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Problem: Two hanging objects connected by a rope

We have two objects — a cube of 13 kg and a sphere of 39 kg. 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:

The cube hangs from a rope attached to the ceiling, while the sphere hangs from a rope under the cubemM

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 T1, and the magnitude of the tensions in the second rope with T2.

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, T1 (which prevents the cube from falling)
  • the tension in the lower rope, T2 (pulling the cube downward)
  • and the gravitational force, mg

The sphere is subject to only 2 forces:

  • the tension in the lower rope, T2 (which prevents the sphere from falling)
  • and the gravitational force, Mg

Let's draw a free-body diagram for the cube, and another for the sphere:

The free-body diagram of the cubemgT2T1
The free-body diagram of the sphereMgT2

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

We want to find the tensions in the two ropes.

We know

m = 13 kg
M = 39 kg

We want to know

T1 = ?
T2 = ?

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 (T2 and Mg). Since the resultant force is zero, these two forces must be equal in magnitude:

T2 = Mg
T2 = (39 kg) (9.8 N/kg)
T2 = 380 N

The cube is subject to three parallel forces (T1, T2, and mg). T1 is directed upward, T2 and mg are directed downward. Again, since the resultant force is zero, the magnitude of T1 must be equal to the sum of the magnitudes of T2 and mg:

T1 = T2 + mg
T1 = 380 N + (13 kg) (9.8 N/kg)
T1 = 510 N

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

Tips & Tricks

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