How Do Tensions on Either Side of a Rotating Pulley Compare?

[Sean77771]

Homework Statement

A rope passes over a 10-cm-diameter, 2.0 kg pulley that rotates on frictionless bearings. A graph shows the pulley's angular velocity as a function of time. The graph increases from (0,0) up to (3,120) linearly. Basically, there is a pulley with a rope thrown over it such that there are two sides to the rope, and two separate tensions, T_L (on the left side of the pulley), and T_R (on the right)

a) Is the tension T_L in the left rope larger, smaller, or equal to the tension in the right rope? Explain.

b) If you answered "equal" in part a, what is the magnitude of the tension T_L? If you answered "larger" or "smaller" in part a, what is the difference abs(T_L - T_R) between the two tensions?

Homework Equations

T = mr^2*alpha

The Attempt at a Solution

I thought that the tension in the left rope was larger, but this turned out to be wrong. I thought it must be, because omega was positive and constantly increasing, meaning the pulley was rotating counter clockwise. Also, just from the angular velocity/acceleration, I'm not sure how to relate T_L to T_R.

Please, I could really use some help!

 

[aq1q]

is it possible for you to have the diagram? I, personally, like to have a strong grasp on understanding the problem. By the way, your equation isn't correct for torque. Torque is basically the cross product of a Force vector and the Radius (displacement vector).

P.S. you don't HAVE to show a diagram. But I don't really understand the problem. sorry :(

 

[ogb p]

I would approach this problem by first reviewing the basic principles of pulleys and torque. In this case, the rope is passing over a pulley with a diameter of 10 cm, which means that the radius of the pulley is 5 cm. The pulley is also rotating on frictionless bearings, which means that there is no external force acting on it to slow down its rotation.

To answer part a, we need to consider the forces acting on the pulley. On the left side of the pulley, we have the tension force T_L from the rope pulling down, and on the right side, we have the tension force T_R from the rope pulling up. Since the pulley is rotating in a counter-clockwise direction, we can assume that the net torque on the pulley is in the clockwise direction. This means that the tension force on the right side (T_R) must be larger than the tension force on the left side (T_L) in order to balance out the torque.

In other words, the tension in the left rope (T_L) is smaller than the tension in the right rope (T_R). This can also be seen from the equation T = mr^2*alpha, where m is the mass of the pulley and r is the radius. Since both m and r are constant, the only variable that can affect the tension is the angular acceleration (alpha). Since the pulley is rotating with a constant angular acceleration, the tension on the right side (T_R) must be larger to produce a greater torque and maintain the constant rotation.

For part b, since we have determined that the tension in the left rope (T_L) is smaller than the tension in the right rope (T_R), we can calculate the difference between the two tensions by taking the absolute value of the difference between T_L and T_R. This would give us the magnitude of the difference between the two tensions.

In conclusion, understanding the principles of pulleys and torque can help us solve this problem and determine the relationship between the tensions on the two sides of the rope.