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Connections between Linear and Rotational Quantities

  1. Feb 7, 2015 #1
    1. The problem statement, all variables and given/known data
    A wheel of radius R starts from rest and accelerates with a constant angular acceleration α about a fixed axis. At what time t will the centripetal and tangential acceleration of a point on the rim have the same magnitude?

    2. Relevant equations
    acp=r x ω2

    at= r x α

    ω= 2π / T → T=2π/ω

    3. The attempt at a solution
    The problem states that the centripetal and tangential acceleration will have the same magnitude at time t.
    So I listed the equation for centripetal acceleration and tangential acceleration and thought that I can put them equal to each other since the magnitudes are the same.

    R x ω2 =R x α I canceled out the R

    ω2= α Here I thought that since ω equals 2π/T I could substitute it for ω since it has T for period
    which relates to time​
    (2π/T)2 = α I took the square root of that.

    2π/T = √α Solving for T

    T = 2π/(√α)

    So would this be the time? The AP Physics Book (3rd Edition by James S. Walker doesn't give me a solution since the problem number is even (only gives for odd).
     
  2. jcsd
  3. Feb 7, 2015 #2

    Nathanael

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    The period isn't important. You have ω2=α. You know that α is constant. You know that ω starts at zero. Can you then find ω as a function of time?
     
  4. Feb 7, 2015 #3
    Hmm. So would that be w(final)=w(initial)+ a(alpha) t ?
    Only one that doees not invovle angular distance. Hmm so I could substitute w^2 for alpha in the equation.
     
  5. Feb 7, 2015 #4

    Nathanael

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    And what do you get?
     
  6. Feb 9, 2015 #5
    I apologize for replying so late. An assignment turned out to be more time consuming than I thought it would be.
    Anyways so if I plug in w2,

    I get W(final)=W(initial) + W2 * t W(initial) is zero and solving for t give me

    t = W(final)/ W2

    Would it be possible to cancel out the W to get 1/W ?
     
  7. Feb 9, 2015 #6

    Nathanael

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    Yes that would be okay. But it may be better to write the final answer in terms of α, since α is a known quantity.
     
  8. Feb 10, 2015 #7
    How would write it in terms of alpha since I substitute it with w^2
     
  9. Feb 10, 2015 #8

    lightgrav

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    substitute the alpha back in?
     
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