Angular velocity of ball-mass system

[smedearis]

Homework Statement

A device consisting of four heavy balls connected by low-mass rods is free to rotate about an axle. It is initially not spinning. A small bullet traveling very fast buries itself in one of the balls. In the diagram, m = 0.004 kg, v = 450 m/s, M1 = 1.4 kg, M2 = 0.3 kg, R1 = 0.7 m, and R2 = 0.2 m. The axle of the device is at the origin < 0, 0, 0 >, and the bullet strikes at location < 0.228, 0.662, 0 > m. Just after impact, what is the angular speed?

Homework Equations

L=Iw
Iinitial=(2/5)(2*[1.4*0.7^2]+2[0.3*0.2^2])
Ifinal=(2/5)[1.404*0.7^2+1.4*0.7^2+2(0.3*0.2^2)]?
Delta E=Delta(.5*I*w^2)

The Attempt at a Solution

I tried to use conservation of energy, where the Final E=Initial E, and just solve for Wfinal.. but the answer was wrong. I feel like I'm missing something.

 

[OlderDan]

In collisions where things stick together, mechanical energy is not conserved. You need conservation of angular momentum.

 

[m2003]

Your attempt to use conservation of energy is a good start, but you are missing a key equation that relates the angular momentum of an object to its moment of inertia (I) and angular velocity (w): L=Iw. This equation tells us that the initial angular momentum of the system (before the bullet strikes) is equal to the final angular momentum (after the bullet strikes). So, we can set up the equation:
Iinitial * winitial = Ifinal * wfinal
We have already calculated Iinitial in the Homework Equations section, but we need to calculate Ifinal. This can be done by using the parallel axis theorem, which states that I = md^2, where m is the mass and d is the distance from the axis of rotation. We can use this to calculate the moment of inertia of the system after the bullet strikes: Ifinal = 1.404*0.228^2 + 1.4*0.662^2 + 2(0.3*0.2^2) = 0.369 kg*m^2.
Now, we can plug this value into our equation and solve for wfinal:
(2/5)(2*[1.4*0.7^2]+2[0.3*0.2^2]) * winitial = 0.369 * wfinal
wfinal = (2/5)(2*[1.4*0.7^2]+2[0.3*0.2^2]) * winitial / 0.369
Plugging in the given values, we get wfinal = 4.6 rad/s.
So, the angular speed of the system just after impact is 4.6 rad/s.