Two identical dragsters, starting from rest, accelerate side by side along a straight track. The wheels have identical angular acceleration. The wheels on one of the cars roll without slipping, while the wheels on the other slip during part of the time.
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(a) For which car, the winner or the loser, do the wheels roll without slipping?
The winner's wheels roll without slipping.
* Both dragsters reach the finish line at the same time.
The loser's wheels roll without slipping.
(b) For the dragster whose wheels roll without slipping, is there a relationship between its linear speed v (in m/s) and the angular speed (in rad/s) of its wheels?
Yes. The relationship is = rv, where r (in m) is the radius of a wheel.
There is no relationship between its linear speed and angular speed.
Yes. The relationship is v = r, where r (in m) is the radius of a wheel.
*Yes. The relationship is = v2/r, where r (in m) is the radius of a wheel.
(c) For the dragster whose wheels roll without slipping, is there a relationship between the magnitude a (in m/s2) of its linear acceleration and the magnitude (in rad/s2) of the angular acceleration of its wheels?
There is no relationship between the magnitude of its linear acceleration and the magnitude of the angular acceleration of its wheels.
*Yes. The relationship is a = 2/r, where r (in m) is the radius of a wheel.
Yes. The relationship is = ra, where r (in m) is the radius of a wheel.
Yes. The relationship is a = r, where r (in m) is the radius of a wheel.
The ones I selected for the answer has the * by them. Are the correct?
Say the angular velocity of the wheel is w and the linear velocity of the car is v
Say the angular acceleration of the wheel is alpha and the linear acceleration of the car is a
Say radius of wheel is r
Then for the no slip car:
forward velocity of axle v = w r
forward acceleration of axle a = alpha r
But if the wheels slip on the pavement
forward velocity of axle v < w r
forward acceleration of axle a < alpha r
In other words if you want to win, you want tires that do not spin. There are however more detailed question such as the torque versus rpm curve of you engine. You might want to let the wheels slip as you start up so you can get the rpm of the engine up quickly. That is not what whoever wrote this question is thinking about though.
Yes, the answers you selected for parts (a) and (b) are correct. The winner's wheels roll without slipping, and for the dragster whose wheels roll without slipping, there is a relationship between its linear speed v (in m/s) and the angular speed (in rad/s) of its wheels, which is v = rω, where r (in m) is the radius of a wheel.
However, for part (c), the answer you selected is incorrect. The correct relationship between the magnitude a (in m/s^2) of its linear acceleration and the magnitude α (in rad/s^2) of the angular acceleration of its wheels is a = rα, where r (in m) is the radius of a wheel. So the correct answer would be "Yes. The relationship is a = r, where r (in m) is the radius of a wheel."