Finding Angular Velocity and Center of Mass Velocity for a Free Rotating Rod

[physicsisgreat]

Homework Statement

I have a rod of mass m and length l on a table without any kind of friction. I give it an impulse J in any point of distance d from the center of the rod, parallel to the table and perpendicular to the rod.
Find the angular velocity ω and the velocity of the center of mass v₀.

Homework Equations

Moment of inertia of the rod rotating around its center: I = m l² / 12
L = I · ω

The Attempt at a Solution

From the impulse theorem:
J = ΔP = P'
I can calculate ω from the angular momentum relations:
L = d x J = I · ω
ω = d J / I = 12 d J / (m l²),
which is 0 if I hit the rod on its center and max if I hit it on d = l/2.
Now I fail to calculate v₀ :P

Thank you in advance :)

 

[BvU]

Hello P.i.G., welcome to PF :)

Your relevant equation needs one or two colleagues: currently v₀ doesn't feature there, $\omega$ doesn't, etc.

Your relevant equation also needs improvement. There is a factor missing.

 

[dean barry]

Should you not apply a couple of equal torques (one clock and one anti-clock) to stop any motion of the centre of mass?

 

[BvU]

Should you not apply a couple of equal torques (one clock and one anti-clock) to stop any motion of the centre of mass?

Not a good idea. 1. It doesn't work. 2. There is nothing that can be considered a cause for these $\tau$. 3. The center of mass is not "stopped"

 

[physicsisgreat]

Hi, I edited my post adding the factor missing and an equation about ω (which I actually used later in my attempt at a solution).
I cannot give another equation about v₀ though, as it is exactly what I am looking for! :)

 

[physicsisgreat]

Mmh, following the impulse-momentum theorem
J = ΔP
I would say that
v₀ = J / m.

At the same time I wonder: can this be true? Can the velocity of the center of mass of the rod be independent from the point in which the impulse is applied?

 

[haruspex]

Mmh, following the impulse-momentum theorem
J = ΔP
I would say that
v₀ = J / m.

At the same time I wonder: can this be true? Can the velocity of the center of mass of the rod be independent from the point in which the impulse is applied?

Yes, it's correct. It feels wrong, until you realize that to impart the same impulse further from the centre of mass of the rod you have to 'work' harder. The rod tends to swing out of the way, so takes less momentum off the impacting object. To achieve the same imparted momentum the impacting object has to start with more momentum.

 

[physicsisgreat]

Thank you very much haruspex! :)