Question

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?

- a) A solid sphere
- b) A thin spherical shell
- c) A solid cylinder of length L
- d) A cylindrical shell of length L
- e) All will reach bottom with same energy

Group of answer choices

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

Answer #1

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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