The effect of a spinning wheel on a turntable

[windazz]

Homework Statement

A person stands at rest on a 5.0kg, 50cm diameter frictionless turntable.
He receives spinning bicycle wheel, 180rpm counterclock wise,
diameter 64cm, and the rim weighs 4.0kg.
The wheel was held in a horzontal plane, the handles extend outward from the center (rotation axis is vertical).

This person's legs and torso are 70kg, have an average diameter of 25cm and a height of 180cm. His arms are 2.5kg each, and he hold the handles of the wheel 45cm from the center of his body.

This person turns the spinning wheel over 180degrees so that the handle that had been pointing up now points down.
Describe this person's motion and calculate any relevant numerical quantities.

Homework Equations

Inertia, axis is r distance away: I = m r^2
Inertia of loop: I= m r^2
Inertia of disk: I= (1/2) m r^2
Inertia of rod, axis about center: I = (1/12) m L^2
Inertia of rod, axis about edge: I = (1/3) m L^2

Conservation of angular momentum:
initial L = final L
change in L would be 0 if L is conserved.
angular momentum: L = I W

The Attempt at a Solution

Bicycle wheel by itself:
- axis is the handles at the center of wheel

-inertial of wheel loop = I 'loop' = m'loop' r'loop'^2
-initial angular momentum of wheel loop = Li = I'loop' W = m'loop' r'loop'^2

-After turnning the wheel 180degrees, the spinning direction is inverted.
counter clock wise --> clock wise
final L = negative initial L
Lf = - Li
- change in L = 2Li = 2 I'loop' W = 2 m'loop' r'loop'^2 W


The person standing on turntable holding the wheel:
- axis is now the body of the person standing on the turntable disk.

-inertial total = I disk + I body + I arms + I loop
= (1/2) m'disk' r'disk'^2 + (1/12) m'body' r'body'^2
+ (1/3) m'arms' r'arms'^2 + m'loop' r'loop'^2

note: r'body' = the body's average diameter 25cm, NOT the height of the person.

- angular momentum of this whole system is the change in L:
L = 2Li = I 'total' W'final'
We can now calculate the final angular velocity of the system.


I'm just confused of why my calculations didn't include the person's height?
Is that just an extra information not necessary to use? or did I do something wrong with my calculations?~~~
Thanks alot! =)

 

[anathema]

Thank you for your post. It appears that you have made some calculations and have a good understanding of the concepts involved in this scenario. However, I would like to offer some additional insights and clarifications to further your understanding.

Firstly, when calculating the moment of inertia for the person's body and arms, it is important to note that the distance from the axis of rotation is not simply the person's height, but rather the distance from the axis to the center of mass of the person's body and arms. This is because the moment of inertia is a measure of an object's resistance to changes in rotational motion, and the center of mass is the point around which an object's mass is evenly distributed. Therefore, the distance from the axis to the center of mass is a more accurate representation of the body's contribution to the moment of inertia.

Additionally, when calculating the final angular velocity of the system, you should take into account the conservation of angular momentum, which states that the initial angular momentum of a system is equal to the final angular momentum. Therefore, the final angular velocity can be calculated by equating the initial angular momentum of the bicycle wheel (before it is turned over) to the final angular momentum of the entire system (after the wheel is turned over). This will give you a more accurate representation of the system's motion.

I hope this helps to clarify any confusion you may have had. Keep up the good work in your scientific endeavors!

 

[baba89]

I would like to point out that the effect of a spinning wheel on a turntable can be described using the principles of angular momentum and inertia. In this scenario, the person standing on the turntable is holding a spinning bicycle wheel and then turns it over 180 degrees. This action causes a change in the direction of the angular momentum of the system.

To calculate the relevant numerical quantities, we can use the formula for angular momentum, L=Iω, where I is the moment of inertia and ω is the angular velocity. In this case, the moment of inertia of the system will be the sum of the moments of inertia of the turntable, the person's body, arms, and the spinning wheel.

The person's height is not necessary to use in these calculations as it does not affect the moment of inertia or the angular velocity of the system. The person's height only affects the overall mass of the person, which is already taken into account in the calculations for the moment of inertia.

Additionally, it is important to note that the change in the direction of the spinning wheel will not affect the overall angular momentum of the system, as angular momentum is conserved. This means that the final angular momentum of the system will be equal to the initial angular momentum.

In conclusion, the person's motion can be described as a change in the direction of the spinning wheel, while the relevant numerical quantities can be calculated using the principles of angular momentum and inertia. The person's height is not necessary to use in these calculations.