Classical Mechanics (Kibble & Berkshire) · Problem 11.7

Lagrange's equations and conservation principles — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · hoop-mounting-step

Problem

Hoop mounting a step A uniform hoop of mass

M

and radius

a

is rolling with speed

V

along level ground when it meets a step of height

h

(

h < a

). The particle

C

of the hoop that makes contact with the step is suddenly brought to rest. Find the instantaneous speed of the centre of mass, and the instantaneous angular velocity of the hoop, immediately after the impact. Deduce that the particle

C

cannot remain at rest on the edge of the step if

V^2 > (a-h)g\left(1-\frac{h}{2a}\right)^{-2}.

Suppose that the particle

C

does remain on the edge of the step. Show that the hoop will go on to mount the step if

V^2 > hg\left(1-\frac{h}{2a}\right)^{-2}.

Deduce that the hoop cannot mount the step in the manner described if

h > a/2

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