Question

3) A Solid Ball, of mass M and radius R rolls from rest down a table-top ramp of height H and then rolls (horizontally) across a table until it gets to the edge and rolls off the edge and drops a distance h to the floor.

(a) At what horizontal distance D (from the table edge) does the ball hit the floor?

(b) Rank the following objects from least D to greatest D :

Solid Ball, Hollow Sphere, Solid Cylinder, Hollow cylinder. (Assuming, of course, they all have the same mass M and radius R ).

Answer 1

Assuming rolling without slipping

v = rw

so,

using conservation of energy, we have

mgH = 1/2mv² + 1/2Iw²

mgH = 1/2mv² + 1/2 * 2/5mr² * (v/r)²

mgH = 1/2mv² + 1/5mv²

gH = 1/2v² + 1/5v²

gH = 7v² / 10

v = sqrt ( 10gH / 7)

so,

time taken to fall height 'h' to the ground

t = sqrt ( 2h / g)

so,

Horizontal distance,

D = vt

D = sqrt ( 10gH / 7) * sqrt ( 2h / g)

or

______________________________________________________

time of fall will be same for all the objects, so D will be greater for object with greater 'v'

more the moment of inertia, slower will be the object.

Solid Ball = 0.4mr²

Hollow Sphere = 0.66mr²

Solid Cylinder = 0.5mr²

Hollow cylinder = mr²

so,

ranking will be (least to greatest)

Hollow cylinder < Hollow sphere < solid cylinder < solid sphere