Suppose I have a disc which doesn't move, just rotate around the axis going through its centre of mass perpendicular to its surface. The disk has a stick perpendicular to its surface at the edge. I also have a block. The block hits the stick tangentially with certain velocity and then stops. How can I calculate the angular velocity of the disc after collision.
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asked Dec 9, 2013 at 5:31
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2 Answers
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It is not immediately obvious, but the block has calculable angular momentum at the point just before impact. the block has velocity $v$ tangential to the disk's center of rotation which is a distance $r$ away., and so has angular velocity $\omega=v/r$. the block also has calculable moment of inertia around that center, $I=mr^2$. Then, it is simply
$(I\omega)_{disc}=(I\omega)_{block}$. if you have $I_{disc}$ you can calculate $\omega_{disc}$
This equation is simple because because of the crucial fact that the block has transferred all its angular momentum to the disc. If the block stuck onto the edge of the disc and ended up rotating together with it, then the equation would be slightly longer, but conservation of angular momentum would still apply just the same.
Additionally, it would be useful to know that linear momentum would also be conserved independently of the conservation of angular momentum. This means while the block has stopped after impact, the entire disc now has both linear and rotational velocity. $(mv)_{disc}=(mv)_{block}$
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$\begingroup$ I was thinking something like it. It is not very intuitive, its such confusing. Anyway thanks. $\endgroup$ Commented Dec 9, 2013 at 7:48
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I suppose you know the mass and extent of the disk. Let's just ignore the mass of the stick. (We don't have to do that, but it makes everything simpler).
We can then just use conservation of energy: You can calculate the kinetic energy of the block. Once it stops, this energy will be transferred completely to the rotational energy of the disk.
Using the formulas for kinetic energy of a moving object and the rotational energy of a disk, you can figure out the angular velocity. I don't provide the full answer since this sounds a bit like a homework problem and leave it to you to figure out the details.
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1$\begingroup$ I know the mass and radius. And stick is massless. Yes it actually is a part of homework. Problem is they told me to determine how much energy is lost in collision. So I wanted to determine the angular velocity so that I can subtract rotational kinetic energy from the previous translational kinetic energy. I suppose conservation of energy is not applicable here. $\endgroup$ Commented Dec 9, 2013 at 6:36
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$\begingroup$ In that case, use conservation of angular momentum as in the other answer :) $\endgroup$ Commented Dec 9, 2013 at 16:42