Conservation of Angular Momentum: angular speed

[mickellowery]

Homework Statement

A student sits on a freely rotating stool holding 2 3.00kg dumbbells. When the students arms are extended horizontally the dumbbells are 1.00m from the axis of rotation and the student rotates with an angular speed of 0.750 rad/sec. The moment of inertia of the student and the stool is 3.00 kgm² and it is assumed to be constant. The student pulls the dumbbells in to a position 0.300m from the rotation axis. What is the new angular speed?

Homework Equations

I was going to try I_i$$\omega$$_i=I_f$$\omega$$_f but the problem with this is that the problem says I is constant at 3.00 kgm². I assume that I need to factor in the radius somehow but I'm not sure how to do it.

The Attempt at a Solution

 

[rock.freak667]

When the dumbells are at 1m, what is the moment of inertia of them?

Add that to the inertia of the stool and you have the initial moment of inertia.

initial angular momentum = I_initialω_initial.


When the dumbells are at 0.3m, what is the moment of inertia then? (Add this to get inertia of the stool to get the final moment of inertia)

 

[mickellowery]

So can I model the dumbbells as a rod and use 1/12 ML² as the moment of inertia or do I have to integrate $$\int$$$$\rho$$dV? I am facing problems with both ways, I don't know what the volume would be if I integrate, and it seems like modeling as a rod would not be correct.

 

[rock.freak667]

So can I model the dumbbells as a rod and use 1/12 ML² as the moment of inertia or do I have to integrate $$\int$$$$\rho$$dV? I am facing problems with both ways, I don't know what the volume would be if I integrate, and it seems like modeling as a rod would not be correct.

No need to do all of that. Just treat them as point masses such that I=mr²

 

[rdl]

To solve this problem, we can use the conservation of angular momentum equation, L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular speed. In this case, we will use the initial and final states of the student and stool system to solve for the final angular speed.

Initial state:
Ii = 3.00 kgm^2
ωi = 0.750 rad/sec

Final state:
Ii = 3.00 kgm^2
ωf = unknown
r = 0.300m

Using the conservation of angular momentum equation, we can set the initial and final angular momentums equal to each other:

Iiωi = Ifωf

Solving for ωf, we get:
ωf = (Iiωi)/If

Substituting in the values from the initial and final states, we get:
ωf = (3.00 kgm^2 * 0.750 rad/sec)/(3.00 kgm^2)

Simplifying, we get:
ωf = 0.750 rad/sec

Therefore, the new angular speed is the same as the initial angular speed, 0.750 rad/sec. This is because the moment of inertia and the angular momentum are constant, and the change in the radius does not affect the final angular speed. This is known as the conservation of angular momentum, where the total angular momentum of a system remains constant unless acted upon by an external torque.