Calculating Linear Momentum of a System with a Bullet and a Rod Collision

[Parallel]

Homework Statement

a rod with mass 'M' and length 'L' is pivoted about a frictionless axle through it's end .a bullet with mass 'm' and speed 'v' is shot and sticks to the rod a distance 'a' from the axle.

I need to find the LINEAR momentum of the system just after the hit.

Homework Equations

moment of inertia of a rod about it's center of mass: I=(1/12)*M*L^2
L = I*(omega)
p = (m+M)*V

The Attempt at a Solution

the intial angular momentum is: mva (about the pivot)
angular momentum after the hit is just: I*omega
where I = (1\3)*M*L^2 + m*(a^2)

equating those two gives omega,by the relation V=omega*L
we can get the speed V,so P = (m+M)*V

and this is not the answer

thanks for the help

 

[OlderDan]

The velocity you calculated is not the velocity you need. Your equation for P should give you a hint about the velocity you do need.

 

[Parallel]

I really don't see,how the equation for P should give me a hint,can you be more specific?

can you tell me what is the velocity I calculated?

 

[OlderDan]

I really don't see,how the equation for P should give me a hint,can you be more specific?

can you tell me what is the velocity I calculated?

You calculated the velocity of the end of the rod. The only thing moving with that velocity is the tiny bit of mass at that end. Everything else is moving slower.

The linear momentum of a system of particles is the sum of the linear momenta of each bit of mass in the system. The concept of the center of mass is important because you can find this total momentum by finding the velocity of the center of mass and treating all the mass as if it had that same velocity [P = (m+M)*V_cm]

 

[Parallel]

I tried to work in the center of mass frame,(although it's messy),but how can I find V_cm?..should I just compute angular momentum before the hit and after the hit,in the CM frame,and equate them?

thanks for your help

 

[OlderDan]

I tried to work in the center of mass frame,(although it's messy),but how can I find V_cm?..should I just compute angular momentum before the hit and after the hit,in the CM frame,and equate them?

thanks for your help

Your initial idea for finding the angular velocity of the system after the collision is correct. Once you have found ω you have two ways to approach the solution. You can either find the center of mass of the whole system and use the distance from the pivot to the CM with ω to find the velocity, or you can find the linear momentum of the bullet using its distance from the pivot (a), ω, and m, and combine that with the linear momentum of the CM of the rod to get the total linear momentum of the system.

 

[Parallel]

thank you very much..(I finally got it..spent like 3 hours on this)

thanks again :)