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Rotation motion problem

  1. Jul 31, 2014 #1
    1. The problem statement, all variables and given/known data
    A mass m1 is connected by a light cord to a mass m2, which slides on a smooth surface. The pulley rotates about a friction less axle and has a moment of inertia I and radius R. Assuming the cord does not slip on the pulley, find (a) the acceleration of the two masses, (b) the tensions T1 and T2, and (c) numerical values for a, T1 and T2 if I = 0.5 kg * m^2, R = 0.3 m, m1 = 4 kg, and m2 = 3 kg. (d) What would your answers be if the inertia of the pulley was neglected?



    2. Relevant equations



    3. The attempt at a solution
    I drew a free body diagram and tried to find the acceleration of the first block. So I did
    -9.6m1 + T = m1a1
    -9.8 + T/m = a1
    unfortunately this is not even close to being the right answer. How should I approach this problem?
     
  2. jcsd
  3. Jul 31, 2014 #2
  4. Jul 31, 2014 #3

    Nathanael

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    No, that is indeed the correct answer. It's just that you haven't figured out what [itex]T_1[/itex] is yet.

    To find [itex]T_1[/itex] you're going to have to draw the free body diagram of the pulley.

    (Note: The two blocks have the same acceleration which is equal to R times the angular acceleration of the pulley)
     
  5. Jul 31, 2014 #4
    OK I did that and did
    Ta = r∂
    Torque = I∂
    Torque = 9.8m
    9.8 m = I∂
    Ta = r(9.8m/I)
    What do I do from here to find the acceleration of both the blocks?
     
  6. Jul 31, 2014 #5

    Nathanael

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    The torque on the pulley will not be 9.8m

    The torque on the pulley comes from the two tensions, [itex]T_1[/itex] and [itex]T_2[/itex]

    The torque on the pulley will be the difference of the two tensions, since they act in opposite directions.



    Note: Since you also don't know [itex]T_2[/itex], you will have to draw the free body diagram of the other block too (isn't this fun! hehe)
     
    Last edited: Jul 31, 2014
  7. Jul 31, 2014 #6
    How do I find the acceleration of the second block. I feel like I'm going in circles. I must find the acceleration of the circle to find the acceleration of the second block. I must find the acceleration of the block to find the acceleration of the circle lol :)
     
  8. Jul 31, 2014 #7

    Nathanael

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    Yes it is circular in a way, but it will still be solvable because you know that the acceleration of the two blocks and the pulley are all related.

    The acceleration of the two blocks is exactly the same, and that acceleration is proportional to the angular acceleration of the pulley.


    So you will end up with 3 equations, but you can still use the same "a" (for acceleration) in all the equations because they are all related.


    Once you write those 3 equations the problem will be almost solved.
    (The 3 equations are the "net force equations" for both blocks and the pulley)
     
  9. Jul 31, 2014 #8
    If the two accelerations of the blocks are against each other shouldn't they cancel out?
     
  10. Jul 31, 2014 #9

    Nathanael

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    What do you mean against each other? They both accelerate together (one goes to the right, one goes down)
     
  11. Jul 31, 2014 #10
    OK I am confused. I understand everything in parts but I am not getting the big picture of the forces
     
  12. Jul 31, 2014 #11

    Nathanael

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    Ok I'll do my best to explain the big picture.

    Block 1 accelerates downwards causing the pulley to rotate and the block 2 to accelerate. Since they are all attatched together by a rope, they all have the same (linear) acceleration.

    The rope pulls up on the block 1 with some tension, T1, and so (by newton's 3rd law) the block pulls down on the rope, which pulls down on the pulley (with the same force, T1).

    The rope also pulls the block 2 forward with some tension T2, and so the block 2 pulls on the pulley (via the rope) with a tension T2

    You don't know what either force of tension is, but you do know one crucial fact: They all accelerate at the same rate!

    So you must write an equation for each object (block 1, block 2, and the pulley) that relates the tension with the acceleration. Once you write all 3 equations you'll be able to solve the problem for the acceleration by using algebra to get rid of ("cancel out") the forces of tension.
    (Once you know the acceleration you can go back and find the forces of tension, since the problem asked for them)


    Did this help at all?
     
