To analyze the situation where a body of mass M is resting on a rough horizontal surface and is subjected to a horizontal pulling force without moving, we need to consider the forces acting on the body and the role of friction. The key here is to understand how friction interacts with the applied force and the weight of the body.
Understanding the Forces at Play
When a force is applied to the body, it experiences two main forces in the horizontal direction: the applied force (let's call it F_applied) and the frictional force (F_friction). The frictional force opposes the motion and is determined by the coefficient of friction (μ) and the normal force (N), which, on a horizontal surface, is equal to the weight of the body (Mg).
Frictional Force Calculation
The maximum static frictional force can be calculated using the formula:
- F_friction = μN
- Since N = Mg (the weight of the body), we have:
- F_friction = μMg
This means that the maximum force of friction that can act on the body before it starts to move is μMg. If the applied force is less than or equal to this maximum frictional force, the body will remain stationary.
Analyzing the Options
Now, let's evaluate the provided options based on our understanding:
- a) F = Mg: This suggests that the force exerted by the surface (normal force) is equal to the weight of the body. This is true, but it does not account for friction.
- b) F = μMg: This indicates that the force by the surface equals the maximum static friction. This is also true but only at the threshold of motion.
- c) Mg ≤ F ≤ Mg √(1+μ²): This option introduces a range that is not applicable in this context. The upper limit does not relate to the forces acting on the body.
- d) Mg ≥ F ≥ Mg√(1- μ²): This option suggests a range that is not relevant to the situation since it implies a relationship that does not hold true for static friction.
Correct Interpretation
Given that the body is not moving, the applied force must be less than or equal to the maximum static friction force. Therefore, the correct relationship is:
Thus, while the normal force is indeed equal to Mg, the force by the surface (friction) is what prevents the body from moving, and it can be expressed as μMg at maximum. Therefore, the most relevant answer in the context of the problem is that the applied force must be less than or equal to the maximum static friction force, which is μMg.
Conclusion
In summary, the force exerted by the surface (friction) is crucial in determining whether the body moves or remains stationary. The applied force must be balanced by the frictional force, which is limited by the coefficient of friction and the weight of the body. Hence, the correct interpretation of the forces leads us to conclude that the applied force must be less than or equal to μMg for the body to remain at rest.
