How many revolutions per second does a bike wheel make?

[ortegavs]

Homework Statement

If the radius of the wheel on a bike is .25m, and has a velocity of 10m/s, how many revolutions does the wheel make each second?

Homework Equations

2πr/rev

The Attempt at a Solution

I solved this problem in terms of pi. First each rev moves the wheel 2πr which is 2π(1/4)=1/2π meters. I then divided 10m/s by 1/2π meters and that is where I get stuck. Here is what I did.

I first setup the problem and divided the bottom part by 2revs/2revs to cancel it out from the bottom. I then divided the bottom again this time by π m/ π m to remove π from bottom. I then divide 10 by 2 to get the final answer which is wrong because it has the wrong units.
=10m/s/1π/2rev x2rev/2rev
=10m/s*2rev/1π x π/π
=10m *π/s*2rev
= 5m*π/s*rev
My guess is that I have to use order of operations and do the top division first, then the bottom, then divide them by each other. 10/1= 10 , .5/1=.5 and 10/.5 =20. The answer given is 20/ π revs.

 

[Gear.0]

I didn't really understand the second part of what you were doing as it seems overly complicated for this simple problem.

But, actually just based on your initial attempt (first paragraph) I think that is the correct answer.
Because you had (1/2)pi meters/revolution. So when you then divide 10m/s by this you get:
20/pi and the units are: (meters/second) / (meters/revolution)
= (meters*revolution) / (meters*second)
= (revolutions) / (second)
which is what you want, so the answer should be 20/pi revolutions in a single second.
However, the easiest way to do this, is to just realize that if we assume the bike moves without the tires sliding, then the velocity of the bike should be equal to the tangential velocity of any point on the wheel. Then you can convert from tangential velocity to angular velocity by dividing by the radius. Then all you have to do is find how many rotations it goes through in 1 second using that angular velocity.

(which also gives 20/pi BTW)

 

[ortegavs]

You can also use unit analysis (rev/2πr)(10m/s) which gives 20 rev/π s. Its so frustrating that I spend 30 minutes on such a simple problem. The other solution posted also checks, thanks.