θ to the horizontal, it can be broken down into two components: the horizontal component (F_x) and the vertical component (F_y). The weight of the block (W) acts vertically downward, and the normal force (N) acts vertically upward from the surface of the table.

Breaking Down the Applied Force

Let’s denote the applied force as F. The components of this force can be expressed as:

• F_x = F * cos(θ) (horizontal component)
• F_y = F * sin(θ) (vertical component)

Normal Force Calculation

The normal force is affected by the vertical component of the applied force. It can be calculated as:

N = W - F_y = W - F * sin(θ)

Frictional Force

The frictional force (f) opposing the motion is given by:

f = μ * N

Here, μ is the coefficient of friction, which is related to the angle of friction (Φ) by the equation:

μ = tan(Φ)

Substituting for N, we get:

f = μ * (W - F * sin(θ))

Condition for Motion

For the block to start moving, the horizontal component of the applied force must overcome the frictional force:

F_x ≥ f

Substituting the expressions we derived:

F * cos(θ) ≥ μ * (W - F * sin(θ))

Solving for the Required Force

Rearranging the inequality gives:

F * cos(θ) + μ * F * sin(θ) ≥ μ * W

Factoring out F from the left side:

F * (cos(θ) + μ * sin(θ)) ≥ μ * W

Now, we can solve for F:

F ≥ (μ * W) / (cos(θ) + μ * sin(θ))

Final Expression

Thus, the magnitude of the force required to move the block is:

F ≥ (tan(Φ) * W) / (cos(θ) + tan(Φ) * sin(θ))

This formula allows you to calculate the minimum force needed to initiate movement, taking into account both the weight of the block and the angle at which the force is applied. Understanding these relationships is crucial in physics, especially in mechanics, where forces and motion are fundamental concepts.