In the system in figure M=13.4 kg, m1=1 kg, m2= 2 kg and m3 = 3 kg. the coefficient of static friction between m1 and m2 is 0.75 and between m2 and m3 is 0.6. all other surfaces are frictionless.

1. what minimum horizontal force F must be applied to M , so that m2 doesn’t slip on M?
2. what maximum value of horizontal force F must be applied for which m1 doesn’t slip on m2?
3. if F=100N , does m1 slip with respect to m2 or m2 slips with respect to M?

To solve this problem, we need to analyze the forces acting on each mass and determine the conditions for slipping based on the coefficients of static friction provided. Let's break it down step by step.

Understanding the Forces Involved

We have three masses: M (the larger mass), m1, m2, and m3. The coefficients of static friction between m1 and m2, and between m2 and m3, will help us find the critical forces that prevent slipping. The forces acting on each mass will depend on the applied horizontal force F and the gravitational forces acting on the masses.

Calculating the Maximum Force for m2 on M

First, we need to ensure that m2 does not slip on M. The maximum static friction force that can act between m2 and M can be calculated using the formula:

• F_friction_max = μ_s * N

Where:

• μ_s is the coefficient of static friction (0.75 between m1 and m2).
• N is the normal force, which is equal to the weight of m2 (m2 * g).

Calculating the normal force:

• m2 = 2 kg
• g = 9.81 m/s² (acceleration due to gravity)
• N = m2 * g = 2 kg * 9.81 m/s² = 19.62 N

Now, we can find the maximum static friction force:

• F_friction_max = 0.75 * 19.62 N = 14.715 N

This means that the horizontal force F must be at least 14.715 N to prevent m2 from slipping on M.

Calculating the Maximum Force for m1 on m2

Next, we need to ensure that m1 does not slip on m2. The maximum static friction force between m1 and m2 can be calculated similarly:

• μ_s = 0.75 (between m1 and m2)
• N = m1 * g = 1 kg * 9.81 m/s² = 9.81 N

Now, calculating the maximum static friction force for m1:

• F_friction_max_m1 = 0.75 * 9.81 N = 7.3575 N

This means that the horizontal force F must not exceed 7.3575 N to prevent m1 from slipping on m2.

Analyzing the Given Force of 100 N

If we apply a force F = 100 N, we need to check the conditions for slipping:

• Since 100 N is much greater than 14.715 N, m2 will definitely slip on M.
• Also, since 100 N is greater than 7.3575 N, m1 will slip on m2 as well.

In summary, with a force of 100 N applied to M, both m1 will slip with respect to m2, and m2 will slip with respect to M. This indicates that the applied force exceeds the maximum static friction forces available at both interfaces.

Final Thoughts

In practical terms, when dealing with multiple masses and friction, it's crucial to calculate the maximum static friction forces at each interface to determine the conditions for slipping. This approach allows us to understand how forces interact in a system and predict the behavior of the masses involved.