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Help! Basic Rotational Dynamics Question

  1. Jun 24, 2010 #1
    Can anyone help me figure out this question? I swear this isnt homework! Im actually from a car forum and would like to understand some basic principles behind car engine pulleys.

    Pulley A drives another pulley Bx via a belt.

    Pulley A
    diameter:15inches
    weight: 1lbs
    applied torque: 10lbs*1inch

    Pulley B1
    diameter: 5inches
    weight: 0.5lbs

    Pulley B2
    diameter: 5inches
    weight: .75lbs

    Pulley B3
    diameter: 7inches
    weight: .25lbs

    Pulley B4
    diameter: 7inches
    weight: .75lbs

    Which of the Bx pulleys will require more torque from Pulley A to rotate? How do you figure this out? What is the resulting torque on each of the Bx pulleys?

    thanks a lot!
     
  2. jcsd
  3. Jun 24, 2010 #2

    K^2

    User Avatar
    Science Advisor

    Assuming all pulleys are geometrically similar, here is how you go about it.

    Belts ensure that surface speed of all pulleys is identical. Id est,

    [tex]R_A \omega_A = R_i \omega_i[/tex]

    And consequently,

    [tex]R_A \dot{\omega}_A = R_i \dot{\omega}_i[/tex]

    And of course, the equation for angular acceleration and torque,

    [tex]I_i \dot{\omega}_i = \tau_i[/tex]

    The only problem is that there is no specific equation for moment of inertia, Ii for pulleys. It's going to be close to (1/2)MR² for solid cylinder, but it can be a little higher or lower depending on the geometry. This is where argument for similar geometries should come in. If all pulleys have similar geometries, then

    [tex]I_i = c M_i R_i^2[/tex]

    What that c is, doesn't really matter. It's a dimensionless constant. The important bit is that it should be the same for all pulleys. Using that, you should be able to express torque for each pulley in terms of angular acceleration of A, masses and radii of the pulleys, and this coefficient c. Then compare results. Good luck.
     
  4. Jun 24, 2010 #3
    wow thanks a lot for the reply!

    so if i understand correctly as long as the geometries on the pulleys are similar its basically I = MR^2

    B1
    I = 3.125 lbs * inch^2
    B2
    I = 4.6875 lbs * inch^2
    B3
    I = 3.0625 lbs * inch^2
    B4
    I = 9.1875 lbs * inch^2

    i'm going to see if i can find out from the companies actual geometries but this is really interesting as B4 is my impression of a overweight underdrive pulley and I'm actually trying to decide between B3(underweight oversize) and B1 (stock) pulleys. thank you very much!
     
  5. Jun 24, 2010 #4

    K^2

    User Avatar
    Science Advisor

    If you actually need an estimate on torque, it's closer to (1/2)MR². Whether it's a little more or a little less depends on exact geometry.
     
  6. Jun 25, 2010 #5
    Last edited by a moderator: Apr 25, 2017
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