Can someone check my answer to a rotational problem?

[toesockshoe]

Homework Statement

A stick of length L and mass M is in free space and not rotating. A point mass m has an initial velocity v heading in a trajectory perpendicular to the stick. The mass collides and adheres to the stick a distance b from the center of the stick. Find the resulting motion of the two together in terms of their center of mass velocity and final angular velocity.

Homework Equations

L=IW
L=rxp

The Attempt at a Solution

I set the origin at the center of rotation (in the middle of the stick)
LET M STAND FOR THE SMALL MASS AND S STAND FOR THE STICK.
this is a momentum problem so i said L_i=L_f at the time of contact.
L_{mi} + L_{si} = <b>L_{mf} + L_{sf} </b>
initial momentm of the stick is 0.
mv_ib = I_{stick}w+I_{mass}w
m_pv_ib=\frac{m_sL^2}{12}w_f + mb^2w
solve for w_f
w_f = \frac{m_pv_ib}{\frac{mL^2}{12}+m_pb^2}

i know i didnt solve for center of mass velocity but would we just do a regular momentum problem? (not rotational) becasue there are no external forces and the center of mass wouldn't really be affected by the rotations right? [/B]

 

[haruspex]

It is important to take a non-accelerating reference point. The mass centre of the stick gets accelerated by the impact. You can use the position in space of the mass centre before impact as your reference point. If you do that, the mass m has a post-impact moment which is more than just $mb\omega$.

 

[toesockshoe]

It is important to take a non-accelerating reference point. The mass centre of the stick gets accelerated by the impact. You can use the position in space of the mass centre before impact as your reference point. If you do that, the mass m has a post-impact moment which is more than just $mb\omega$.

ok so the origin would be the center of the stick and stay at that position (so it doesn't move along with the stick right)? would the post impact of the mass be: r x P + mbw?

 

[haruspex]

ok so the origin would be the center of the stick and stay at that position (so it doesn't move along with the stick right)? would the post impact of the mass be: r x P + mbw?

It depends how you are defining r and P.

 

[toesockshoe]

It depends how you are defining r and P.

well, P=mv right? and m would be the velocity of the mass (which is the same as the velocity of the stick) and we can take rsin(\theta) (where r is the distance from the origin to the mass )as b (because rsin(theta) is always b). would that be correct?

 

[haruspex]

well, P=mv right? and m would be the velocity of the mass (which is the same as the velocity of the stick) and we can take rsin(\theta) (where r is the distance from the origin to the mass )as b (because rsin(theta) is always b). would that be correct?

If P is the momentum of the point mass in an inertial frame, you don't need to add $mb^2\omega$ as well. $b\omega$ is the velocity of the mass relative to the rod's centre. Alternatively, if you were to define v as the velocity after impact of the rod's centre then you would use $mbv+mb^2\omega$

 

[toesockshoe]

If P is the momentum of the point mass in an inertial frame, you don't need to add $mb^2\omega$ as well. $b\omega$ is the velocity of the mass relative to the rod's centre. Alternatively, if you were to define v as the velocity after impact of the rod's centre then you would use $mbv+mb^2\omega$

wait do you mean the total momentum of the stick would be: mbv + mb^2\omega ? isn't that what I said when I said you simply add mvb?

 

[haruspex]

wait do you mean the total momentum of the stick would be: mbv + mb^2\omega ? isn't that what I said when I said you simply add mvb?

Are you mixing up m with M?
Your post #5 is somewhat confusing because you wrote

m would be the velocity of the mass (which is the same as the velocity of the stick)

I assume you meant v, not m, and the velocity of the mass is not the same as the velocity of the stick. If v is the velocity of the stick's mass centre then the velocity of the mass is $v+b\omega$. Its angular momentum about the reference point is mb times that. The total angular momentum about the reference point is $ML^2\omega/12+mbv+mb^2\omega$.

 

[toesockshoe]

Are you mixing up m with M?
Your post #5 is somewhat confusing because you wrote

I assume you meant v, not m, and the velocity of the mass is not the same as the velocity of the stick. If v is the velocity of the stick's mass centre then the velocity of the mass is $v+b\omega$. Its angular momentum about the reference point is mb times that. The total angular momentum about the reference point is $ML^2\omega/12+mbv+mb^2\omega$.

ugh yes i messed up. in post 7, i mean the total angular momentum of the mass... not the stick. that would be correct right?

