Force exerted on ground from a slipping wheel

Question

I am trying to do some car calculations for a game.

I have a tyre (radius $r$, inertia $I$) rotating around it's axle with an angular velocity $\omega$ faster than the center of mass is moving across the surface. So the tyre is not rolling along the ground, but slipping. The surface has a friction coefficient $\mu$. The tyre is pressed down by the car with a weight $N$

I need to calculate two things.

1. The force the tyre exerts on the ground.
2. The resulting (de)acceleration of the tyre.

Imagine a car doing a wheelspin. I need to find the force the excess speed of the tyre transfers to the ground, giving an acceleration to the car, and thereby also de deacceleration of the tyre.

I have calculations for the external forces (engine, brakes, drag, lateral slipping) and this is the final piece to complete the puzzle. So help is much appreciated, as i have been trying to solve this in a myriad of ways, with no success.

//Edited - It is the wheels connected to the enigne i am trying to calculate. However i am in this case not interested in the torque coming from the engine, only actions from the extra energy stored in wheel from building up the spin.

The engine provides a torque $\tau$ on the tyre

Answer 1

To answer the first part of your question, I'm going to assume that the center of mass of the car is at the center of the car itself. This is usually not true of real cars - the engine block is quite heavy and is usually positioned well forward of the center of the car. In the absence of any information about how heavy the engine is and where it is, I'm going to assume a symmetrical car.

The force that the tyre exerts on the ground is equal to the weight of the tyre itself plus one-fourth the weight of the car. Any other force that the tyre might exert on the ground would result in a car that jumps up off the ground or digs into the ground, which is clearly not the case.

There is not enough information to answer the second part of your questions. Once you break the rolling-without-slipping assumption, we need to have some information about the torque on the tyre.

To clarify why we need the torque on the tyre, consider this: for all we know the tyre has a torque of say one billion N-m on it, and will begin accelerating much faster than the speed of sound or anything else. Once the rolling-without-slipping condition is discarded, no assumptions can be made about the rotational rate of the tyre.