Find the angular speed of the smaller gear

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Discussion Overview

The discussion revolves around the relationship between the angular speed of a smaller gear and a larger gear in a mechanical system, particularly focusing on the influence of linkage length and the conditions under which the gears operate. Participants explore theoretical and practical aspects of gear mechanics, including angular velocity, tangential speed, and the effects of chain tension.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question why the angular speed of the smaller gear depends solely on the larger gear, suggesting that the linkage length should have an influence.
  • Others argue that the angular speed is determined by the fixed ratio of angular velocities of the gears, which is independent of the linkage length.
  • There is a discussion about the tangential velocity of the gear centers and whether it affects the angular speed, with some asserting that the center of the gear does not have tangential velocity.
  • Participants mention that the peripheral speed of both gears must be the same due to the chain's constraints, regardless of the linkage length.
  • Some participants propose analyzing the system from a rotating frame of reference to better understand the mechanics involved.
  • There are inquiries about the relevance of string tension in the analysis of the gear's behavior, with differing opinions on its significance.
  • The relationship between torque and angular speed is discussed, with some clarifying that rapid rotation does not necessarily require torque, but acceleration does.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the influence of linkage length and string tension on angular speed, and the discussion remains unresolved with no consensus reached on these points.

Contextual Notes

Some limitations include assumptions about the behavior of the chain and gears, the conditions under which torque is considered, and the specific definitions of angular and tangential velocities. The discussion does not resolve these complexities.

katya
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Thread moved from the technical forums, so no Homework Template is shown
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?

The first question can be answered by looking at slack and tightening of chain caused by rotation of link. Can someone please help to find the answer to the second question?
 

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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.
 
Baluncore said:
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.
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.
 
katya said:
... Long Linkage means high tangential velocity for the center of the small gear, but its angular speed remains independent ...
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.
 
katya said:
Long Linkage means high tangential velocity for the center of the small gear
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.
 
A.T. said:
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.

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.
 
Baluncore said:
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.
Baluncore said:
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.
Baluncore said:
The peripheral speed of the small gear must be the same as the peripheral speed of the large gear
Large gear does not rotate..only the small gear.
 
katya said:
since the end of the linkage is glued to center of the gear
...
Usually for pure rolling on ground,
...
Large gear does not rotate..only the small gear.

Please provide a complete diagram/description of the mechanism.
 
A.T. said:
Please provide a complete diagram/description of the mechanism.
 
  • #10
katya said:

It helps to consider the rotating frame where both gear centres are at rest, and then transform back to the inertial frame.
 
  • #11
A.T. said:
It helps to consider the rotating frame where both gear centres are at rest, and then transform back to the inertial frame.
Can you please elaborate.
 
  • #12
katya said:
Can you please elaborate.
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
jbriggs444 said:
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.
Thank you.I got it by changing frames.Can you do the analysis using ground frame only, considering the string tension?
 
  • #14
katya said:
Thank you.I got it by changing frames.Can you do the analysis using ground frame only, considering the string tension?
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
jbriggs444 said:
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.

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
katya said:
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?
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
jbriggs444 said:
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.
Thank you for replying.
 

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