IIT JEE , AIEEE Forum - Physics , Mechanics

shubham_sachdeva

Q. with what minimum linear speed should a sphere of radius r be set rolling on the horizontal part so that it completely goes round the circle on the cylindrical part.

karthik2007

Let the sphere be given an initial velocity v.

Now, at the topmost part of the cylinder (see figure) , the sphere goes in a circle of radius of R-r.
Let the sphere have a velocity of v' at the top of the cylinder.
So, we have mv'²/(R-r) = mg + N

As we need the minimum speed, N = 0

So mv^'2/(R-r) - mg

or v'² = g(R-r) - (1)

Now, conserving energies, we have:

1/2mv² + 1/2Iw² = mg(2R-2r) + 1/2mv'² + 1/2Iw'²

As it undergoes pure rolling, w = v/r, and I = 2/5mr².

Putting these in the energy equation, we get

7/10mv² = 2g(R-r) + 7/10g(R-r)

or 7/10v² = g(27R-27r)/10

Which gives v =

27g(R-r)/7

Do tell me if I have gone wrong somewhere, as the final answer seems to be a little scary :)
Note: Click the image for better clarity.
Cheers!

karthik2007

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