Calculating Velocity of m1 After Falling a Distance of d Meters over a Pulley

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Homework Help Overview

The problem involves two masses, m1 and m2, connected by a rope over a massless and frictionless pulley. The scenario describes the release of the masses from rest and seeks to determine the velocity of m1 after it has fallen a distance of d meters. The context is rooted in classical mechanics, specifically dealing with forces, acceleration, and motion under gravity.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss the calculation of acceleration and the forces acting on each mass, including gravitational force and tension. There are attempts to derive equations using Newton's second law and to clarify the roles of tension and gravity. Some participants question the correctness of the initial assumptions and calculations.

Discussion Status

There is ongoing exploration of the forces involved, with some participants correcting each other's equations and clarifying the relationships between the variables. Guidance has been provided on drawing Free Body Diagrams and applying Newton's laws, leading to a more refined understanding of the acceleration and tension in the system.

Contextual Notes

Participants are working under the assumption that the pulley is ideal (massless and frictionless) and that the rope does not slip. There is a focus on ensuring the correct application of physics principles without reaching a definitive solution yet.

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Homework Statement


Two masses, m1 and m2, are hung over a pulley as shown. Assume that m1 is heavier, that the pulley is massless and frictionless, and that the rope does not slip. The blocks are held motionless and then released. Determine the magnitude of the velocity of m1 after it has fallen a distance of d meters. You may enter "m_1" for m1 and "m_2" for m2.


Homework Equations





The Attempt at a Solution



Found the acceleration of the 2 objects, (m_1+m_2)*g, but got stuck after that.
 

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You'll have to show your work. Your acceleration doesn't look right.
 
Well the total mass of the 2 objects is (m_1+m_2) and then you multiply it by g.
 
postfan said:
Well the total mass of the 2 objects is (m_1+m_2) and then you multiply it by g.

No, that's not correct (even the units do not yield acceleration... g is an acceleration, and multiplying it by mass gives you force).

Start by drawing the Free Body Diagrams (FBDs) for each mass. There's a force in common provided via the rope.
 
Is the common force tension?
 
postfan said:
Is the common force tension?
That's correct.
 
So now what do I do?
 
postfan said:
So now what do I do?
You follow gneill's advice. Draw Free Body Diagrams for each mass. What forces act on each?
 
On the block with mass m1 there is a gravitation force of m1*g and a tension force of m2*g, on the block with mass m2 there is a gravitation force of m2*g and there is a tension force of m1*g. Is that right?
 
  • #10
postfan said:
On the block with mass m1 there is a gravitation force of m1*g and a tension force of m2*g, on the block with mass m2 there is a gravitation force of m2*g and there is a tension force of m1*g. Is that right?
No. You have the gravitational forces correct, but the tension is something you'll have to solve for. It's an unknown, so label it "T". (And since the tension is the same throughout the rope, you cannot have it exerting different forces on each mass.)
 
  • #11
How do you solve for tension?
 
  • #12
postfan said:
How do you solve for tension?
Once you have your FBDs, you'll apply Newton's 2nd law to each mass. That will give you two equations. You'll solve those equations to find the tension, which will be one of the two unknowns.

That's if you want to know the tension. Since the question doesn't ask for it, you can just solve for the acceleration, which is the second unknown.
 
  • #13
I got g for the acceleration. Assuming it's right, what do I do next?
 
  • #14
postfan said:
I got g for the acceleration. Assuming it's right, what do I do next?
How did you get g for the acceleration? It's not correct. g would be the acceleration of something in free fall, not something attached to ropes and pulleys.

You haven't completed step 1: Draw FBDs for each mass.
 
  • #15
Yes I did. There are 2 forces acting on each object, gravity and tension.

Using Newtons 2nd I got 2 equations : m1*g-T=m1*a and m2*g-T=m2*a, solving for a gives a=g.

If I did something wrong please tell me what.
 
  • #16
postfan said:
Yes I did. There are 2 forces acting on each object, gravity and tension.

Using Newtons 2nd I got 2 equations : m1*g-T=m1*a and m2*g-T=m2*a, solving for a gives a=g.

If I did something wrong please tell me what.
Ah, you made a sign error in your equation for m2. If m1 accelerates down, then m2 must accelerate up.

(The only way that a = g would be if T = 0. In other words, if the rope were cut and the masses were in free fall.)
 
  • #17
Ok ,fixing the sign I got (m1-m2)*g/(m1+m2). Is that right?
 
  • #18
postfan said:
Ok ,fixing the sign I got (m1-m2)*g/(m1+m2). Is that right?
Good!
 
  • #19
Ok , for the next step I used the formula v^2=2*a*x and got v=sqrt(2*g*(m_1-m_2)/(m_1+m_2)*d). Is that right?
 
  • #20
postfan said:
Ok , for the next step I used the formula v^2=2*a*x and got v=sqrt(2*g*(m_1-m_2)/(m_1+m_2)*d). Is that right?
Looks good.

Just for fun, try solving it again using energy methods.
 

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