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A Gear Ratio Torque Problem

by TheAustrian
Tags: gear, ratio, torque
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TheAustrian
#1
Jan10-13, 08:24 PM
P: 165
Hi all... I'm new here...

1. The problem statement, all variables and given/known data
this is not a homework question, I'm just solving practice questions for exam preparation
Two wheels with masses M1 = 2 kg and M2 = 4 kg are connected.
The ratio is R1 = 5 cm and R2 = 10 cm
Considering an angular velocity of ωo = 10rads-1
for the small wheel and a constant angular acceleration of α = 1
rads-2

What will be the torque τ required to stop the system after 20 s within a
period of 2 s?
2. Relevant equations
T1ω2=T2ω1

and

P = E/t
3. The attempt at a solution

I already found the energies of the Gears,
E1= 1.125J and E2=2.25J
but I have no idea how to find the torque needed to stop them in 2 seconds. I'm not even sure if I'm on the right track. I have like 3-4 A4 sheets worth of work all scribbled on/crossed out...

actually my problem is that if I work out the Torques, their ratio comes out at 0.25 rather than 0.5, so I'm not sure what to do.
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voko
#2
Jan10-13, 11:50 PM
Thanks
P: 5,687
I think you could use the simple relationship between torque and angular momentum. Energy makes this more complicated than it needs to be.
TheAustrian
#3
Jan10-13, 11:56 PM
P: 165
Quote Quote by voko View Post
I think you could use the simple relationship between torque and angular momentum. Energy makes this more complicated than it needs to be.
But how will that allow me to calculate the question?

voko
#4
Jan11-13, 12:25 AM
Thanks
P: 5,687
A Gear Ratio Torque Problem

In 20 seconds the system will have a certain amount of angular momentum. You need to apply some constant torque to bring it down to zero in 2 seconds.
TheAustrian
#5
Jan11-13, 12:36 AM
P: 165
Quote Quote by voko View Post
In 20 seconds the system will have a certain amount of angular momentum. You need to apply some constant torque to bring it down to zero in 2 seconds.

So I should work out time taken to get angular momentum to zero and work on from there?
voko
#6
Jan11-13, 12:37 AM
Thanks
P: 5,687
You are GIVEN this time: 2 seconds.
TheAustrian
#7
Jan11-13, 01:00 AM
P: 165
Quote Quote by voko View Post
You are GIVEN this time: 2 seconds.

Sorry, I phrased myself wrongly here
I meant I should work out the Angular momentum to get to zero in the time-span of two seconds, and then I need to work from there? Is there a relationship that connects torque and angular momentum?
voko
#8
Jan11-13, 01:02 AM
Thanks
P: 5,687
Torque is the rate of change of angular momentum.
TheAustrian
#9
Jan11-13, 01:19 AM
P: 165
Quote Quote by voko View Post
Torque is the rate of change of angular momentum.
With respect to time?
voko
#10
Jan11-13, 01:40 AM
Thanks
P: 5,687
Yes, with respect to time. Since you are preparing for an exam, I suggest that you review the fundamentals of rotary motion. The relationship between angular momentum and torque is the equivalent of Newton's law, and you must know it even if awaken in the middle of the night!
TheAustrian
#11
Jan11-13, 03:37 PM
P: 165
I think I will just surrender, I do not see how Angular Momentum should come into the question, if it's Torque I must know...

Thanks for trying to help though.
voko
#12
Jan12-13, 12:44 AM
Thanks
P: 5,687
This is a pity.

The equation you should remember is ## \displaystyle \tau = \frac {d L } {d t } ##, where ## \tau ## is torque, and ## L ## is angular momentum. This is analogous to Newton's second law: ## \displaystyle F = \frac {d p} {d t} ##. Assuming the torque is constant, as the problem requires, this simplifies to ## \displaystyle \tau = \frac {\Delta L} {\Delta T} ##, where ## \Delta L ## is the change of angular momentum and ## \Delta T## is the period of time during which the change occurs.

## L = I \omega ##, where ## I ## is the moment of inertia and ## \omega ## is angular velocity. Knowing the moments of inertia of the two wheels and their angular velocity, you know the net angular momentum, and the rest is simple.


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