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Circular Motion with constant angular acceleration

  1. Sep 22, 2013 #1
    1. The problem statement, all variables and given/known data

    An object travels counterclockwise on a circular path with radius R and constant angular
    acceleration α , so that

    vector r(t) = R cos(αt^2/2) i^+ R sin(αt^2/2) j^
    

    2. Relevant equations

    b. Find the time T when the object made a single revolution and returned to its
    original position. Evaluate vectors r, v, and a at both t = 0 and t = T.
    c. Show by computation that at t = T, the acceleration vector is the sum of
    a part parallel to the velocity vector with magnitude dv/dt , and a part perpendicular to the
    velocity vector with magnitude v^2/R

    3. The attempt at a solution

    I am calculating based on the fact that the object will travel a distance of 2πR at the time it made a revolution, but it doesn't work !
     
  2. jcsd
  3. Sep 22, 2013 #2
    A revolution brings the object where it was originally. Express that mathematically.
     
  4. Sep 22, 2013 #3
    I am sorry I really don't know how to express that mathematically. I have just calculate its speed to be Rαt but I can not make an equation because the object has an increasing acceleration (because its angular accel is constant). This is quite new to me.
     
  5. Sep 22, 2013 #4

    SteamKing

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    Science Advisor
    Homework Helper

    Well, take your equation for r(t) from the OP.

    What are the coordinates for the object at time t = 0?

    At time t = T, you will have these same coordinates. Knowing that sine and cosine are periodic functions, use this fact to figure out what T must be to return the object to its original position.
     
  6. Sep 22, 2013 #5
    r(t) that you were given is the position of the object. As one revolution brings the object where it started from, you should have r(0) = r(T).
     
  7. Sep 22, 2013 #6
    Oh thank you SteamKing and Voko I know how to do it now. My problem is that I was stuck with the idea that the magnitude of r(t) is always R so I thought I must use another equation rather than r(t). Thanks a ton!
     
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