 

[toesockshoe]

Are you mixing up m with M?
Your post #5 is somewhat confusing because you wrote

I assume you meant v, not m, and the velocity of the mass is not the same as the velocity of the stick. If v is the velocity of the stick's mass centre then the velocity of the mass is $v+b\omega$. Its angular momentum about the reference point is mb times that. The total angular momentum about the reference point is $ML^2\omega/12+mbv+mb^2\omega$.

also, you would just set that equal to the inital momentum, which I said was m_pv_ib and solve for w, and that would be the correct answer right?

 

[haruspex]

ugh yes i messed up. in post 7, i mean the total angular momentum of the mass... not the stick. that would be correct right?

If you are defining v as the post impact velocity of the stick's centre, yes.

 

[haruspex]

also, you would just set that equal to the inital momentum, which I said was m_pv_ib and solve for w, and that would be the correct answer right?

Yes, but you also need to use linear momentum to find the post-impact velocity of the stick. I fear there is confusion here because you have used v for that unknown, but v is the given initial velocity of the mass.

 

[toesockshoe]

Yes, but you also need to use linear momentum to find the post-impact velocity of the stick. I fear there is confusion here because you have used v for that unknown, but v is the given initial velocity of the mass.

yes yes, idk why I am making so many stupid mistakes. ok, so to do that would just just use p_f=p_i and this would give you the linear velocity right?

 

[haruspex]

yes yes, idk why I am making so many stupid mistakes. ok, so to do that would just just use p_f=p_i and this would give you the linear velocity right?

Yes.

 

[toesockshoe]

Yes.

ok, so doing that I get : v_f = \frac{m_pv_i}{m_p+m_s} , then its a simple plug into the rotational mometum problem. Also, this would be the velocity of the center of mass correct?

 

[toesockshoe]

also, do you mind explain to me why only the momentum of the small mass is different than what I posted initially? How come the angular momentum of the stick has no linear velocity factored into it?

 

[haruspex]

ok, so doing that I get : v_f = \frac{m_pv_i}{m_p+m_s} , then its a simple plug into the rotational mometum problem. Also, this would be the velocity of the center of mass correct?

Yes, but rather than work out where the mass centre of the whole is, it's simpler to let v_f be the velocity of the mass centre of the rod and write an expression for the linear velocity of the mass (just after impact) in terms of that.

 

[haruspex]

also, do you mind explain to me why only the momentum of the small mass is different than what I posted initially? How come the angular momentum of the stick has no linear velocity factored into it?

You are taking moments about the point where the mass centre of the rod was just before impact. The velocity of the mass centre of the rod just after impact is along a line passing through that point, so has no angular momentum about it.

 

[toesockshoe]

Yes, but rather than work out where the mas centre is, it's simpler to let v_f be the velocity of the mass centre of the rod and write an expression for the linear velocity of the mass (just after impact) in terms of that.

oh is it because the center of the rod is moving in a straight line whereas the mass is moving in a circle?
would the equation be: v_{fp} = \frac{m_pv_i-m_sv_{fs}}{m_p}... now i have 2 unknowns in this equation. how would be able to solve that?

 

[toesockshoe]

You are taking moments about the point where the mass centre of the rod was just before impact. The velocity of the mass centre of the rod just after impact is along a line passing through that point, so has no angular momentum about it.

hey haruspex... i had another question on this problem. why are we using L = I\omega^2 for the momentum of the point mass after the collision? in another thread I created, some said that Iw^2 is only used for spinning and L= r x p is used for rotating around another object (which the bullet is doing). why isn't the final angular momentum of the mass r x p instead of Iw^2?

 

[haruspex]

would the equation be:$v_{fp} = \frac{m_pv_i-m_sv_{fs}}{m_p}$

No, I mean express $v_{fp}$ in terms of $v_{fs}$, $\omega$ and b.

 

[toesockshoe]

No, I mean express $v_{fp}$ in terms of $v_{fs}$, $\omega$ and b.

i think there is a problem with my understand of rotations first. i want to see where I am going wrong there. do you mind answering the following:

had another question on this problem. why are we using L=Iω2 for the momentum of the point mass after the collision? in another thread I created, some said that Iw^2 is only used for spinning and L= r x p is used for rotating around another object (which the bullet is doing). why isn't the final angular momentum of the mass r x p instead of Iw^2?

