Bicycle Type Chain [Mechanics] Problem

In summary, Vivek, a new member of physicsforums.com, is seeking advice on a problem involving motion of a chain along a circular track in classical mechanics. The problem is from an old book and Vivek provides the diagram and equation he has already derived. He asks for help in finding the correct expression for tension and provides his own solution for reference.
  • #1
maverick280857
1,789
4
Hi guys..

I am a new member of physicsforums.com (I am a science student at the pre-undergrad level). I am unable to solve a problem involving motion of a chain along a circular track, in classical mechanics. The problem is from an old book on mechanics. Please read the problem (below) and offer your advice...

The problem:

"A flexible bicycle-type chain of length (1/2)*Π*r (0.5 * pie * radius) has a mass RHO per unit length and is released after being held by its upper end in an initial rest condition in the smooth circular channel. Determine the acceleration 'a' which all links experience just after release. Also find the expression for the tension T in the chain as a function of θ (theta)¸ for the condition immediately following release. Isolate a differential element of the chain as a free body and apply the appropriate motion equation."

The Diagram (drawn in Paint: the red dots represent the chain, the diag is otherwise vague and not as good as the original book version) is on the link,

Please visit www.geocities.com/fsairin/physics.html [Associated Diagram]

I have gone as far as getting T (The tension) as a function of θ¸ but the function also includes 'a' which has to be eliminated.

Any ideas/solutions/approaches are appreciated...

Cheers
Vivek

---------------------------------------------
"Experiment without theory is
lame. Theory without
experiment is blind."
 
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  • #2
For the Moderator

(FOR THE MODERATOR)

HI

I think I posted this in a wrong place...can you please shift it to the homework forum?

Cheers
Vivek
 
  • #3
Hi

I am including my solution so that you can help me better...

[tex]
dm = \rho r d\phi
[/tex]

Forces

[tex]
\sum dF_{\phi} = T(\phi + d\phi) + g dmsin\phi - T(\phi) = dma_{t}
[/tex]

which gives,

[tex]
dT = dm(a_{t} - gsin\phi) = \rho r (a_{t} - gsin\phi)d\phi
[/tex]

Assuming

[tex]T(\phi = 0) = T(\phi = \frac{\pi}{2}) = 0[/tex]

(I think there could be a mistake in the choice of these boundary values.)

[tex]
\int_{0} ^{T(\phi)} dT = \rho r a_{t} \int_{0}^\phi d\phi - g\int_{0}^\phi sin\phi d\phi
[/tex]

or

[tex]
T(\phi) = \rho r a_{t} \phi - \rho r g(1-cos\phi)
[/tex]

Also

[tex]
T(\phi = \frac{\pi}{2}) = 0 \Rightarrow \rho r a_{t}\frac{\pi}{2} = \rho r g \Rightarrow a_{t} = \frac{2g}{\pi}
[/tex]

So substituting for the acceleration in the expression for tension, we have

[tex]
T(\phi) = \rho r g(\frac{2\phi}{\pi} - (1 - cos\phi))
[/tex]

instead of

[tex]
T(\phi) = \rho r g(\frac{2\phi}{\pi} - sin\phi)
[/tex]

which is the answer given in the book.

Any help is greatly appreciated...

Cheers
Vivek
 

1. What is a bicycle type chain?

A bicycle type chain is a mechanical device that transfers power from the pedals to the bike's wheels. It is made up of inner and outer plates, rollers, and pins, and is responsible for propelling the bike forward.

2. Why do bicycle chains break?

There are several reasons why a bicycle chain may break. Some common causes include wear and tear from regular use, improper maintenance, and shifting gears too quickly or forcefully. Additionally, if the chain is not properly lubricated, it can become stiff and prone to breaking.

3. How do I know when it's time to replace my bicycle chain?

It is recommended to replace your bicycle chain every 1,000-2,000 miles of riding. However, factors such as riding conditions and maintenance can affect the lifespan of your chain. Signs that it may be time to replace your chain include skipping gears, difficulty shifting, and visible wear or rust on the chain.

4. Can I repair a broken bicycle chain?

In most cases, a broken bicycle chain cannot be repaired and will need to be replaced. If the chain has broken due to a damaged link, it may be possible to remove the damaged link and reconnect the chain, but this should only be done as a temporary fix until a new chain can be installed.

5. How can I prevent bicycle chain problems?

Regular maintenance is key to preventing bicycle chain problems. This includes keeping the chain clean and properly lubricated, avoiding shifting gears too forcefully, and replacing the chain before it becomes too worn. It is also important to regularly check the chain for any signs of damage or wear and address any issues as soon as possible.

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