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.

THIS IS Figure 1:

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.

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.

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. THIS IS Figure 1: wf =? wf? 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. 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.

Explanation / Answer

Part A: It = 1/2 mt R^2     moment of inertia of disk
wi It = 1/2 mt * R^2 * wi    initial angular momentum of disk
-md * vd * R = initial angular momentum of dart
If = 1/2 mt R^2 + md * R^2    final combined momet of inertia
If * wf = 1/2 mt * R^2 * wi - md * vd * R
Substitute for If and rearrange to get wf   the final angular speed
Part B:
If = 1/2 mt R^2 + md * R^2    again the final moment of inertia
1/2 mt * R^2 * wi the initial total angular momentum
1/2 mt * R^2 * wi = (1/2 mt R^2 + md * R^2) * wf
wf = (1/2 mt) wi / (1/2 mt + md)
because the dart has no angular momentum relative to the axis of rotation

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