Question

During a TV report about ground water contamination, a reporter stands next to an old style well which works by lowering a bucket at the end of a rope into a deep hole in the ground to get water. At the top of the well, the rope is wound around a vertical pulley consisting of a heavy steel ring supported by light spokes. To demonstrate the depth of the well, the reporter completely wraps the light rope around the pulley, then releases it. The bucket, which starts at rest near the pulley, unwinds the rope from the pulley as it falls, taking 2.5 seconds to hit the bottom of the well. Although the depth of the well is never mentioned, you decide to calculate it. You estimate that the pulley has the same mass of the bucket and was very well-oiled, since you didn’t hear any squeaking. The moment of inertia of the pulley is one half of the product of its mass and the square of its radius. Detail process of each and every points can only get full credit.

Answer 1

Mass of the pulley = m

mass of the bucket = m

Moment of inertia of pulley = 1/2mr^2

Time taken by the bucket = 2.5 s

Using Free body diagram of the bucket we can write

mg -T = ma

             --------------------- equation (1)

and the free body diagram of pulley gives us

using this value of T in equation (1) we get.

Using this acceleration and time = 2.5 s

we can calculate the depth of the well as

So the depth of the well is 20.42 m