Physics
posted by Sam .
The differential gear of a car axle allows the wheel on the left side of a car to rotate at a different angular speed than the wheel on the right side. A car is driving at a constant speed around a circular track on level ground, completing each lap in 19.9 s. The distance between the tires on the left and right sides of the car is 1.63 m, and the radius of each wheel is 0.340 m. What is the difference between the angular speeds of the wheels on the left and right sides of the car?

linear velocity=w*radius
inner velocity=w*(r)
outer velocity=w(r+.340)
So the difference in linear velocities at the track is
outerinner=w(r+.340r)=.340 w
Now consider the tires: they have to have the same linear velocity at the pavement. So the difference in tire velocitys is .340w
Inner tire=wit*radtire
outer tire=wot*(radtire)
difference=(wotwit)radius=w*.340
so the difference in the angular speeds of the two tires is: difference=2Pi*radiustrack*.340/(19.9*radiuswheels) 
w = 2*pi*f
f = 1/19.9
where f is the frequency of the car going around the track, w is the angular speed around the track, pi is 3.14
w = 2*pi/19.9 =0.315
The speed traveled by one tire is
w*r
where r is the radius of the wheel
The speed traveled by the other tire is
w*(r+1.63)
The difference in speeds of the wheels is w*(r+1.63)w*r = 1.63w = .513 = w(tire)*r(tire)
where w(tire) is the angular speed of the tire, and r(tire) is the radius of the tire
w(tire)*0.340 = 0.513
Solve for w(tire)
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