Find the angular speed of the smaller gear

In summary, the angular speed of a small gear is dependent only on the larger gear that it is connected to. The length of the linkage connecting the two gears does not affect the angular speed of the smaller gear. This is because the chain locks the peripheral speed of the gears, and the angular velocity of the gears is inversely proportional to their radii. The center of the gear has the same tangential velocity as the end of the linkage, and the speed of the chain is measured on the circumference of the gear.
  • #1
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?

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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  • #2
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
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.
 
  • #4
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.
 
  • #5
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.
 
  • #6
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.
 
  • #7
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.
 
  • #8
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.
 
  • #9
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.
 

1. What is angular speed?

Angular speed is a measure of how fast an object is rotating around a fixed point. It is usually measured in radians per second (rad/s) or degrees per second (deg/s).

2. How is angular speed calculated?

Angular speed is calculated by dividing the angular displacement (change in angle) by the time it takes to make that change. This can be expressed as ω = Δθ/Δt, where ω is the angular speed, Δθ is the angular displacement, and Δt is the time interval.

3. What is the relationship between angular speed and linear speed?

Angular speed and linear speed are related by the formula v = ωr, where v is the linear speed, ω is the angular speed, and r is the radius of the circular motion. This means that as the angular speed increases, the linear speed also increases, and vice versa.

4. How can I find the angular speed of a smaller gear?

To find the angular speed of a smaller gear, you can use the formula ω = (n1/n2)ω2, where n1 and n2 are the number of teeth on the gears and ω2 is the angular speed of the larger gear. This formula applies when the gears are in contact and rotating together.

5. What are some factors that can affect the angular speed of a gear?

The angular speed of a gear can be affected by factors such as the size and number of teeth on the gear, the force applied to the gear, and the friction between the gear and its surroundings. Additionally, the type of gear (e.g. spur gear, bevel gear) can also affect the angular speed.

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