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Homework Help: Non-uniform circular motion and tangential acceleration

  1. Oct 2, 2008 #1
    1. The problem statement, all variables and given/known data
    An object of mass m is constrained to move in a circle of radius r. Its tangential acceleration as a function of time is given by [tex]a_{tan} = b + ct^2[/tex], where b and c are constants.

    A) If [tex]v = v_0[/tex] at t = 0, determine the tangential component of the force, [tex]F_{\tan }[/tex], acting on the object at any time t > 0.
    Express your answer in terms of the variables m, r, [tex]v_0[/tex], b, and c.

    B) Determine the radial component of the force [tex]F_{\rm{R}}[/tex].
    Express your answer in terms of the variables m, r, [tex]v_0[/tex], b, t, and c.

    2. Relevant equations
    [tex]a_{tan} = b + ct^2[/tex]
    [tex]a_r=\tfrac{v^2}{r}[/tex]
    Newton's Laws


    3. The attempt at a solution
    A. was not a problem for me:
    [tex]F_{\tan}=ma_{\tan}=m(b+ct^2)[/tex]

    For B.:
    [tex]F_R=ma_r[/tex]
    [tex]a_r=\tfrac{v^2}{r}[/tex]
    It seems to make sense that because v is tangential speed we could use...
    [tex]v(t)=v_0+a_{\tan}t=v_0+(b+ct^2)t[/tex]
    So that...
    [tex]a_r=\frac{(v_0+(b+ct^2)t)^2}{r}[/tex]
    Finally giving...
    [tex]F_R=m(\frac{(v_0+(b+ct^2)t)^2}{r}[/tex]

    Which is not correct. What did I do wrong?
     
  2. jcsd
  3. Oct 3, 2008 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi Symstar! Welcome to PF! :smile:
    That only works if atan is constant, doesn't it? :wink:

    Hint: dv/dt = … ? :smile:
     
  4. Oct 3, 2008 #3
    Re: Welcome to PF!

    dv/dt = atan correct?

    So would I need to integrate?
    [tex]\int a_{tan} = \int b + ct^2[/tex]
    [tex]\frac{dv}{dt}= bt+\tfrac{1}{3}ct^3[/tex]

    And it seems logical in our case that +C would actually be +v0

    Which would end up giving me:
    [tex]F_R=m\frac{(v_0+bt+\tfrac{1}{3}ct^3)^2}{r}[/tex]

    Which I just confirmed to be the correct answer... thanks for you your help tim.
     
    Last edited: Oct 3, 2008
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