A sphere, cylinder, and hoop are positioned at the top of an incline ramp and th

Question

A sphere, cylinder, and hoop are positioned at the top of an incline ramp and then all three are released from rest at the same time. Each object rolls without slipping down the ramp. Which object reaches the bottom first?

Sphere reaches the bottom of the ramp first

Cylinder reaches the bottom of the ramp first

Hoop reaches the bottom of the ramp first

Not enough information - for instance, the masses and sizes of the objects were not given.

Sphere reaches the bottom of the ramp first

Cylinder reaches the bottom of the ramp first

Hoop reaches the bottom of the ramp first

Not enough information - for instance, the masses and sizes of the objects were not given.

Explanation / Answer

Total kinetci energy K =(1/2)Iw^2 +(1/2)mv^2

I =mk^2, v =Rw

K = (1/2)mk^2v^2/R^2 +(1/2)mv^2

K = (1/2)mv2 [1+ (k2/R2) ]

If the incline ramp is h heoght from ground

From coservation of energy

Kf =Ui

(1/2)mv2 [1+ (k2/R2) ] = mgh

v = [2gh/[1+ (k2/R2) ]1/2

For sphere k2 = (2/5)R2

vsphere = [10gh/7]1/2

For cylinder k2 = (1/2)R2

vcylinder = [4gh/3]1/2

For hoop k2 = R2

vhoop = [gh]1/2

for all objects g ,h is same.

vsphere>vcylinder>vhoop

From the result obtained it is clear that among thee objects, the sphere reaches bottom of ramp first.

Correct option is (A)

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