Solving Rotational Mechanics: Minimum Friction & Kinetic Energy

[prasanna]

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].<br /> <b>(a) What should be the minimum coefficient of friction between the sphere and the plane to prevent skidding?</b><br /> I did this:<br /> <br /> mgsin[tex]\theta[\tex] - f = ma ________ 1<br /> (f is frictional force)<br /> <br /> torque = I[tex]\alpha[\tex] = r X f <br /> (symbols stand for their usual meanings)<br /> <br /> r*f = [tex]\frac{2mr^2}{\frac{a}{r}}[\tex]<br /> <br /> f = (2/3)ma _______2<br /> <br /> Subst. in 1 ,<br /> a= (3/5)gsin[tex]\theta[\tex] ______ 3<br /> <br /> subst. both 2 and 3 in 1,<br /> <br /> f = (2/5) mgsin[tex]\theta[\tex]<br /> but,<br /> [tex]\mu[\tex]mgcos[tex]\theta[\tex] = f = (2/5)mgsin[tex]\theta[\tex]<br /> <br /> [tex]\mu[\tex] = (2/5) tan [tex]\theta[\tex]<br /> <br /> this matches with the textbook answer.<br /> <br /> (<b>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).</b><br /> I did this :<br /> <br /> [tex]\mu[\tex] = (1/5) tan [tex]\theta[\tex]<br /> <br /> f = [tex]\mu[\tex]mgcos[tex]\theta[\tex] <br /> <br /> putting this in 1(of part a)<br /> <br /> a = (4/5) g sin[tex]\theta[\tex]<br /> <br /> torque = I[tex]\alpha[\tex] = [tex]\frac{2mr^2}{3}[\tex]<br /> [tex]\alpha[\tex] = [tex]\frac{3gsin\theta}{10r}[\tex]<br /> <br /> KE = (1/2)mv^2 + (1/2)I[tex]\omega^2[\tex]<br /> <br /> v^2 = 2aL = (8/5) gLsin[tex]\theta[\tex]<br /> <br /> [tex]\omega^2[\tex] = 2[tex]\alpha\theta[\tex]<br /> <br /> i found [tex]\omega^2[\tex] = (6gLsin[tex]\theta[\tex]) / (20[tex]\pi\r^2[\tex]<br /> <br /> I get <br /> <br /> KE = (4/5) mgLsin[tex]\theta[\tex] + mmgLsin[tex]\theta[\tex] / 2[tex]\pi[\tex]<br /> <br /> KE = <b>0.831 </b> * mgLsin[tex]\theta[\tex] <br /> <br /> But the answer in the book is (7/8) mgLsin[tex]\theta[\tex] <br /> which is <b>0.875 </b> mgLsin$$\theta[\tex] <br /> <br /> <br /> Where did I go wrong?$$[/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex][/tex]

 

[ponjavic]

Hey !
Please help me out with this problem.

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

mgsin$$\theta$$ - f = ma ________ 1
(f is frictional force)

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

r*f = $$\frac{2mr^2}{\frac{a}{r}}$$

f = (2/3)ma _______2

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

subst. both 2 and 3 in 1,

f = (2/5) mgsin$$\theta$$
but,
$$\mu$$mgcos$$\theta$$ = f = (2/5)mgsin$$\theta$$

$$\mu$$ = (2/5) tan $$\theta$$

this matches with the textbook 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 :

$$\mu$$ = (1/5) tan $$\theta$$

f = $$\mu$$mgcos$$\theta$$

putting this in 1(of part a)

a = (4/5) g sin$$\theta$$

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

KE = (1/2)mv^2 + (1/2)I$$\omega^2$$

v^2 = 2aL = (8/5) gLsin$$\theta$$

$$\omega^2$$ = 2$$\alpha\theta$$

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

I get

KE = (4/5) mgLsin$$\theta$$ + mmgLsin$$\theta$$ / 2$$\pi$$

KE = 0.831 * mgLsin$$\theta$$

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


Where did I go wrong?

Might help a bit ;)

 

[prasanna]

Might help a bit ;)

Thank a lot !
How did you do that?

What happened that my original thread did not show LaTEX ??

 

[spacetime]

Hey Prasanna, your name is like that of a cricketer.
Anyway, use [/tex] instead of [\tex] for ending LaTeX code.



spacetime
www.geocities.com/physics_all

 

[Justin Lazear]

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

Backslashes for TeX characters, forward slashes for tags.

--J