Does a car get heavier as its wheels spin faster, clasically?

Question

A parked car has weight equal to its mass times gravity, in other words, it exerts a force of magnitude mg in the direction normal to the ground, and if you were to weigh the car on a scale, it would show that weight. For a moment, imagine the car at rest in empty space and apply a torque to its wheels. By conservation of momentum, as the wheels begin to spin, the car would begin to spin in the opposite direction. Back on Earth, as the car's wheels begin to spin, the car also tries to spin in the opposite direction, but the ground keeps it from actually spinning, in other words, the car exerts a torque of some magnitude in the direction perpendicular to the ground and the ground exerts an equal and opposite torque that keeps the car from spinning. So the car with spinning wheels is exerting two forces directed perpendicular to the ground, and if you were weigh the car with spinning wheels on a scale, would it show a weight higher than for the parked car?

For this question, please ignore relativity.

Answer 1

The sum of the wheel pressures on the surface that is supporting the car will not change.

The torque that is associating with the wheels spinning up will shift the distribution of the weight of the car over the wheels, but the sum of the wheel pressures will not change.

Answer 2

Angular momentum is conserved on the earth without the car "spinning" because when the drive wheel spins (without slipping) during acceleration it technically causes the earth to spin in the opposite direction (though the change in angular velocity of the earth is infinitesimal because the moment of inertia of the earth is so huge compared to that of the wheel).

This all comes about because of the static friction force the earth exerts forward on the wheel that is equal and opposite to the force the wheel applies backward on the earth due to its torque, per Newton's 3rd law. Because both of these forces are parallel to the surface of the earth they have no effect on the weight of the car. That said, the weight of the car will shift on acceleration since the suspension system is not rigid.

Hope this helps.