1if an object moves with velocity v in a horizontal circle its kinetic energy = 1/2mv2
A uniform sphere of mass m and radius r rolls without sliding over a horizontal plane, rotating about a horizontal axle OA. in the process, the center of the sphere moves with velocity v along a circle of radius R. Find the kinetic energy of the sphere.
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1there we would take radius of rotaion as r+R then add KE trans.+ KErot.
62There is a mistake in prashant's solution... something is wrong.. i dont know exactly what!!!
My solution is 1/2 I1ω12+1/2 I2ω22
where I1= 2/5mr2
ω1 = v/r
I2= mR2+2/5mr2
ω2 = v/R
Thus, total energy = 1/2(mr2+2/5mr2)(v/r)2 + 1/2(mR2+2/5mr2) (v/R)2
= 7/10mv2 + 1/2(2/5m(r/R)2)(v)2
= 7/10mv2 + 1/5m(r/R)2)(v)2
1K.E.=1/2 mv2 + 1/2 I1ω12+1/2 I2ω22
ω1=v/r
ω2=v/R
I1=2/5mr2
I2=2/5mr2+mR2
Hence, KE=1/2 mv2 + 1/2 (2/5mr2)v2/r2 + 1/2(2/5mr2+mR2)v2/R2
=6/5 mv2[1+r2/6R2]
62Hint: THis is made of 2 pure rotations only
1) at point O
2) At the point where the sphere touches the ground
62ok.. then what is teh difference between.. a sphere .. moving along a straight line and moving in a circular orbit..
(forget rotation about the axis!)
Will the kinetic energy be the same??
1but .. at any moment of time the velocity of the centre of mass =v
and the angular velocity = w
so the result .. but i cant understand where im going wrong!!
62yeah i know it is a bit hard to digest.. even when i was preparing for jeee it took me a lot of lot of time to digest this solution!! *to be true i could not solve it myself!
(It is another one from IRODOV!!!!)
1but that will just contribute to its translational energy!! (it is moving with v so simply 1/2mv2!!)
62do u realise that there are 2 different omeagas!
one is of the ball along its axis.. another from the center!
1if r=0 then the velocity of the bottom most point cant be 0 it will slide!!
1as it rolliing without sliding
v=wr
translational energy =1/2mv2
rotational energy = 2/5mr2w2/2=mv2/5
hence ans ..
62There is a lot more to it..!!!
how about r .. dont u think if r is zero.. the answer will be very very differnt!
how are r and R related!
9ive included translational and rotational KE wat else is there