Rotational Motion Problem (a bike)

[Carpe Mori]

Homework Statement

How is the angular speed of the rear wheel (Wr) of a bicycle related to that of the pedals and front sprocket (Wf)? That is derive a formula for Wr/Wf. Let Nf and Nr be the number of teeth on the front and rear sprockets respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly. Then evaluate the ratio of Wr/Wf when fron and rear sprockets have 52 and 13 teeth respectively

Homework Equations

W = v/r

Circumference (c) = 2pi*r (i am not sure if this is relevant)

The Attempt at a Solution

v for both front and back sprocket is the same (note W is not)

Wf = v/Rf

Wr = v/Rr

Since distances between teeth are same in back and front:

C = 2pi*Rr/Nr = 2pi*Rf/Nf

so...

Wr = v/(Nr*Rf/Nf*Rr)

Wf = v/(Nf*Rr/Nr*Rf)

Wr/Wf = Nf^2*Rr^2/Nr^2*Rf^2

but i do not think that is right because the question implies that your only variable should be N...

HELP! (if i seem unclear about anything...do tell)

 

[mezarashi]

Your equations for angular velocity look sound. I think your confusion starts when you substitute: C = 2pi*Rr/Nr. That to me doesn't make much sense. I would agree that the circumference C = kNr, where k is the spacing or pitch (some constant) of the teeth. And C is also equal to C=2pi*Rr. Which would lead to Rr being expressed in terms of Nr.

Give that a try.

 

[Shooting Star]

Your equations for angular velocity look sound. I think your confusion starts when you substitute: C = 2pi*Rr/Nr. That to me doesn't make much sense.

Nr/(2pi*Rr) is just the number of teeth per unit length, and so would be the same on both. He has just taken the reciprocal, which is constant C.

Wr = v/(Nr*Rf/Nf*Rr)

Wf = v/(Nf*Rr/Nr*Rf)

Wr/Wf = Nf^2*Rr^2/Nr^2*Rf^2

What is all this? Number of teeth on each is proportional to the circumference and hence the radius. Find Wr/Wf as some ratio of Rr and Rf and then put Nr and Nf in the ratio.

 

[Carpe Mori]

well i took my C = 2pi*Rr/Nr = 2pi*Rf/Nf and solved for both Rr and Rf and then plugged those values into my initial angular velocity equations.

and i resulted with

Wr/Wf = Nf^2*Rr^2/Nr^2*Rf^2

is this correct yes or no?? it means angular velocity depends on both Radius and number of teeth of both sprockets...which makes sense to me

 

[Shooting Star]

well i took my C = 2pi*Rr/Nr = 2pi*Rf/Nf and solved for both Rr and Rf and then plugged those values into my initial angular velocity equations.

and i resulted with

Wr/Wf = Nf^2*Rr^2/Nr^2*Rf^2

is this correct yes or no?? it means angular velocity depends on both Radius and number of teeth of both sprockets...which makes sense to me

I don't know. You check it once more. But I'll show you the easy and natural way.

v = Rf*Wf = Rr*Wr =>
Wf/Wr = Rr/Rf = 2pi*Rr/2pi*Rf = C*Nr/C*Nf = Nr/Nf =>
Wr/Wf = Nf/Nr.

There was no need to introduce the C, since it's obvious that the number of teeth is directly proportional to the radius.