Question

A hollow cylinder with inner radius R, outer radius 2R and mass M is rolling with speed of its axis v.what is its kinetic energy?

Answer 1

Answer:

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Explanation:

KE=12Mv2+12Iω2

KE=12Mv2+12I(v2R)2 ….(1)

Let we have solid cylinder of radius 2R and mass M from which cylinder of radius R of mass M'=M'4 is taken out from common axis

Mas of hollow cylinder M=M'−M''

M=M'−M'm=2M'4⇒M=3M'4

⇒M'=4M3&M''=M3

I2R=IR+I⇒I=I2R−IR

I=12M'(2R)2−12M''R2

I=[12×4M3×4R2−12×M3R2]

I=52MR2 ....(2)

KE=1316Mv2

Alternatively : Let the length of cylinder is l

I=∫2RRdmr2⇒I=∫2RRM[π(2R)2−πR2]l×2πrdrl×r2

I=2M3R2∫2RRr3dr⇒I=52MR2.