Determining the minimum amount of torque to make wheels start rolling

Question

Context: I am building a cart with mass m (including the cart and the payload) that has 4 wheels. The 2 back wheels will be driven by motors. I am trying to determine the minimum amount of torque motors need to supply to the back wheels so that they starting rolling (not spinning or in other words rotating stationarily) from rest, moving the cart forward

My current thought process: Motor applies a torque to the wheel, the wheel in turn generates a force onto the road surface which, in return, applies an opposite and equal force onto the wheel surface. This F_friction (road on wheel) causes a counter torque opposing the motor torque (FBD is in figure 1)

So with reference to a block sliding on a surface (figure 2) where one needs to apply a force that exceeds maximum static friction for the block to start moving forward from rest.

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My question is, is it the same thing for the wheels where the motor needs to supply a torque that exceeds the torque caused by F_friction (road on wheels)so that angular acceleraton can increase angular speed of the wheels from 0 (at rest)? This also leads to another question. When the wheels have gained a certain angular speed, how do we know if they spin (rotating stationarily without moving foward) or roll (rotating and moving forward)?

All the questions above lead to my final confusion, is the condition (motor torque exceeds torque caused by F_friction (road on wheels)), that helps the wheels gain angular speed from rest, the very condition that causes the wheels to spin (or slip, term used by many people) instead of rolling? Can anyone help me with some thoughts on my analysis so far and derive equations to determine the minimum amount of torque? Thank you.

Answer 1

There may be a confusion about what would happen if there were different amounts of torque from the motor.

If there were a low torque this would try and rotate the wheels, but the wheels wouldn't spin or skid, as the the friction from the road on the wheels would be large enough to prevent it.

In this case the cart would start rolling slowly forward.

So there isn't a minimum amount of torque.

Some more on this case: Imagine you were sitting in a wheelchair and tried to move forward by grabbing the top of the wheels and moving them forward. If the wheels don't skid, the force you provided goes into moving you and the wheelchair forward.

However there is a maximum amount of torque.

For the cart example, if the total torque $T$ from the motor is so large that it causes the wheels to try and rotate fast, this can cause the wheels to skid. That happens as follows:

$$T=F\times r$$

$$F=\frac{T}{r}$$

if the friction force from the road surface can't match this force the wheels will skid i.e if

$$F=\mu N \lt \frac{T}{r}$$

where $\mu$ is the coefficient of friction and $N$ is the normal force $\frac{mg}{2}$

so the wheels will skid if $$T \gt \frac{\mu mg r}{2}$$

Answer 2

If you have ever tried to move your car by pushing it, you know that a significant amount of force is required. The weight of the car deforms the bottom of the tires and to a much smaller extent the surface below the tires (and also the two surfaces of the bearings which support the axle). This is called “rolling friction”. Its like the wheels are trying to roll uphill. Even the steel wheels of a train car rolling on a steel track are subject to this effect. So there is a minimum of torque required to get and keep a vehicle moving which depends a lot on the material of the wheels and the surface below.