Q22DQ - University Physics with Modern Physics

Question

A large crate is suspended from the end of a vertical rope. Is the tension in the rope greater when the crate is at rest or when it is moving upward at constant speed? If the crate is traveling upward, is the tension in the rope greater when the crate is speeding up or when it is slowing down? In each case, explain in terms of Newton’s laws of motion.

Step-by-Step Solution

Answer

The tension equals the crate’s weight when at rest or moving up at constant speed. The tension in the rope will be greater when speeding up than when slowing down.

Step 1: Newton’s Second law

A body accelerates when a net force acts on it and travels in the same direction as the net force. The product of mass and acceleration vector is equal to the force vector.

$\sum \overrightarrow{F} =m\overrightarrow{a}$

Step 2: Determine the tension in the rope at rest or at a constant speed

Here tension in the rope will pull the crate upward, and the force of gravity will pull it downward. At rest or moving at constant speed upward, the acceleration is zero.

T - mg = 0

         T= mg

The tension is equal to the weight of the crate.

Step 3: Determine the Tension in the rope when speeding up and slowing down.

When speeding up at an acceleration a, the tension in the rope is

T + (-mg)=ma

T = m(g + a)

When it slows down, the crate is de-accelerating; therefore, it will have a negative sign.

T + (-mg)=m(- a)

T = m(g - a)

It is apparent from the obtained results that tension in the rope will be greater when speeding up than when slowing down.