Calculating Rotational Speed After Dart Strikes Target Disk

[xxphysics]

Homework Statement

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 ω 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.

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.

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.

Homework Equations

Really not too sure. But since there is mass and velocity I'm assuming energy equations?
ΔK = (1/2)Iω^2 - (1/2)Iωi^2

where I = mr^2

The Attempt at a Solution

So you end up with ω = (ω)^1/2 ? Is that correct? It doesn't seem right, any help would be great thanks.

 

[haruspex]

I'm assuming energy equations?

As a matter of principle, you should not appeal to conservation of mechanical energy unless you have a good reason to believe it is conserved. Since there is an impact here, it quite likely is not conserved.
What other conservation laws can you think of that might be appropriate?

 

[xxphysics]

As a matter of principle, you should not appeal to conservation of mechanical energy unless you have a good reason to believe it is conserved. Since there is an impact here, it quite likely is not conserved.
What other conservation laws can you think of that might be appropriate?

Conservation of angular momentum ? L = Iw

 

[TomHart]

I haven't tried to work this problem out, but consider intuitively how the angular velocity may be affected by the weight of the dart relative to the disk - for example, an extremely heavy dart as opposed to an extremely lightweight dart. And then look at your solution in light of that. For me, it sometimes helps to take things to extremes to make it easier for me to see how things may be affected. And then I would think about how the point at which the dart strikes would affect the rotation of the disk. For that I would think about an extremely heavy dart striking a light disk. One more thing: For these types of problems, it really comes down to two things - energy and/or momentum - well, and the things that make up energy and momentum.

 

[haruspex]

Conservation of angular momentum ? L = Iw

That's a distinct possibility. What is the criterion for that to be conserved?

 

[herecomedatboi]

(A) We can solve this problem using conservation of angular momentum. The dart has some amount of angular momentum with respect to the axis of the disk. When the dart strikes the disk, it adds that angular momentum, and also increases the rotational inertia of the disk.
Initial Momentum: m_d * v_d * R - 1/2 * m_t * R_t² * Omega_i
Final Momentum: (1/2 * m_t * R_t² + m_d * R_t²) * Omega_f
Set the final and initial angular momenta equal to each other and isolate Omega_f
(B) In this case, the dart’s initial angular momentum about the disk’s axis is zero.
So same equation, but the initial angular momentum for the dart this time is zero.

 

[PeroK]

(A) We can solve this problem using conservation of angular momentum. The dart has some amount of angular momentum with respect to the axis of the disk. When the dart strikes the disk, it adds that angular momentum, and also increases the rotational inertia of the disk.
Initial Momentum: m_d * v_d * R - 1/2 * m_t * R_t² * Omega_i
Final Momentum: (1/2 * m_t * R_t² + m_d * R_t²) * Omega_f
Set the final and initial angular momenta equal to each other and isolate Omega_f
(B) In this case, the dart’s initial angular momentum about the disk’s axis is zero.
So same equation, but the initial angular momentum for the dart this time is zero.

You might be 16 months too late!

 

[herecomedatboi]

You might be 16 months too late!

You're quite right! I had the same problem in my class and I struggled for a while so I figured I might post a solution.