AP Physics 1 Exam Question:

A block of mass m is suspended by two strings as shown in the figure below. The strings make angles θ and φ with the vertical axis. The block is in equilibrium.

1. Determine an expression for the tension in the first string (T1) in terms of m, g, θ, and φ.
2. Determine an expression for the tension in the second string (T2) in terms of m, g, θ, and φ.

Assume that the strings have negligible mass and the gravitational force is directed vertically downwards.

Answer:

1. We will consider the forces acting on the block in the vertical direction. The gravitational force acting on the block is given by:

   Fg = m * g

   The vertical component of T1 will counteract the gravitational force, while the vertical component of T2 will add to the gravitational force. Therefore, we can write the equation for the vertical forces as:

   T1 * cos(θ) + T2 * cos(φ) = m * g

   Solving this equation for T1, we get:

   T1 = (m * g - T2 * cos(φ)) / cos(θ)

2. Next, we will consider the forces acting on the block in the horizontal direction. The horizontal components of T1 and T2 must balance each other to keep the block in equilibrium. Therefore, we can write the equation for the horizontal forces as:

   T1 * sin(θ) = T2 * sin(φ)

   Solving this equation for T2, we get:

   T2 = (T1 * sin(θ)) / sin(φ)

These expressions for T1 and T2 provide a way to calculate the tensions in the two strings given the mass of the block, the angles of the strings, and the acceleration due to gravity.