 

[haruspex]

hey haruspex... i had another question on this problem. why are we using L = I\omega^2 for the momentum of the point mass after the collision? in another thread I created, some said that Iw^2 is only used for spinning and L= r x p is used for rotating around another object (which the bullet is doing). why isn't the final angular momentum of the mass r x p instead of Iw^2?

Are we using L = I\omega^2 for the point mass? I'm using r x p, where r = b, $p = m v_{fp}$, and $v_{fp}$ is ... some function of $v_{fs}$, b and ω.
Of course, they're the same really. Using the original location of the stick's mass centre as the reference point, you can write $I_p= mb^2$, but then you have to work out the instantaneous angular velocity of the particle about that point. It will not be just ω.

 

[toesockshoe]

Are we using L = I\omega^2 for the point mass? I'm using r x p, where r = b, $p = m v_{fp}$, and $v_{fp}$ is ... some function of $v_{fs}$, b and ω.
Of course, they're the same really. Using the original location of the stick's mass centre as the reference point, you can write $I_p= mb^2$, but then you have to work out the instantaneous angular velocity of the particle about that point. It will not be just ω.

is v_{fp} = v_{fs}b\omega ? the point mass is moving in a circle so I'm not sure how the linear velocity would look. also, you didnt answer my question on when we have to use L=Iw^2 and when we have to use L=p x r... i seem to have some confusion here... are both the exact same? Do we get the same answer if we use either one of them? is there a general rule other than use Iw^2 if the object is spinning and use p x r if the object is rotating around an axis?

 

[haruspex]

is v_{fp} = v_{fs}b\omega ? the point mass is moving in a circle so I'm not sure how the linear velocity would look. also, you didnt answer my question on when we have to use L=Iw^2 and when we have to use L=p x r... i seem to have some confusion here... are both the exact same? Do we get the same answer if we use either one of them? is there a general rule other than use Iw^2 if the object is spinning and use p x r if the object is rotating around an axis?

I assume you mean v_{fp} = v_{fs}+b\omega , in which case, yes.
L=Iw (not w^2) is generally used in regard to an object rotating about its mass centre. If we break the object into tiny mass elements, we can apply L=mr^2w to each and add them up. I is just the sum of the mr^2 terms, and all have the same w.
Alternatively, each of those mass elements has velocity v=rw orthogonal to its radius vector, so we can obtain its moment about the mass centre by L=r x p = mvr = mr^2w, same thing.
If we are considering the moment about some other point, we can either use the parallel axis theorem and consider rotation about the instantaneous centre of rotation, or treat the motion as the combination of a rotation about the mass centre plus a linear motion of the mass centre. Suppose the mass centre is distance r from the instantaneous centre of rotation, O. With the first approach, I_O = I+mr^2, L=I_Ow. With the second, L=Iw+mrv=Iw+mr^2w. Same thing.

 

[toesockshoe]

I assume you mean v_{fp} = v_{fs}+b\omega , in which case, yes.
L=Iw (not w^2) is generally used in regard to an object rotating about its mass centre. If we break the object into tiny mass elements, we can apply L=mr^2w to each and add them up. I is just the sum of the mr^2 terms, and all have the same w.
Alternatively, each of those mass elements has velocity v=rw orthogonal to its radius vector, so we can obtain its moment about the mass centre by L=r x p = mvr = mr^2w, same thing.
If we are considering the moment about some other point, we can either use the parallel axis theorem and consider rotation about the instantaneous centre of rotation, or treat the motion as the combination of a rotation about the mass centre plus a linear motion of the mass centre. Suppose the mass centre is distance r from the instantaneous centre of rotation, O. With the first approach, I_O = I+mr^2, L=I_Ow. With the second, L=Iw+mrv=Iw+mr^2w. Same thing.

oh ok, so how would it be possible to use L=Iw for the point mass's momentum here? It would not be possible right... becuase it is a single point that is not rotating around its center.

