How Does a Woman Walking on a Turntable Affect Its Rotation?

[BrainMan]

Homework Statement

A woman whose mass is 70 kg stands at the rim of a horizontal turn table that has a moment of inertia of 500 kg * m/s and a radius of 2 m. The system is initially at rest, and the turntable is free to rotate about friction less vertical axle through its center. The woman starts walking clockwise around the rim at a constant speed of 1.5 m/s relative to the earth. (a) In what direction and with what angular speed does the turntable rotate? (b) How much work does the woman do to set the system into motion?

Homework Equations

The Attempt at a Solution

I am not sure how to approach this problem. I was going to try to use the conservation of angular momentum and find the momentum of just the girl walking in the circle. Then I was going to find the angular momentum including the turntable and solve for the angular velocity of the system. Unfortunately the problem gives her speed in m/s and I am not sure how to convert that to rad/sec.

 

[Simon Bridge]

Hint: $v_{\perp}=r\omega$

 

[BrainMan]

Hint: $v_{\perp}=r\omega$

OK the answer I am getting is really close to the right answer but I am still not getting it right. So
(1) Find the girls moment of inertia
I = Mr^2
I = 70(2)^2
I = 280
(2) find the girls angular momentum as if there was no platform
L = IW
V = 1.5 m/s
V = rw
1.5 = 2W
W = .75
280 * .75 = 210
(3) find with the platform
210 = (280 + 500)w
210/780 = w
w = .26923 rad/s
the correct answer is .420 rad/s

 

[Nathanael]

210 = (280 + 500)w

Your error is here.

You should not have included the moment of inertia of the woman.

 

[BrainMan]

Your error is here.

You should not have included the moment of inertia of the woman.

Why?

 

[Nathanael]

Why?

Well for one, they are not moving at the same angular speed, so it doesn't really make sense to do that.


But it's because of the way you're solving the problem.

The way you're solving the problem is this:
You know that the net angular momentum must be zero, so you are finding the angular momentum of the woman (by herself) and then setting it equal to the (negative of the) angular momentum of the turn table (by itself)

That was the plan right? (If that's not what you were doing, sorry for ruining it )

 

[BrainMan]

Well for one, they are not moving at the same angular speed, so it doesn't really make sense to do that.


But it's because of the way you're solving the problem.

The way you're solving the problem is this:
You know that the net angular momentum must be zero, so you are finding the angular momentum of the woman (by herself) and then setting it equal to the (negative of the) angular momentum of the turn table (by itself)

That was the plan right? (If that's not what you were doing, sorry for ruining it )

OK I see. That was totally what I was trying to do. :)

 

[Simon Bridge]

It helps to take conservation problems one step at a time.
i.e. given
v (woman's tangential speed)
m (woman's mass)
I (turntable moment of inertia) and
r (turntable radius)
... it is best practice to list your variables like this, with what they stand for.
Start with the data you are given, then you can introduce variables for the quantity being calculated and any "missing" data needed to find them.

(a) ω (angular momentum of turntable)
conservation of angular momentum means that:
$$\vec L_{before}= \vec L_{after}$$... treating "clockwise" as positive:$$\implies 0 = L_{woman}+ L_{turntable}$$... you have to say which conventions you are using. Don't expect the person marking your work to get it.
Notice the shift from (pseudo)vectors to magnitudes?

Angular momentum does not add the same way momentum does, except when the rotational movement is about the same axis. That is the situation here and it will probably be the situation for everything you come across for a while, so the maths will be just like regular momentum calculations done in one dimension.

Modelling the woman as a point mass gives you:
$$L_{woman} = I_{woman}\omega_{woman}=(mr^2)(v/r) =mrv$$... that last one is what you use in the equation above.
Note: you are not told to treat the woman as a point mass, so you have to say what you are doing.
Even if it was instructed, it is good practice to say anyway.

Women are not normally 1D creatures - and many will object to this characterization. You could attempt the problem modelling the woman as something else: a cylinder perhaps or an hourglass. That would involve bringing in data that is not provided, like the height and girth or a woman (say: an "average" woman - so you can use national statistics). Later on, using your initiative and good judgement to find missing data will be part of the exercise; you are probably not expected to do that for this problem though ;)

(b) hint: conservation of energy.
You'll be making some assumptions about where the energy in the motion comes from.