To solve this problem, we need to analyze the forces acting on the cube and determine the distance between the line of action of the weight (mg) and the normal reaction force. Let's break it down step by step.
Understanding the Forces at Play
We have a cube with mass \( m \) and side length \( a \). The forces acting on the cube are:
- The weight of the cube, \( W = mg \), acting downwards at the center of the cube.
- The normal force, \( N \), acting upwards at the base of the cube.
- A horizontal force \( F = \frac{mg}{3} \) applied at the upper surface of the cube.
- The frictional force, which opposes the motion due to the applied force.
Calculating the Normal Force
Since the cube is resting on a rough surface, the normal force \( N \) must balance the weight of the cube. Therefore, we have:
N = mg
Frictional Force Consideration
The maximum static frictional force can be calculated using the coefficient of friction \( u \):
F_{friction} = uN = \frac{1}{2}mg
Since the applied force \( F = \frac{mg}{3} \) is less than the maximum static friction, the cube will not slide. Thus, the frictional force will equal the applied force:
F_{friction} = \frac{mg}{3}
Finding the Distance Between Forces
Next, we need to find the distance between the line of action of the weight \( mg \) and the normal reaction force \( N \). The weight acts at the center of the cube, which is located at a height of \( \frac{a}{2} \) from the base. The normal force acts at the base of the cube.
To find the distance between these two lines of action, we can visualize the cube:
- The weight \( mg \) acts downward through the center of the cube, which is at a height of \( \frac{a}{2} \).
- The normal force \( N \) acts upward at the bottom of the cube.
The distance between the line of action of the weight and the normal force is simply the height from the base to the center of the cube:
Distance = \frac{a}{2}
Conclusion on Options
Given the options:
- (a) \( \frac{a}{2} \)
- (b) \( \frac{a}{3} \)
- (c) \( \frac{a}{4} \)
- (d) none of these
The correct answer is (a) \( \frac{a}{2} \), as this represents the distance between the line of action of the weight and the normal reaction force acting on the cube.
