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
outer-inner=w(r+.340-r)=.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=(wot-wit)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)
To find the difference between the angular speeds of the wheels on the left and right sides of the car, we need to first determine the angular speed of each wheel.
Given:
- Time taken to complete each lap: 19.9 s
- Distance between the tires on the left and right sides of the car: 1.63 m
- Radius of each wheel: 0.340 m
Let's start by calculating the circumference of the circular track using the formula:
Circumference = 2 * π * radius
Circumference = 2 * 3.1416 * 0.340 m
Circumference ≈ 2.135 m
Now, we can calculate the linear speed of the car using the formula:
Linear Speed = Distance / Time
Linear Speed = Circumference / Time
Linear Speed = 2.135 m / 19.9 s
Linear Speed ≈ 0.107 m/s
Since the linear speed is the same for both wheels (as the car is driving at a constant speed), we can use the formula for angular speed:
Angular Speed = Linear Speed / Radius
Angular Speed = 0.107 m/s / 0.340 m
Angular Speed ≈ 0.315 rad/s
Now, to find the difference between the angular speeds of the wheels, we need to find the speed difference between the left and right sides of the car. Since we know the distance between the tires on the left and right sides of the car (1.63 m), we can calculate the linear speed difference using the formula:
Linear Speed Difference = (Distance Between Tires) * (Difference in Angular Speeds)
Linear Speed Difference = 1.63 m * (Difference in Angular Speeds)
We want to find the Difference in Angular Speeds, so rearranging the formula:
Difference in Angular Speeds = Linear Speed Difference / (Distance Between Tires)
Now, substituting the known values:
Difference in Angular Speeds = (1.63 m * (Difference in Angular Speeds)) / 1.63 m
Simplifying the equation, we get:
Difference in Angular Speeds = Difference in Angular Speeds
Therefore, the difference between the angular speeds of the wheels on the left and right sides of the car is 0 rad/s, indicating that they are rotating at the same angular speed.