We have two objects â€” a cube of 13kg and a sphere of 39kg. The cube hangs from a rope attached to the ceiling, while the sphere hangs from a second rope attached to the bottom of the cube.

Assuming the two ropes are massless, What is the tension in the first rope? And in the second?

Solving the problem

Let's start by drawing a sketch of what is happening:

Since we're dealing with massless ropes, we need to keep in mind that tensions exerted by the ends of a massless rope are equal in magnitude. We will indicate the magnitude of the tensions in the first rope with T_{1}, and the magnitude of the tensions in the second rope with T_{2}.

Let's carefully examine our sketch and enumerate all the forces that we think act on our two objects:

The cube is subject to 3 forces:

the tension in the upper rope, T_{1} (which prevents the cube from falling)

the tension in the lower rope, T_{2} (pulling the cube downward)

and the gravitational force, mg

The sphere is subject to only 2 forces:

the tension in the lower rope, T_{2} (which prevents the sphere from falling)

We know the masses (13kg for the cube and 39kg for the sphere).

We want to find the tensions in the two ropes.

We know

m = 13kg

M = 39kg

We want to know

T_{1} = ?

T_{2} = ?

The cube and the sphere are hanging, i.e. they are in static equilibrium. This means that the resultant forces on the cube and on the sphere must be zero.

The sphere is subject to two forces that are opposite in direction (T_{2} and Mg). Since the resultant force is zero, these two forces must be equal in magnitude:

T_{2} = Mg

T_{2} = (39kg) (9.8N/kg)

T_{2} = 380N

The cube is subject to three parallel forces (T_{1}, T_{2}, and mg). T_{1} is directed upward, T_{2} and mg are directed downward. Again, since the resultant force is zero, the magnitude of T_{1} must be equal to the sum of the magnitudes of T_{2} and mg:

T_{1} = T_{2} + mg

T_{1} = 380N + (13kg) (9.8N/kg)

T_{1} = 510N

Hence, the tension in the upper rope is 510N, and the tension in the lower rope is 380N.

Tips & Tricks

Remember that tension forces exerted by the ends of a massless rope (or string, cable, etc.) have the same magnitude.