Rotational mechanics

  1. Hey !
    Please help me out with this problem.

    A hollow sphere of mass m is released from top on an inclined plane of inclination [tex]\theta[\tex].
    (a) What should be the minimum coefficient of friction between the sphere and the plane to prevent skidding?
    I did this:

    mgsin[tex]\theta[\tex] - f = ma ________ 1
    (f is frictional force)

    torque = I[tex]\alpha[\tex] = r X f
    (symbols stand for their usual meanings)

    r*f = [tex]\frac{2mr^2}{\frac{a}{r}}[\tex]

    f = (2/3)ma _______2

    Subst. in 1 ,
    a= (3/5)gsin[tex]\theta[\tex] ______ 3

    subst. both 2 and 3 in 1,

    f = (2/5) mgsin[tex]\theta[\tex]
    [tex]\mu[\tex]mgcos[tex]\theta[\tex] = f = (2/5)mgsin[tex]\theta[\tex]

    [tex]\mu[\tex] = (2/5) tan [tex]\theta[\tex]

    this matches with the text book answer.

    (b) Find the Kinetic Energy of the ball as it moves down a length L on the incline if the friction coefficient is half the value calculated in part(a).
    I did this :

    [tex]\mu[\tex] = (1/5) tan [tex]\theta[\tex]

    f = [tex]\mu[\tex]mgcos[tex]\theta[\tex]

    putting this in 1(of part a)

    a = (4/5) g sin[tex]\theta[\tex]

    torque = I[tex]\alpha[\tex] = [tex]\frac{2mr^2}{3}[\tex]
    [tex]\alpha[\tex] = [tex]\frac{3gsin\theta}{10r}[\tex]

    KE = (1/2)mv^2 + (1/2)I[tex]\omega^2[\tex]

    v^2 = 2aL = (8/5) gLsin[tex]\theta[\tex]

    [tex]\omega^2[\tex] = 2[tex]\alpha\theta[\tex]

    i found [tex]\omega^2[\tex] = (6gLsin[tex]\theta[\tex]) / (20[tex]\pi\r^2[\tex]

    I get

    KE = (4/5) mgLsin[tex]\theta[\tex] + mmgLsin[tex]\theta[\tex] / 2[tex]\pi[\tex]

    KE = 0.831 * mgLsin[tex]\theta[\tex]

    But the answer in the book is (7/8) mgLsin[tex]\theta[\tex]
    which is 0.875 mgLsin[tex]\theta[\tex]

    Where did I go wrong???
  2. jcsd
  3. Might help a bit ;)
  4. Thank a lot !!
    How did you do that?

    What happened that my original thread did not show LaTEX ??
  5. Hey Prasanna, your name is like that of a cricketer.
    Anyway, use [/tex] instead of [\tex] for ending LaTeX code.

  6. You used backslashes in the [/tex] instead of the forward slashes.

    Backslashes for TeX characters, forward slashes for tags.

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