Question

Two children are playing with a roll of paper towels.One child holds the roll between the index fingersof her hands so that it is free to rotate, and the second child pulls at constant speed on the free end ofthe paper towels. As the child pulls the paper towels,the radius of the roll of remaining towels decreases.

(a) How does the torque on the roll change with time?(b) How does the angular speed of the roll change in time? (c) If the child suddenly jerks the end paper towel with a large force, is the towel more likely to break from the others when it is being pulled from an early full roll or from a nearly empty roll?

Short answer

Therefore, the solution is

If roll is suddenly jerked, the angular acceleration may not be sufficient to get it going with the paper, and it will break.

When the radius is large, this is more likely.

Step-by-step solution

of 1

• As the roll unwinds, the force from the child remains constant, but the point of application of the force (the radius) decreases. As a result, the torque decreases with time.
• Although the torque is decreasing, it is constantly positive, allowing the roll to accumulate angular momentum. In addition, the roll's moment of inertia decreases as it unwinds. The angular velocity will almost surely increase as the angular momentum grows and the moment of inertia falls.

If the roll is tiny, the large force may be able to angularly accelerate it quickly enough to prevent the towel from breaking. However, if the roll is too large, there will be too much angular momentum for the towel to accelerate quickly, and it will break.