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Whenever a car accelerates, friction is pushing it forward. However, because the contact point doesn’t move relative to the earth, friction does no work. By conservation of angular momentum, the earths rotation speed should change, meaning the rotational kinetic energy of earth changes. Therefore, work was done on the earth. But where did the work come from if friction doesn’t do work?

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  • $\begingroup$ The contact point does move, as Kelly Shepphard writes (it is the geometrical point of contact, which moves along with the car). However, this motion is not relevant for work; instead it is the motion of the mass point of the Earth where the force acts. This mass point almost does not move, so work of car on Earth is indeed very close to zero. $\endgroup$ – Ján Lalinský Nov 27 '18 at 23:23
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It seems very common to be reluctant to accept that in an interaction between two bodies, it is possible to transfer a meaningful amount of momentum but a negligible amount of energy, but such is the case when the two bodies have vastly different masses.

Let's look at the car/Earth example. Say the car has a mass of $6 \times 10^2$ kg (it's a small car) and the Earth has a mass of $6 \times 10^{24}$ kg, and both begin perfectly at rest. The car begins applying torque to its wheels, and thus a frictional force arises between the car and Earth. After some time, the car is going $10$ m/s. It has gained $6 \times 10^3$ kg m/s of momentum. We know by conservation of momentum that the Earth has gained the same amount of momentum in the opposite direction. But because the Earth is $10^{22}$ times more massive, the velocity, and therefore the displacement of the car during the acceleration is $10^{22}$ times greater (ignoring, for simplicity, the rotation that would be imparted to the Earth, and only considering the center of mass motion--if we added that in, it would up the amount of energy transferred to the Earth by maybe a factor of 2 or something) than that of the Earth. Since work is force times distance, and the magnitude of the force is the same for car and Earth, the car gains $10^{22}$ times as much energy: the car gets $3\times 10^4$ J, and the Earth gets $3 \times 10^{-18}$ J.

There is a separate paradox also alluded to in this question: friction with the ground can do no work, as the point of application of the force does not move; and yet as it accelerates, friction is the only external force applied to the car. How can the car then gain kinetic energy? The resolution to this quandary is that the car's engine does work: it applies a torque to the wheels, and that torque acts through an angular displacement. Some of this work goes into spinning up the wheels, but because they are not allowed to slip on the ground, most of it goes into the translational kinetic energy of the car.

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In any mechanics problem the first thing to be done is to correctly choose the reference frame. Most mechanical quantities change value from one frame to another, and basic mechanics laws (Newton's) only hold true in an inertial frame.

Because of its mass and slow rotation we are usually authorized to take Earth as an inertial frame, but strictly speaking it isn't. Surely you can't assume Earth as your frame when you at the same time are interested in Earth's motion!

First of all, do not mix in one problem two different frames. This is what you are doing when yu begin saying

because the contact point doesn't move relative to the earth, friction does no work.

thus taking Earth as your frame, and later

the earth's rotation speed should change

Now you have switched to a frame not following Earth's rotation.

Very likely you have anticipated the final answer. In a frame where Earth is rotating, it isn't true that the contact point between tire and ground doesn't move.

Assume, to make things simple, your car is moving along the equator, due east, i.e. in the same direction of Earth's rotation. Then friction force of tire on ground is opposed to earth's motion, its work is negative, Earth's kinetic energy is bound to decrease. In the opposite case of car due west work of friction force is positive and Earth accelerates its rotation.

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Contact point doesn't move if and only if car momentally accelerates having zero speed. When car speeds up/slows down while moving, contact point moves (with a car itself). When real car speeds up, it's increasing speed gradually - so contact point moves.

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