Question

# Consider the objects below, all of mass M and radius R (where appropriate). They are placed...

Consider the objects below, all of mass M and radius R (where appropriate). They are placed on an incline plane at the same height. Which object will roll down the incline and reach the bottom with the greatest total energy?

1. a) A solid sphere
2. b) A thin spherical shell
3. c) A solid cylinder of length L
4. d) A cylindrical shell of length L
5. e) All will reach bottom with same energy

A solid sphere

A thin spherical shell

A solid cylinder of length L

A cylindrical shell of length L

All will reach the bottom with the same total energy

moment of inertia of solid sphere = 2/3 mr^2 = kmr^2

thin spherical shell = 2/3 mr^2 = kmr^2

splid cylinder = 1/2 mr^2 = kmr^2

cylindrical shell = mr^2 = kmr^2

here k is a proportional constant it varies from object to object

from conservation energy

mgh = 1/2mv^2+1/2Iw^2

I = moment of inertia

mgh = 1/2mv^2+1/2kmr^2*w^2

v = rw

mgh = 1/2mv^2+1/2kmv^2

(1+k)v^2 = 2gh

v = sqrt(2gh/1+k)

now substitute k values

v solid sphere = sqrt(2gh/1+2/5) = sqrt(1.428gh)

v sphericla shell = sqrt(1.2gh)

v cylinder = sqrt(1.33gh)

v cylindrical shell = sqrt(gh)

solid sphere has gratest speed and energy

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