please show all work am studying for a test so the formulas and why you choose i

Question

please show all work am studying for a test so the formulas and why you choose it is very important thanks in advance.

Two point masses, each of mass 5kg are joined by a light straight rod of length 1m. This “dumbbell” is placed on a horizontal frictionless surface, and set into rotation, about a vertical axis through the center of mass, with an angular speed of 4 rad/s. Suddenly, the rod doubles in length (as a car antenna can). What is the initial moment of inertia of the system?

What is the initial angular momentum of the system?

What is the initial kinetic energy of the system?

What is the final angular velocity of the system?

What is the final kinetic energy of the system?

Explanation / Answer

1) Each point mass will have its own moment of inertia about the center of the rod.

So, I=mr^2+mr^2, which is 2mr^2.
m=5 kg, and r=0.5 m (distance to center of rod). This means that I is 2(5 kg)(0.5m^2), which is 2.5 kg*m^2.

2) L = I which is just (2.5)(4)=10 kg*m^2/s

3) KE = 0.5*I^2 which is just (0.5)(2.5)(4^2)=20 J

4) Now, we repeat the same followed in step 1, but remember that your radius is twice as large (1 m now). So, your I=2(5)(1^2)=10 kg*m^2
Use this value of I and the conservation of angular momentum to solve for (I=I, =(2.5*4)/10, = 1 rad/s)

5) KE = 0.5*I^2, =(0.5)(10)(1^2)=5 J

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