 

[toesockshoe]

I assume you mean v_{fp} = v_{fs}+b\omega , in which case, yes.
L=Iw (not w^2) is generally used in regard to an object rotating about its mass centre. If we break the object into tiny mass elements, we can apply L=mr^2w to each and add them up. I is just the sum of the mr^2 terms, and all have the same w.
Alternatively, each of those mass elements has velocity v=rw orthogonal to its radius vector, so we can obtain its moment about the mass centre by L=r x p = mvr = mr^2w, same thing.
If we are considering the moment about some other point, we can either use the parallel axis theorem and consider rotation about the instantaneous centre of rotation, or treat the motion as the combination of a rotation about the mass centre plus a linear motion of the mass centre. Suppose the mass centre is distance r from the instantaneous centre of rotation, O. With the first approach, I_O = I+mr^2, L=I_Ow. With the second, L=Iw+mrv=Iw+mr^2w. Same thing.

also, we don't know v_{fs} or v_{fp}. to find the final linear velocity of the stick, do we do p_i = p_f?

in which case: m_pv_i = (m_p+m_s)v_f and thus: v_f = \frac{m_pv_i}{m_p+m_s} ?

 

[haruspex]

how would it be possible to use L=Iw for the point mass's momentum here?

If I here is the MoI about the particle's mass centre, that will be zero. The particle's angular momentum about the reference point, post collision, will be $m_pv_{fp}b$.

we don't know $v_{fs}$ or $v_{fp}$.

No, but as I keep pointing out there is a relationship between them. It involves w and b, but does not involve any masses.

$v_f = \frac{m_pv_i}{m_p+m_s}$

No. That would be true if $v_{fs}=v_{fp}$.

We seem to be going around in circles here.
First, find the relationship between $v_{fs}$, $v_{fp}$, w and b. It's simply a relative velocity relationship. Just after collision, what is the particle's relative velocity compared with the mass centre of the stick?
When you have that, write out the equation for conservation of linear momentum.

 

[toesockshoe]

If I here is the MoI about the particle's mass centre, that will be zero. The particle's angular momentum about the reference point, post collision, will be $m_pv_{fp}b$.

No, but as I keep pointing out there is a relationship between them. It involves w and b, but does not involve any masses.

No. That would be true if $v_{fs}=v_{fp}$.

We seem to be going around in circles here.
First, find the relationship between $v_{fs}$, $v_{fp}$, w and b. It's simply a relative velocity relationship. Just after collision, what is the particle's relative velocity compared with the mass centre of the stick?
When you have that, write out the equation for conservation of linear momentum.

the relationship is v_{fp} = v_{fs} + \omega b

this is the conservation of linear momentum:
p_i = p_f
m_pv_i^2 = m_pv_p + m_sv_s
m_pv_{pi}^2 = m_p(v_{fs}+\omega b) + m_sv_{fs}

 

[haruspex]

the relationship is v_{fp} = v_{fs} + \omega b

this is the conservation of linear momentum:
p_i = p_f
m_pv_i^2 = m_pv_p + m_sv_s
m_pv_{pi}^2 = m_p(v_{fs}+\omega b) + m_sv_{fs}

Good, except for the v² on the left.
Fix that, then use the above to determine v_{fs}.

 

[toesockshoe]

Good, except for the v² on the left.
Fix that, then use the above to determine v_{fs}.

uhhh, that gets really messy cause there's a quadratic doesn't it?
anyway, after I find v_{fs} ... I find v_{fp} using the vfs + wb forumla right?

also, my friend did the priblem this way:

L_i = L_f
m_pv_{pi}b = i\omega
m_pv_{pi}b = (\frac{m_sL^2}{12} + m_pb^2) w_f because the angular velocity are the same for both masses right?

isnt this method much easier?

 

[haruspex]

that gets really messy cause there's a quadratic doesn't it?

But it shouldn't be quadratic. I said that was an error in your equation:

Good, except for the v² on the left.

Correct that equation and try again.

isnt this method much easier?

Oh yes, that's much easier. Unfortunately it's wrong. It's the same error you made in your initial post on this thread.

 

[toesockshoe]

Oh yes, that's much easier. Unfortunately it's wrong. It's the same error you made in your initial post on this thread.

you can't do it because the point is not rotating on its around its own mass center right?

it shouldn't be quadratic. I said that was an error in your equation.

I did try solving it after I fixed it... I got m_p v_{pi}^2 = m_p(v_{fs}+\omega b)^2 + m_sv_{fs}^2... and that gets messy because you need to square the v_{fs} + wb and that makes it difficult to factor out v_{fs} becasue there is two v_fs terms to the 2nd power and one v_fs term to the first power. where am i going wrong?

 

[haruspex]

you can't do it because the point is not rotating on its around its own mass center right?

