Why Are Tensions Different in the Same Rope?

[Legendon]

Same Rope, DIFFERENT tensions?

Homework Statement

Consider the system shown with m1 = 20.0 kg, m2 = 12.5 kg, R = 0.200 m, and the mass of pulley M = 5.00 kg. Object m2 is resting on the floor, and object m1 is 4.00 m above the floor when it is released from rest. The pulley axis is frictionless. The cord is light, does not stretch, and does not slip on the pulley. (a) Calculate the time interval required for m1 to hit the floor. (b) How would your answer change if the pulley were massless?

Homework Equations

The Attempt at a Solution

The solution is fine but i cannot comprehend in the first place why the tensions are not equal. I thought in the same string the tension is the same? And in part b, i think the tension is the same for both T1 and T2.

 

[PhanthomJay]

I don't understand what you mean when you say

The solution is fine but i cannot comprehend in the first place why the tensions are not equal.

Since you're OK with the solution, you must realize that the tensions must be unequal in order to get an unbalanced torque on the pulley so that it angularly/tangentially accelerates consistent with the linear acceleration of the blocks. The tensions are equal, as you noted, only if the pulley is massless and frictionless (an ideal pulley) or if it is of negligible mass and friction such that the results are close enough by making the assumption of an ideal pulley.

 

[Legendon]

Thanks, i can understand that part. That it is the difference in torque that causes the acceleration when pulley has mass now this one is giving problems.
The motor right powers the system causing the rigid object left to spin clockwise. So when my prof drew the FBD of the rigid object, the tensions are both pointing away to the right. I expected the tension on top to point to the right and bottom to point left. Why are both pointing right ?

 

[PhanthomJay]

Thanks, i can understand that part. That it is the difference in torque that causes the acceleration when pulley has mass now this one is giving problems.
The motor right powers the system causing the rigid object left to spin clockwise. So when my prof drew the FBD of the rigid object, the tensions are both pointing away to the right. I expected the tension on top to point to the right and bottom to point left. Why are both pointing right ?

This same question could be asked for the original problem. Tension forces ALWAYS pull away from the object on which they act. If you hoist up a pail of water using a pulley attached to the ceiling (an Atwood machine), the side of the rope with the pail moves up, and the side that you are pulling on moves down, but on both sides, the rope tension force on the pulley acts down, right? And by Newton 3, the rope tension force on the pail and the rope tension force on your pulling hand acts up, right?

 

[rwisz]

As a scientist, it is important to understand the concept of tension and how it applies to different situations. In this case, the tension in a rope can vary depending on the forces acting on it. In the given system, the tension in the rope connecting m1 and the pulley is different from the tension in the rope connecting m2 and the pulley. This is because m1 is accelerating downwards due to gravity while m2 is at rest on the floor. This difference in acceleration creates a difference in tension in the two ropes.

If we were to imagine the pulley as a massless object, the tension in both ropes would be equal because the pulley would not have any effect on the acceleration of the objects. However, in reality, the pulley has a mass and therefore adds to the overall system's inertia, affecting the acceleration of m1.

To answer the given questions, we can use the equations of motion to calculate the time interval required for m1 to hit the floor. For part a, we can use the equations of motion for constant acceleration to calculate the time taken by m1 to travel 4.00 m downwards. For part b, we can use the same equations but assume the pulley to be massless, which would result in an equal tension in both ropes.

In conclusion, the concept of tension can vary in different situations and it is important to take into account all the forces acting on a system to accurately determine the tension in a rope.