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?
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 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 sphere is subject to only 2 forces:
Let's draw a free-body diagram for the cube, and another for the sphere:
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.
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:
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:
Hence, the tension in the upper rope is 510 N, and the tension in the lower rope is 380 N.