Not exactly. It's wrong because the particle's angular momentum about the point in space where the mass centre of the stick had been is not just $m_pb^2\omega_f$. There is also the $m_pv_{fs}b$ term. Look at your relative velocity equation.

I did try solving it after I fixed it... I got m_p v_{pi}^2 = m_p(v_{fs}+\omega b)^2 + m_sv_{fs}^2

I'll try once more... IT SHOULD NOT BE QUADRATIC! You appear to be adding up energies, not angular momenta.

 

[toesockshoe]

Not exactly. It's wrong because the particle's angular momentum about the point in space where the mass centre of the stick had been is not just $m_pb^2\omega_f$. There is also the $m_pv_{fs}b$ term. Look at your relative velocity equation.

I'll try once more... IT SHOULD NOT BE QUADRATIC! You appear to be adding up energies, not angular momenta.

OHH STUPID ME. yes sorry. i added up energies.

 

[toesockshoe]

Not exactly. It's wrong because the particle's angular momentum about the point in space where the mass centre of the stick had been is not just $m_pb^2\omega_f$. There is also the $m_pv_{fs}b$ term. Look at your relative velocity equation.

I'll try once more... IT SHOULD NOT BE QUADRATIC! You appear to be adding up energies, not angular momenta.

ok so v_{fs} = \frac { m_pv_{fs}- m_p\omega b}{m_p + m_s}

 

[toesockshoe]

Not exactly. It's wrong because the particle's angular momentum about the point in space where the mass centre of the stick had been is not just $m_pb^2\omega_f$. There is also the $m_pv_{fs}b$ term. Look at your relative velocity equation.

I'll try once more... IT SHOULD NOT BE QUADRATIC! You appear to be adding up energies, not angular momenta.

ok, so let's say that the mass hit the stick at an angle... thus sending the stick in a somewhat diagonal direction. in this case, you would need to add the L=r x p to the stick as well right? (in addition to doing hte similar thing to the point mass_

 

[haruspex]

ok so v_{fs} = \frac { m_pv_{fs}- m_p\omega b}{m_p + m_s}

Not quite. The $v_{fs}$ on the right-hand side should be a different variable.

lets say that the mass hit the stick at an angle... thus sending the stick in a somewhat diagonal direction. in this case, you would need to add the L=r x p to the stick as well right? (in addition to doing hte similar thing to the point mass

Your question is not precise enough for me to answer.
As I say, there are several approaches that work - the trick is to use one consistently and not count any momentum contributions twice over or not at all.
If you care to post the algebra for such an oblique impact I'm happy to check it.

 

[toesockshoe]

Not quite. The $v_{fs}$ on the right-hand side should be a different variable.

v_{fs} = \frac { m_pv_{pi}- m_p\omega b}{m_p + m_s}

Not quite. The $v_{fs}$ on the right-hand side should be a different variable.
Your question is not precise enough for me to answer.
As I say, there are several approaches that work - the trick is to use one consistently and not count any momentum contributions twice over or not at all.
If you care to post the algebra for such an oblique impact I'm happy to check it.

sorry i copied it wrong to latex... this is what i have:

Disregard me other question... basically we did r x p on the mass and v happened to be v_stick + wb... would you say that in problems where there is translational speed its a good idea to always use r x p for a point mass?

 

[toesockshoe]

Not quite. The $v_{fs}$ on the right-hand side should be a different variable.

Your question is not precise enough for me to answer.
As I say, there are several approaches that work - the trick is to use one consistently and not count any momentum contributions twice over or not at all.
If you care to post the algebra for such an oblique impact I'm happy to check it.

also can you explain to me how to find that v_{pf} = v_{sf} + \omega b ? to be i just used it cause it made sense conceptually, but how would you show your work to get that naswer?

 

[haruspex]

... basically we did r x p on the mass and v happened to be v_stick + wb... would you say that in problems where there is translational speed its a good idea to always use r x p for a point mass?

Yes.

also can you explain to me how to find that v_{pf} = v_{sf} + \omega b ? to be i just used it cause it made sense conceptually, but how would you show your work to get that naswer?

It's just relative velocity. This only applies here at the instant after collision, note. In the direction of the particle's original motion, the stick gets a velocity $v_{fs}$, and the particle's velocity relative to that is $\omega b$, in the same direction, so the two add to give the particle's total velocity. Later, of course, as the stick rotates, the relative velocity will be in a different direction from the system's mass centre, but we don't need to worry about that.