  13. Aug 1, 2014 #12
    It helped me visualize the big picture but I am still struggling with the mathematics so the tension in block 1 should be m1a and the tangential force on the block should be m1a so the tension in the second block should be m2a. How do I relate all of this?
     
  14. Aug 1, 2014 #13

    Nathanael

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    No those are not the tensions. You don't know what the tensions are. Don't even try to figure out the tension because at this point it could really be anything (it's not something simple like m1a)

    For the first block you've already wrote the equation correctly in your very first post:
    "-9.8 + T/m = a"
    That is equation number one.

    Now do the same thing for the other block, then do the same thing for the pulley (with torque)
     
  15. Aug 1, 2014 #14

    ehild

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    The tension is the force exerted by the string. The string pulls the objects connected at its ends by the same force. You have two pieces, the horizontal and vertical one, with two different tensions. These tensions act also at the rim of the pulley, and exert opposite torques RT1 and RT2.

    Show the equation you get for each object.

    The hanging ball moves vertically with acceleration a. The forces applied are gravity and the tension T1. So m1a = ???
    The ball pulls the string. The vertical piece gets longer, the horizontal piece gets shorter with the same amount. The string pulls the box on the horizontal surface, with the same speed and acceleration as the ball moves. But the tension is different in the horizontal piece of the string, it is T2. So m2a2=T2

    The tensions act also at the rim of the pulley and rotate it. The string does not slide on the pulley, sot the linear acceleration of the rim is the same as the acceleration of both the box and the ball. As the ball moves downward, the pulley rotates clockwise. Show the equation you get for the relation between the torque and the angular acceleration of the pulley. You know that the angular acceleration is equal to a/R.

    You have three equations for the unknowns T1, T2, a. Cancel the tensions. Show your work.


    ehld
     

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  16. Aug 1, 2014 #15
    You could work with pulley equivalent mass, use an example velocity for m1 (say 1.0 m/s), then calculate the KE values for m1 and the pulley.
    The equivalent mass of the pulley = ( KE pulley / KE (m1) ) * m1
    Now you can treat the pulley as a linear mass to solve some of this problem.
     
  17. Aug 1, 2014 #16
    I'm sorry but I don't understand.
     
  18. Aug 1, 2014 #17

    ehild

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    You do not need any equivalent mass. The moment of inertia I and the radius of the pulley R are given. How is the angular acceleration of the pulley related to the tensions, R, and I?


    ehild
     
  19. Aug 2, 2014 #18
    Sorry, i misled you there
     
  20. Aug 4, 2014 #19
    equivalent mass

    I found this anomaly when helping solve a problem involving a non - slipping sphere rolling down an incline. )

    For the wheel eqivalent mass calculation , use either :
    = ( KE wheel / Ke m1 ) * m1
    or :
    = ( KE wheel / Ke m2 ) * m2

    ( The result is the same )

    Because the wheel rotates without slipping its KE is always proportional to the KE of either m1 or m2 (use either, the result is the same), if you could define the wheels impact on the system as a linear mass ( as shown previously, lets call it m3 ), then the acceleration (a) of the system could be (more easily) defined as :
    ( a = f / m )
    a = ( m * g ) / ( m1 + m2 + m3 )

    As far as string tension is concerned, the tension T2 would then be :
    = m2 * a
    And the tension T1 :
    = m3 * a

    Ignoring the wheel :
    a = ( m1 * g ) / ( m1 + m2 )
    T1 ( & T2 ) = m2 * a


    You can check part of this theory by dropping m1 an arbitrary 1.0 metre (h), calculating the final velocity with the pre calculated acceleration, then figure the KE's of all : m1, m2 and the wheel, and added together this should equal :
    m1 * g * h
    ( final total system KE = original potential energy (PE) of m1 )
     
  21. Aug 4, 2014 #20

    ehild

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    I do not understand what you suggest. There is no wheel in the problem. If you mean pulley, its moment of inertia is given. You can define an equivalent mass as I/R2 for the pulley, but the problem is easy to solve, why do you complicate it with something nobody understands what it is? And what is the fourth mass m?
    You ignore the wheel (pulley), why do you calculate its equivalent mass then? And the problem requires that the moment of inertia of the pulley should be taken into account.
    The acceleration of any of the masses is determined by the sum of all forces acting on it and the tension is only one of the forces.


    ehild
     
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