Angular momentum vs energy

In summary, the problem is that the conservation of angular momentum is not always correct, and this can lead to incorrect calculations.
  • #1
uq_civediv
26
0
the problem is the following:

we have a vertical wooden bar pivoted from the top end, length [tex] 2 l [/tex], mass [tex] M [/tex]

a bullet with mass [tex] m [/tex] hits it in the middle at velocity [tex] v [/tex] and gets stuck

i am asked to find the angular velocity [tex] \omega [/tex] of the system bar+bullet immediately after the hit

i do know this calls for applying the conservation of energy or angular momentum, for some reason however i get different results

Both if them involve the moment of inertia of the combined system, [tex] I_{\Sigma}=\frac{1}{3} M (2l)^2 + m l^2 = \frac{4}{3} M l^2 + m l^2[/tex]

Conservation of angular momentum gives me [tex]m v l = \omega I_{\Sigma} [/tex], from which [tex] \omega = \frac{m v l}{I_{\Sigma}} = \frac{m v l}{\frac{4}{3}M l^2+m l^2} = \frac{m}{\frac{4}{3}M+m} \cdot \frac{v}{l} [/tex]

Whereas conservation of energy says [tex] \frac{m v^2}{2} = \frac{\omega^2 I_{\Sigma}}{2} [/tex], which gives [tex] \omega = \sqrt{\frac{m}{I_{\Sigma}}} v = \sqrt{\frac{m}{\frac{4}{3}M + m}}\cdot \frac{v}{l} [/tex]

So the big question is where did I mess up this time. I know it's something really basic because I can't see it. Usually i ask a deskmate or someone to have a look if they spot something really simple but since nobody's around I had to come here.

P.S. while you're at it, why do my [tex] m [/tex], [tex] v [/tex] and [tex] \omega[/tex] look superscripted but [tex] M [/tex] and [tex] 2 l [/tex] don't ?
 
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  • #2
Your only mistake is in thinking that mechanical energy is conserved--it's not. This is an example of a perfectly inelastic collision.

uq_civediv said:
P.S. while you're at it, why do my [tex] m [/tex], [tex] v [/tex] and [tex] \omega[/tex] look superscripted but [tex] M [/tex] and [tex] 2 l [/tex] don't ?
To use Latex in the middle of a line of text and have it appear even, use "itex" as your delimiter, not "tex". It gives you [itex]m[/itex] instead of [tex]m[/tex].
 
Last edited:
  • #3
so the angular momentum one is correct ? (just clarifying...)

and the loss in energy is the good ol' [tex] \int F ds[/tex] over the distance the bullet travels into the rod !
 
  • #4
uq_civediv said:
so the angular momentum one is correct ? (just clarifying...)
Yes.

and the loss in energy is the good ol' [tex] \int F ds[/tex] over the distance the bullet travels into the rod !
Good luck calculating that! To find the loss in energy, just calculate the final KE and compare it to the initial.
 
  • #5
Doc Al said:
Good luck calculating that! To find the loss in energy, just calculate the final KE and compare it to the initial.

o no wasn't going to do that, good luck indeed
just realising where the energy went.
case closed anyway
 

1. What is the difference between angular momentum and energy?

Angular momentum and energy are both physical quantities that describe the motion of an object. However, they are different in their definitions and how they are calculated. Angular momentum is a measure of the rotational motion of an object, while energy is a measure of the ability of an object to do work.

2. How are angular momentum and energy related?

Angular momentum and energy are related through the law of conservation of angular momentum, which states that the total angular momentum of a closed system remains constant. This means that any changes in one quantity will cause a corresponding change in the other, as long as there is no external torque acting on the system.

3. Can angular momentum be converted into energy?

No, angular momentum cannot be directly converted into energy. However, changes in angular momentum can cause changes in energy, and vice versa. For example, when an object's rotational speed increases, its kinetic energy also increases.

4. How does the conservation of angular momentum affect the motion of objects?

The conservation of angular momentum affects the motion of objects by maintaining a constant rotation speed and direction, unless an external torque is applied. This means that objects will continue to rotate at a constant rate unless acted upon by an external force, such as friction or a change in mass distribution.

5. What are some real-life examples of the relationship between angular momentum and energy?

One example is a spinning top, where the angular momentum and energy are both conserved as the top spins. Another example is an ice skater performing a spin, where their arms are pulled in to decrease their moment of inertia, causing them to spin faster and conserve their angular momentum and energy.

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