# Conservation of angular momentum on a piano stool

• map7s
In summary, the student initially rotates freely on a piano stool with an angular speed of 3.33 rev/s and holds 1.44 kg masses in each outstretched arm 0.759 m from the axis of rotation. The combined moment of inertia of the student and stool is 4.34 kgm2. When the student pulls his arms inward, his angular speed increases to 3.54 rev/s and the masses are now considered to be points. The equation for calculating the kinetic energy of the system is KE=1/2 (I_total)w^2=1/2 [(I_ss)+2mr^2](w^2), where I_ss is the constant moment of inertia of the student and
map7s
A student on a piano stool rotates freely with an angular speed of 3.33 rev/s. The student holds a 1.44 kg mass in each outstretched arm, 0.759 m from the axis of rotation. The combined moment of inertia of the student and the stool, ignoring the two masses, is 4.34 kgm2, a value that remains constant.
(a) As the student pulls his arms inward, his angular speed increases to 3.54 rev/s. How far are the masses from the axis of rotation at this time, considering the masses to be points?
(b) Calculate the initial and final kinetic energy of the system.

For the first part, I tried doing mr^2w(initial)=mr^2w(final) and it said that my answer was within 10% of the correct answer. I tried doing different variations, playing around with the given moment of inertia, but I couldn't figure out how to factor it in along with the varying radius.

You realize that angular momentum is conserved. You need to figure out how the rotational inertia of the total system (student+stool+masses) changes as the masses are brought closer to the center. What's the contribution of the masses to the total rotational inertia?

Well, the masses have to be factored in twice since there are two of them but I'm not sure how to combine it with the initial moment of inertia given in the problem...

What's the moment of inertia of a point mass M at a distance R from an axis?

moment of inertia I=mr^2

Exactly. So what's the total moment of inertia of the system at those two points (arms outstretched; arms in)?

arms outstretched: mr^2=(2)(1.44)(0.759)^2
arms in: mr^2=(2)(1.44)r^2

That's just the contribution of the two masses. Now add that to the moment of inertia of the "student+stool", which is given as a constant.

...and so to calculate the kinetic energies I use the equation KE=1/2 Iw^2=1/2 [I+2mr^2](w^2) ?

map7s said:
...and so to calculate the kinetic energies I use the equation KE=1/2 Iw^2=1/2 [I+2mr^2](w^2) ?
That should do it.

I tried using that equation and plugging in my numbers, but apparently it isn't correct...

map7s said:
I tried using that equation and plugging in my numbers, but apparently it isn't correct...
OK.. You have two different I involved. I assumed from what you did earlier you had them distinguished. So let's be more explicit.

KE=1/2 (I_total)w^2=1/2 [(I_ss)+2mr^2](w^2)

where I_ss is the assumed constant moment of inertia of the student and stool combination. You have two calculations to do for the two different positions of the masses with their corresponding angular velocities. If you got the correct r earlier, you should be getting the correct KE.

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## 1. What is conservation of angular momentum?

Conservation of angular momentum is a fundamental law of physics that states that the total angular momentum of a system remains constant as long as there are no external torques acting on it.

## 2. How does conservation of angular momentum apply to a piano stool?

In the case of a piano stool, the stool and the pianist can be considered as a system. As the pianist rotates on the stool, the total angular momentum of the system remains constant.

## 3. What factors affect the conservation of angular momentum on a piano stool?

The two main factors that affect the conservation of angular momentum on a piano stool are the mass and distance of the pianist from the axis of rotation. The greater the mass and the closer the distance, the faster the stool will rotate.

## 4. How does friction affect the conservation of angular momentum on a piano stool?

Friction between the stool and the floor can cause external torques, which can slow down the rotation of the stool and affect the conservation of angular momentum.

## 5. Can the conservation of angular momentum be violated on a piano stool?

No, the conservation of angular momentum is a fundamental law of physics and cannot be violated. However, external factors such as friction can affect the rate of rotation of the stool, but the total angular momentum of the system will remain constant.

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