Problem: A mass hanging from two ropes

A mass of 108 g is hanging from two massless ropes attached to the ceiling.

One rope makes an angle of 50° with the ceiling, while the other makes an angle of 29°.

Find the tensions in the two ropes.

Solving the problem

Let's begin by drawing our mass hanging from the two ropes:

Looking at our sketch, we can infer that the mass is subject to 3 forces:

• the tension force exerted by the first rope, T₁
• the tension force exerted by the second rope, T₂
• and the force of gravity, mg

Here's the free-body diagram of our hanging mass:

We know the mass (108 g, which in kilograms is 0.108 kg), and the angles that the two ropes make with the ceiling (50° and 29°).

We are asked to find the tensions in the two ropes.

We know

We want to know

The mass is hanging.

What does this tell us?

This tells us that the acceleration of the mass must be zero.

And because the acceleration is zero, the resultant force acting on the mass is also zero. Indeed, for Newton's 2^nd Law:

Now that we determined that the resultant force acting on the mass is zero, we can find the tensions of the two ropes using the following step by step process:

1. First, we find the x and y components of the resultant force, as a sum of the x and y components of all the forces that act on the mass.
2. Then, since we know that R is zero, R_x and R_y must also be zero, so we substitute them with 0 in the equations that we found in the previous step.
3. Finally, we use the resulting equations to find the tensions, T₁ and T₂.

Let's start with the first step.

We draw the coordinate axes on our free-body diagram. For convenience, we choose the x axis horizontal and the y axis vertical. Then, we determine the x and y components of all the forces that act on the mass.

We need to keep in mind that the angle between each tension force and its x component is equal to the angle that the rope, producing that tension, makes with the ceiling:

Thus, the x and y components of the resultant force will be:

x:

y:

The next step is to substitute R_x and R_y in Eq. (1) and Eq. (2) with 0:

Using these two equations we can easily find T₁ and T₂.

There are multiple ways in which we can do this. One way would be to first solve Eq. (3) for T₂:

Then, we substitute T₂ with 0.735T₁ in Eq. (4):

And we solve it for T₁:

Finally, we substitute the value of T₁ in Eq. (5) to find T₂:

Therefore, the tension in the rope at the 50° angle is 0.946 N, and the tension in the rope at the 29° angle is 0.695 N.

Tips & Tricks

• Remember that when an object hangs from two ropes, the angle between the tension produced by a rope and the x component of that tension, is equal to the angle that the rope makes with the ceiling:

• Next
  Problem: Two hanging objects connected by a rope
• Previous
  Problem: Mass pulled up an incline with friction

• Facebook
• Twitter
• Email