Example 13.4.2

Suppose a mass m is supported by two ropes, as shown below. One rope extends horizontally from the wall, and the other rope goes to the ceiling at an angle q from the vertical. Calculate the tension in each rope.

We begin by making a free-body diagram of the mass m. This is a picture that isolates the mass from its surroundings, and shows only those forces acting on the mass. The forces acting on this mass are the tensions in the two ropes, which we will call T1 and T2, and its weight mg, which of course points down:

We will also need to break the T2 vector into x and y components (the other two force vectors are already on either the x or y axis). This becomes

Now, for the total force to add up to zero, all the vectors along the y axis and all the vectors along the x axis must add up to zero.

1. Sum of forces in the x-direction equals zero

T2 sin q - T1 = 0

2. Sum of forces in the y-direction equals zero

T2 cos q - mg = 0

We now have two equations in two unknowns, T1 and T2 (we are assuming that m and q are known). The second equation can be solved for T2 to give

This can be substituted into the first equation, which can then be solved for T1: