A dart of inertia m d is fired such that it strikes with speed v d , embedding i

Question

A dart of inertia md is fired such that it strikes with speed vd, embedding its tip in the rim of a target that is a uniform disk of inertia mt and radius Rt. The target is initially rotating clockwise in the view shown in (Figure 1) , with rotational speed about an axis that runs through its center and is perpendicular to its plane. Assume that the dart's inertia is concentrated at its tip.

Part A

What is the final rotational speed of the target if the dart strikes tangent to the target rim as in the figure, case (a)? Enter positive value if the rotation is counterclockwise and negative value if the rotation is clockwise.

Express your answer in terms of some or all of the variables mt, md, Rt, , and vd. Hint given: vd is not a part of the final answer.

Part B

What is the final rotational speed of the target if the dart strikes normal to the rim as in the figure, case (b)? Enter positive value if the rotation is counterclockwise and negative value if the rotation is clockwise.

Express your answer in terms of some or all of the variables mt, md, Rt, , and vd. Hint given: Rt and vd are not a part of the final answer

Explanation / Answer

before hitting

angular mometumof disk = L1 = -0.5*mt*Rt^2*w

angular momentum of dark = L2 = +md*vd*Rt

total initial angular moemntum Li = -0.5*mt*Rt^2*w + md*vd*Rt

after hitting

moment of inertia of the disk + dart = 0.5*mt*Rt^2 + md*Rt^2 = (0.5*mt+md)*Rt^2

angular momentum Lf = (0.5*mt+md)*Rt^2*w'

from moentum conservation

Lf = Li

(0.5*mt+md)*Rt^2*w' = -0.5*mt*Rt^2*w + md*vd*Rt

w' = [ -0.5*mt*Rt^2*w + md*vd*Rt ] / ((0.5*mt+md)*Rt^2)

w' = [ md*vd - 0.5*mt*Rt*w ] / ((0.5*mt+md)*Rt)

++++++++

B)

angular mometumof disk = L1 = 0.5*mt*Rt^2*w

angular momentum of dark = L2 = 0

total initial angular moemntum Li = 0.5*mt*Rt^2*w

after hitting

moment of inertia of the disk + dart = 0.5*mt*Rt^2 + md*Rt^2 = (0.5*mt+md)*Rt^2

angular momentum Lf = (0.5*mt+md)*Rt^2*w'

from moentum conservation

Lf = Li

(0.5*mt+md)*Rt^2*w' = -0.5*mt*Rt^2*w

w' = [ -0.5*mt*Rt^2*w ] / ((0.5*mt+md)*Rt^2)

w' = (-0.5*mt*w ) / (0.5*mt+md)

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