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Uniform circular Motion

  1. Jan 31, 2009 #1
    1. The problem statement, all variables and given/known data

    how does velocity and acceleration change in circular moton

    2. Relevant equations



    3. The attempt at a solution
    I know when a object is circular motion the velocity is tangential to the motion also, acceleration centripital, sum of the forces which points to a center seeking force
     
  2. jcsd
  3. Jan 31, 2009 #2
    In Uniform Circular Motion the position vector can be expressed as

    [tex]\vec{r}=Rcos(\omega t)\hat{x}+Rsin(\omega t)\hat{y}[/tex]

    where omega is the frequency of oscillation, t is time , and R is the radius of the circle.

    We calculate velocity and acceleration by taking first and second derivatives with respect to time.

    [tex]\vec{\dot{r}}=-\omega Rsin(\omega t)\hat{x}+\omega Rcos(\omega t)\hat{y}[/tex]

    [tex]\vec{\ddot{r}}=-\omega ^{2} Rcos(\omega t)\hat{x}-\omega ^{2}Rsin(\omega t)\hat{y}=-\omega ^{2}\vec{r}[/tex]

    Also, [tex]R\omega = v[/tex] where v is the tangential velocity (To show this use [tex]Rd\theta =dS[/tex] where dS is an infinitesimal tangential distance and divide both sides by [tex]dt[/tex]) so

    [tex]\vec{\ddot{r}}=-\frac{v^{2}}{R^{2}}\vec{r}=-\frac{v^{2}}{R^{2}}R\hat{r}=-\frac{v^{2}}{R}\hat{r}[/tex]

    So the acceleration is anti parallel to the radius vector (ie. towards the center of the circle) and has a magnitude of [tex]\frac{v^{2}}{R}[/tex]
     
    Last edited: Jan 31, 2009
  4. Jan 31, 2009 #3
    thanks kind of get it
     
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