What are the tension and acceleration in this pulleys and rope problem?

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In the pulley and rope problem involving two masses, the tension in the rope is calculated to be 13.7 N, while the acceleration of mass 1 is determined to be 1.37 m/s². The initial assumption that tension equals half the weight of mass 2 was incorrect, as the dynamics of the system require considering the movement of both masses. It was clarified that mass 2's acceleration is half that of mass 1, leading to a correct formulation of the equations governing the system. The final calculations confirm the textbook answers, demonstrating the importance of analyzing the forces acting on both masses. Understanding the relationship between the masses and their accelerations is crucial for solving such problems accurately.
Teegvin
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Mass 1 = 10.0 kg
Mass 2 = 3.00 kg

The pulleys and rope are massless, and there is no friction.

What is the tension in the rope?

What is the acceleration of mass 1?

If my thinking is correct, the tension is half the weight of mass 2 (14.7 N), and the acceleration of mass 1 is 1.47 m/s^2.

My textbook gives the solutions as 13.7 N and 1.37 m/s^2.
 
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Teegvin said:
If my thinking is correct, the tension is half the weight of mass 2...
What makes you think that? Show your work and we can see what happened.
 
You're right about the tension being evenly distributed on both sides of mass 2, but mass 2 isn't stationary. As mass 1 slides across the table, mass 2 is moving downward, so the net is less than weight of mass 2.

Try thinking about mass 1 first. The force required (10kg * a) equals the tension on either side of mass 2.
 
BobG said:
Try thinking about mass 1 first. The force required (10kg * a) equals the tension on either side of mass 2.

I figured it out after I read this, and after I realized that the acceleration of mass 2 is half that of mass 1, because it goes half the distance in the same time.

m2(a1)/2 = m2g - 2T
= m2g - 2(10a1)
m2(a1)/2 + 20a1 = m2g
(3/2)a1 + 20a1 = 3g
(43/2)a1 = 3(9.80)
a1 = 1.37

Is this reasoning correct?
 
Looks good to me. (As a matter of style, I would solve for the acceleration in terms of m1 and m2. I wouldn't plug in the actual masses until the last step. But you've got it!)
 

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