AP Physics 1 Exam Question

A block of mass $m$ is placed on a horizontal surface with a coefficient of friction $\mu$. It is connected to a hanging mass $M$ through a massless and frictionless pulley, as shown in the figure below.

1. Determine the tension in the string connecting the block and the hanging mass.

Given:

• Mass of the block: $m$
• Mass of the hanging mass: $M$
• Coefficient of friction between the block and the surface: $\mu$

(a) Assume the system is at rest and there is no acceleration. Identify the forces acting on the block and the hanging mass.

(b) Calculate the net force acting on each object.

(c) Write down the equation(s) that represent the equilibrium condition for the block and the hanging mass.

(d) Solve the equation(s) to find the tension in the string.

Answer

(a) For the block:

• Gravitational force ($mg$) acting downwards.
• Normal force ($N$) exerted by the surface in the upward direction.
• Frictional force ($f$) opposing the block's motion.

For the hanging mass:

• Gravitational force ($Mg$) acting downwards.

(b) The net force on the block can be calculated by applying Newton's second law in the horizontal direction:

The net force on the hanging mass is simply its weight:

For the block:

For the hanging mass:

(d) To find the tension in the string, we need to solve the equation from the block's equilibrium condition:

To find the frictional force $f$, we can use the equation:

Since the block is at rest, the vertical forces must balance:

Substituting this value of $N$ into the equation for friction, we get:

Now, we can substitute this value of $f$ into the equation for tension:

Solving for tension:

Therefore, the tension in the string connecting the block and the hanging mass is $\mu \cdot mg$.