Two identical boxes with mass 10kg sit on the horizontal floor of a truck. There is friction between one of the boxes and the floor, and no friction between the other box and the floor. The two boxes are tied together with a massless string. The truck begins to slow down with acceleration -3.75 m/s2.
If neither block slides, how large a coefficient of static friction must there be between the floor and the one box on the left (the one with the force of friction)?
To determine the coefficient of static friction required between the floor and the box on the left, we need to analyze the forces acting on the system.
1. Begin by drawing a free-body diagram for the box on the left. We can label the forces acting on it.
- The weight acting vertically downward with a magnitude of (mass of box) * (acceleration due to gravity).
- The tension force in the string acting to the right, which has the same magnitude for both boxes.
- The force of friction acting opposite to the direction of acceleration, to the left.
2. Consider the forces acting on the box on the right. Since there is no friction, the only force acting on it is the tension force in the string, directed towards the left.
3. Now apply Newton's second law separately to each box:
- For the box on the left: The sum of the horizontal forces is equal to the mass of the box multiplied by its acceleration. From the free-body diagram, this can be written as:
(tension force) - (force of friction) = (mass of box) * (acceleration).
- For the box on the right: The sum of the horizontal forces is equal to the mass of the box multiplied by its acceleration. In this case, the only force is the tension force, so we have:
(tension force) = (mass of box) * (acceleration).
4. Since the tension force is the same for both boxes, we can set the two equations equal to each other:
(tension force) - (force of friction) = (tension force).
Simplifying this equation, we find:
(force of friction) = 0.
5. This means that there is no force of friction acting on the box on the left. However, we know that there is friction between the box and the floor. Thus, the only way this can happen is if the coefficient of static friction between the floor and the box on the left is zero.
Therefore, to make sure that neither box slides, the coefficient of static friction must be zero.