A hoop of mass M and radius R lies motionless on a frictionless surface. Aimed t

Question

A hoop of mass M and radius R lies motionless on a frictionless surface. Aimed towards it, but off center by a distance d << R, is a smaller object of mass m << M and radius
r << R moving at speed v . The two objects undergo a totally inelastic collision.
Note: For m << M , it can be assumed that the center of mass motion of the composite
object is appropriately approximated by being located at the center of mass of the hoop
and that Vr would be parallel to vr.

a. Find an expression in terms of M , R, m, v and d for
i. the final translational speed V , and
ii. the final rotational speed, w, of the composite object.
b. What are the maximum and minimum values of w and V , and what values of d
correspond to each?
c. Find an expression for how much energy was lost in the collision, in terms of m ,
M , d , R, and Ki (=1/2mv^2) .
d. the maximum and minimum energy loss possible by varying the value of d , and
e. the values of w and V for M=2.0 kg, m=25g , R=15 cm, d=10 cm, and v=1.0 m/s

Explanation / Answer

a)

i) MV = mv => V = mv/M

ii) MR2w = mvd =>> w = mvd/MR2

b) Vmax = mv/M = Vmin

wmax = mvR/MR2 = mv/MR (for d = R)

wmin = 0 (for d= 0)

c) energy lost = 1/2mv2 - 1/2MV2 - 1/2MR2w2 = (1/2)(mv2 - M(mv/M)2 - MR2(mvd/MR2)2)

= (1/2)(mv2 - m2v2/M - m2v2d2/MR2) = (1/2)mv2(1 - m/M - md2/MR2) = Ki(1 - m/M - md2/MR2)

d) max loss = Ki(1 - m/M) (for d = 0)

min loss = Ki(1 - 2m/M) (for d = R)

e) w = 0.025 x 1 x 0.1/(2 x 0.152) = 0.055556 rad/s

V = 0.025 x 1/2 = 0.0125 m/s

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