I'm trying to analyse the following situation, which is part of a bigger project, and no, it's not a homework. If I want the wheel with a mass of "m" and radius of "r", to accelerate ("a") uphill until it reaches a predetermined velocity, what would the requiered torque "M" on the COM of the wheel be, if for example, the wheel was powered by an electromotor?

This is the FBD I've drawn:

..where:

• $F_d$ is the dynamic component of the force of gravity
• $F_s$ is the static component
• $F_{tr}$ is the static friction
• $F_N$ is the normal force of the surface upon the wheel
• $m$ is mass
• $r$ is radius
• $a$ is acceleration
• $M$ is torque or momentum (sorry for any confusion)

Don't mind the M on the FBD. It's just a visual representation of the induced torque to drive the wheel uphill. Also, I know for a fact, that the force of friction is pointing in the direction of the movement (University Physics, site 319).

For the situation here, this is the equation I've written:

$\ -F_D+F_{tr}= \ ma_x$

Which equals to:

$\ -mg\sin (\alpha)+F_{tr}= \ ma_x$

I know the acceleration, the mass of the body and the angle of incline. My question is, is it safe to assume that, if I solve the last equation for $\ F_{tr}$ and multiply it by radius "r", I get the necessary torque to drive the wheel uphill with the desired acceleration?

I've read on many occations, also on some other questions that in order for the car, for example, to move uphill, it has to overcome the friction force. Is it really that straightforward?

There are many examples of a body accelerating downhill, but I've never seen a detailed dynamic model, that is also backed by source or an example where the body is forced uphill my an external power source. Is the problem the same, just reversed? The friction which is pointing up the hill in both cases (body going down or uphill) tells me otherwise.