Problem of torque transmission and determining how to achieve a velocity

[Strontium90]

Homework Statement

The problem that I was working on and was not able to resolve. This is the problem straight from the book:

12.13) The sprocket wheel on the rear axle of a certain bicycle is 10 cm diameter, and the sprocket wheel to which the pedals are attached is 20 cm in diameter. The wheels of the bicycle are 65 cm in diameter. At how many revolutions per minute must the pedals be turned to travel at 5 m/s?

Homework Equations

This is a problem in a chapter dealing on how to analyze simple machines. Machines based on Classical Mechanics such as the lever, pulley and the inclined plane. The problem was from a sub-section that dealt with the topic of "Torque transmission". It had man of the same concepts as the rest of the chapter. One of the concepts was that of Ideal Mechanical Advantage. It is the quotient of d_out\d_in. Another concept that was illustrated was that of the ratio of angular velocities. This was said to be the inverse of the IMA. It is the following ω_out\ω_in.

The Attempt at a Solution

I used a previous example of a worked out problem to try to solve the problem and that did not yield the answer. Also, one thing that I saw was that if the problem was looking for the number of revolutions that one would make to get to or maintain a velocity of 5 m/s, you would have to make many revolutions each minute to keep this velocity.

 

[Villyer]

Can you show work for the method you tried already? It makes it easier to help you.

 

[azizlwl]

Find out the relationship between translational and rotational velocity.

 

[Strontium90]

The method that I used to get the answer that was the most similar to the answer in the book was this.

I took the diameter of the wheel, divided it in half and came up with 2.04m for the circumference of the wheel. Then I divided the desired velocity (5 m/s) by the circumference of the wheel (2.04m) and that gave the number of revolutions that the wheel had to make in a second to travel five meters. The number of revolutions that the wheel has to make to achieve the desired velocity is 2.449 revs. Then I divided 5m by .325m (the radius of the wheel) and got 15.38 rad/sec for the angular velocity of the wheel at the desired linear velocity.

From a similar problem in the chapter, the next step that I took was to take a formula and use it to solve for the angular velocity of the sprocket that the pedals use. The formula is:

(omega)_out(d_out/d_in) = (omega)_in

For (omega)_out I used the angular velocity of the wheel. For d_out I used the diameter of the outgoing sprocket, i.e. the output sprocket, the one that drives the rear wheel. For d_in, I used the input sprocket, i.e. the one that is attached to the pedals. When I put in values to this formula, the result that I got was approx. 7.69 rad/sec. Then I multiplied that by (60 sec/ 2pi) and that gave me a result of 73.5 rev/min. The books answer is 7.3 rev/min.

 

[Strontium90]

Azizlwl, would the relationship in question be v = ωr ?

 

[azizlwl]

"At how many revolutions per minute must the pedals be turned to travel at 5 m/s?"

Yes, here we have ω and v.
The sprocket wheel on the rear axle and rear wheel experience same ω.
The sprocket wheel to which the pedals and The sprocket wheel on the rear axle are connected by a chain thus have the same v.
Check the differences of r's, the radius.

 

[Strontium90]

Azizl, would you be referring to the quotient of the radius_out / radius_in? or as it is known as the Ideal Mechanical Advantage, in terms of torque transmission? When I use that in an equation, I will get an answer of 73 revolutions per minute. The book's answer is 7.3 revs per minute. Is the answer in the book incorrect? Because I can not figure out how they arrived at the answer they have.

 

[Strontium90]

One more question, what publishers, authors or books do you recommend for learning the basics of physics Azizl? I am interested in strengthening my skills.

 

[azizlwl]

I always refer to 2 books
1. 3000 solved problems in Physics, Shaum's Solved Problems Series
2. University Physics-Young and Freedman. For more on the concept.

 

[Strontium90]

So was my method of solving the problem correct? Looking at the book's answer, it does not seem right. If one is going five meters a second, your velocity is high and so you need to apply a great deal of revolutions to the wheels via the pedals.

 

[azizlwl]

I got the same answer too.
v=rω
5m/s=0.325mω
ω=15.38 rad/sec

v_chain=r_Srω
v_chain=0.05(15.38)
v_chain=0.769m/s

v_chain=r_Sfω
0.769m/s=0.1mω
ω=7.69 rad/sec=73.4rev/min

 

[Strontium90]

Ugh, so the book has a typo. Ok thank you Azizlwl for the explanation and a good work out.

 

[azizlwl]

Glad I'm able to help. I'm learning too.