# Homework Help: Find the angular speed of the smaller gear

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1. Dec 18, 2017

### katya

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1.Why does the angular speed of small gear depend only on larger gear only?

2.Why does the length of linkage connecting two gears does not have any influence on the angular speed of smaller gear?

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Last edited: Dec 18, 2017
2. Dec 18, 2017

### Baluncore

Welcome to PF.
It will depend on which gear drives the chain.
An angular speed is measured in radians per second. If the big gear is driven then the linear speed of the chain will be proportional to the circumference of the big gear. The gears will have a fixed ratio of angular velocities about their centres. The chain has no centre, so it has a linear speed only, measured in metres per second. The chain simply locks the peripheral speed of the two gears.

3. Dec 18, 2017

### katya

Thank you.Sorry, I meant to ask about the dependency of angular speed on the length of linkage connecting gears[edited].
Long Linkage means high tangential velocity for the center of the small gear, but its angular speed remains independent.

4. Dec 18, 2017

### Baluncore

I think you have got that backwards. The speed of the chain is the tangential or peripheral speed of the gear. That is measured on the circumference of the gear in metres per second. One turn of a gear will advance the chain by 2⋅π⋅gear radius.
If a gear turns once per second it is turning with an angular velocity of 2⋅π⋅radians per second. The chain will be passing at a speed of 2⋅π⋅gear radii per second.
The peripheral speed of the small gear must be the same as the peripheral speed of the large gear because the chain does not hang down, slip or stretch. If you pull any length of chain along at 1 metre per second, it does not matter how long the chain is, it all moves at 1 m/s.
The angular velocity of the gears will be inversely proportional to their radii and must be referenced to the centre of the respective gear.

5. Dec 18, 2017

### A.T.

What? The center of the gear has tangential velocity?

The answers to you original "why" questions depend on the context and what kind of answers are expected here. You could just as well ask, why the angular velocity doesn't depend on the color of the gears.

6. Dec 18, 2017

### katya

since the end of the linkage is glued to center of the gear , centre of gear has the same velocity as the end of the linkage.
Usually for pure rolling on ground, Vcenter=RxW I am asking if such relationship exists here.

7. Dec 18, 2017

### katya

Large gear does not rotate..only the small gear.

8. Dec 18, 2017

### A.T.

Please provide a complete diagram/description of the mechanism.

9. Dec 18, 2017

### katya

10. Dec 18, 2017

### A.T.

It helps to consider the rotating frame where both gear centres are at rest, and then transform back to the inertial frame.

11. Dec 18, 2017

### katya

12. Dec 18, 2017

### jbriggs444

Hold the connecting rod in place and rotate the table (which is still attached to the central gear). Figure out how things rotate. Then translate back to a table-relative viewpoint.

13. Dec 18, 2017

### katya

Thank you.I got it by changing frames.Can you do the analysis using ground frame only, considering the string tension?

14. Dec 18, 2017

### jbriggs444

I do not understand. String tension is irrelevant, surely. One can make the string or belt very tight or leave it almost slack without changing the behavior of the mechanism at all.

15. Dec 18, 2017

### katya

Consider a radius ratio of 50:1 and a small linkage length, small gear will be rotating very fast, but its center of small gear will moving slowly, will there be any slipping?What provides the torque to small gear to rotate that fast?

16. Dec 18, 2017

### jbriggs444

Rotating rapidly does not require torque. Accelerating rapidly to a high rotation rate is what requires torque.

But that is irrelevant to analyzing how many times the small gear would rotate if no slipping occurs. Torque does not enter into that analysis.

17. Dec 18, 2017