# Rotational mechanics

45
Hey !

A hollow sphere of mass m is released from top on an inclined plane of inclination $$\theta[\tex]. (a) What should be the minimum coefficient of friction between the sphere and the plane to prevent skidding? I did this: mgsin[tex]\theta[\tex] - f = ma ________ 1 (f is frictional force) torque = I[tex]\alpha[\tex] = r X f (symbols stand for their usual meanings) r*f = [tex]\frac{2mr^2}{\frac{a}{r}}[\tex] f = (2/3)ma _______2 Subst. in 1 , a= (3/5)gsin[tex]\theta[\tex] ______ 3 subst. both 2 and 3 in 1, f = (2/5) mgsin[tex]\theta[\tex] but, [tex]\mu[\tex]mgcos[tex]\theta[\tex] = f = (2/5)mgsin[tex]\theta[\tex] [tex]\mu[\tex] = (2/5) tan [tex]\theta[\tex] this matches with the text book answer. (b) Find the Kinetic Energy of the ball as it moves down a length L on the incline if the friction coefficient is half the value calculated in part(a). I did this : [tex]\mu[\tex] = (1/5) tan [tex]\theta[\tex] f = [tex]\mu[\tex]mgcos[tex]\theta[\tex] putting this in 1(of part a) a = (4/5) g sin[tex]\theta[\tex] torque = I[tex]\alpha[\tex] = [tex]\frac{2mr^2}{3}[\tex] [tex]\alpha[\tex] = [tex]\frac{3gsin\theta}{10r}[\tex] KE = (1/2)mv^2 + (1/2)I[tex]\omega^2[\tex] v^2 = 2aL = (8/5) gLsin[tex]\theta[\tex] [tex]\omega^2[\tex] = 2[tex]\alpha\theta[\tex] i found [tex]\omega^2[\tex] = (6gLsin[tex]\theta[\tex]) / (20[tex]\pi\r^2[\tex] I get KE = (4/5) mgLsin[tex]\theta[\tex] + mmgLsin[tex]\theta[\tex] / 2[tex]\pi[\tex] KE = 0.831 * mgLsin[tex]\theta[\tex] But the answer in the book is (7/8) mgLsin[tex]\theta[\tex] which is 0.875 mgLsin[tex]\theta[\tex] Where did I go wrong??? 2. ### ponjavic 226 Might help a bit ;) 3. ### prasanna 45 Thank a lot !! How did you do that? What happened that my original thread did not show LaTEX ?? 4. ### spacetime 119 Hey Prasanna, your name is like that of a cricketer. Anyway, use$$ instead of [\tex] for ending LaTeX code.

spacetime
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5. ### Justin Lazear

290
You used backslashes in the [/tex] instead of the forward slashes.

Backslashes for TeX characters, forward slashes for tags.

--J