A car drives anticlockwise (conterclockwise) around a flat, circular curve at constant speed, so that the left, front wheel traces out a circular path of radius R = 10 m. If the width of the car is 2m, what is the ratio of the angular velocity about its axis of the left, front wheel to that of the right, front wheel, of the car as it moves through the curve? Assume the wheels roll without slipping.

Options

A. `(4)/(3)`
B. `(1)/(3)`
C. `(6)/(5)`
D. `(5)/(6)`

Answer

D

Explanation

At constant speed, we can use the average angular velocity : `omega=Deltatheta//Deltat`.
The two wheels will share the same `Deltat`, but `Deltatheta` can be found through `Deltatheta=S//C`,
where S is the arc length of the wheel's path, and C is the circumference of the wheel. Let us consider a convenient arc lenth of `pi` radians, so that `S=piR`. Then, for the left wheel,
`Deltatheta_(l)=(piR)/(2pir)`, and for the right wheel, `Deltatheta_(r)=(pi(R+dR))/(2pir)`
where r is the same radius of the two wheels. We then take the ratio,
`(omega_(l))/(omega_(r))=((piR)/(2pirDeltat))/((pi(R+dR))/(2pirDeltat))" "(omega_(l))/(omega_(r))=(R )/(R+dR),(omega_(l))/(omega_(r))=(1)/(1+(dR)/(R ))`
Plugging int the numbers,
`(omega_(l))/(omega_(r))=(1)/(1+(2m)/(10m)),(omega_(l))/(omega_(r))=(5)/(6)`