Problem:

A block of weight w is placed on a horizontal table and a pushing force is applied at an angle θ to the horizontal. The angle of friction is given as ∅. We need to determine the magnitude of force required to move the block.

Solution:

To solve this problem, we can break down the forces acting on the block and analyze their components in the horizontal and vertical directions.

1. Weight of the block:
The weight of the block acts vertically downwards. Its magnitude is given as w.

2. Normal force:
The normal force acts perpendicular to the surface of the table, opposing the weight of the block. Since the block is not moving vertically, the normal force is equal in magnitude to the weight, i.e., w.

3. Pushing force:
The pushing force is applied at an angle θ to the horizontal. We need to determine the horizontal component of this force, which is responsible for overcoming the friction and moving the block.

4. Friction force:
The friction force acts parallel to the surface of the table, opposing the motion of the block. Its magnitude can be determined using the equation: friction force = coefficient of friction * normal force. In this case, the coefficient of friction is given as ∅. Therefore, the friction force is ∅ * w.

5. Resolving forces:
To determine the magnitude of the force required to move the block, we need to analyze the horizontal forces. We can resolve the pushing force into its horizontal and vertical components.

The horizontal component of the pushing force can be determined using the equation: horizontal component = magnitude of force * cos(θ).

6. Equilibrium in the horizontal direction:
For the block to be in equilibrium in the horizontal direction (not moving), the horizontal component of the pushing force should balance the friction force.

Therefore, we have the equation: horizontal component = friction force.

Substituting the values, we get: magnitude of force * cos(θ) = ∅ * w.

7. Solving for the magnitude of force:
To find the magnitude of force required to move the block, we rearrange the equation:

magnitude of force = (∅ * w) / cos(θ).

Simplifying further, we can rewrite the equation as:

magnitude of force = w * (∅ / cos(θ)).

Since sin(θ) = (∅ / cos(θ)), we can substitute this value:

magnitude of force = w * sin(θ).

Therefore, the magnitude of force required to move the block is w * sin(θ) / cos(θ * ∅).

Conclusion:

The magnitude of force required to move the block is given by the expression w * sin(θ) / cos(θ * ∅).

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