1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Rolling motion in a special ramp

  1. May 16, 2010 #1
    1. The problem statement, all variables and given/known data

    Assuming the sphere roles down without sliding prove that the acceleration of it's center of mass is:

    [tex] a= \frac{g\cdot \sin(\theta)}{1+\frac{2}{5}\cdot \frac{1}{1-\frac{1}{4}\cdot \xi^2}}[/tex]

    [tex]Where \ \xi=\frac{L}{R}[/tex]

    Note that the moment of inertia of the sphere is:

    [tex]I_{sphere}=\frac{2}{5}\cdot M\cdot R^2[/tex]

    SemTtulo-1.jpg

    2. Relevant equations

    [tex]\tau= F\cdot r\cdot \sin(\varphi)[/tex]

    [tex]\alpha\cdot R=a[/tex]

    3. The attempt at a solution

    The forces acting on the sphere are: the normal force, the force of gravity and the frictional force. The sum of the y components of the normal force will be equal to [tex]M\cdot g[/tex] and the sum of the x components will be 0. So that the sum of torques due to the normal force is zero as well as torques due to the sphere's weight.

    [tex]\tau_{a}=F_{a}\cdot R\cdot \sin(\varphi)[/tex]

    Where we have by the figure:

    [tex]\sin(\varphi)= \frac{\sqrt{R^2-(\frac{L}{2}^2)}}{R}= \sqrt{1-\frac{1}{4}\cdot \xi^2}[/tex]

    So we have that:

    [tex] \left\{ \begin{array}{ccc} 2\cdot F_{a}\cdot R\cdot \sqrt{1-\frac{1}{4}\cdot \xi^2} & = & I\cdot \alpha \\ -2\cdot F_{a} + M\cdot g\cdot \sin(\theta) & = & M\cdot a \end{array} \right. [/tex]

    Which will yield:

    [tex]F_{a}= \frac{I\cdot a}{2\cdot R^2\cdot \sqrt{1-\frac{1}{4}\cdot \xi^2}}[/tex]

    After some manipulation you arive to

    [tex]a=\frac{M\cdot g\cdot \sin(\theta)}{M+\frac{I}{R^2\cdot \sqrt{1-\frac{1}{4}\cdot \xi^2}}}[/tex]

    Which after substituting I for it's value leads to:

    [tex]a= \frac{g\cdot \sin(\theta)}{1+\frac{2}{5}\cdot \sqrt{\frac{1}{1-\frac{1}{4}\cdot \xi^2}}}[/tex]

    Can somebody please tell me where I've gone wrong
     
    Last edited: May 16, 2010
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted