Problem: In the diagram below, the (slightly weird but very strong) professor is raising the midget elephant. The elephant has a mass of 560 kg and is accelerating upward at 1.5 m/s². The pulley (a disk or cylinder) has a mass of 28 kg and a radius of 23 cm.

a) What is the tension in the rope just above the elephant?

b) What is the moment of inertia of the pulley?

c) Find the angular acceleration of the pulley.

d) What is the tension in the rope on the professor's side of the pulley?

e) The pulley is 8 m above the ground. At the instant shown below, the rope that the professor makes an angle of 40^o from the ground and causes the elephant to have a speed of 0.85 m/s. How fast must the professor be walking away from the point directly under the pulley?

Solution:

a) Using Newton's Second Law for the free body of the elephant,

You'll notice that the tension is NOT equal to the weight of the elephant!

b) Since the pulley is a disk (or a cylinder), I use the formula from the table:

c) Using the third "connection" equation,

d) Applying Newton's Second Law to the free body of the pulley,

As usual, I carried the unrounded intermediate results on the calculator to avoid round off error.

e) This is a "related rates" problem:

While the 8 m is constant, the rate at which L is changing is the elephant's rate of rising (0.85 m/s). The rate of change of x is the professor's speed. I first find x and L: