How is Maximum Sustained Speed Calculated for a Car Driving Up an Incline?

[daniellionyang]

This is from the 2014 F=ma exam:
The maximum torque output from the engine of a new experimental car of mass m is τ. The maximum rotational speed of the engine is ω. The engine is designed to provide a constant power output P. The engine is connected to the wheels via a perfect transmission that can smoothly trade torque for speed with no power loss. The wheels have a radius R, and the coefficient of static friction between the wheels and the road is μ.

What is the maximum sustained speed v the car can drive up a 30 degree incline? Assume no frictional losses and assume μ is large enough so that the tires do not slip.

The solution is as follows:
The fundamental idea is P = Fv where F is the component of the weight parallel to the incline. Then
v = P/mg sin θ Since θ = 30◦, the answer is v = 2P/mg.

Why does the solution ignore the effects of friction when calculating F. Shouldn't F=mgsin(theta)+mu*Fn?
Thanks!

 

[russ_watters]

The vehicle is rolling up the hill, not sliding up it.

 

[daniellionyang]

So, in rolling without slipping, even if there is friction, we ignore it? (this means it doesn't affect the net force right?)

 

[russ_watters]

It depends on exactly what/where the friction is, but either way, in this case you were instructed to ignore it. The assumptions provided to you may or may not reflect how such problems are done in the real world: they are done to make it easier for you to solve the problem when you are learning it.

 

[daniellionyang]

Thanks! You mentioned that we ignore it "in this case." Are there any cases where we do not ignore this friction?

 

[russ_watters]

Thanks! You mentioned that we ignore it "in this case." Are there any cases where we do not ignore this friction?

The friction between the wheels and road? The problem states the criteria for why in this case it could be ignored. Flip that over...

 

[CWatters]

So, in rolling without slipping, even if there is friction, we ignore it? (this means it doesn't affect the net force right?)

I'm wondering if you misunderstand something... The friction mentioned in this problem is "static friction between the wheels and the road". Static friction doesn't act to slow the car down. Are you confusing this with rolling resistance or air drag?

The problem statement says two separate things..

Assume no frictional losses...

They mean ignore things like friction in the transmission, rolling resistance and air drag etc. Those would affect the net force on the car.

...and assume μ is large enough so that the tires do not slip.

You want high static friction between a tyre and the road to stop the wheels spinning. This is why drag racers like a nice sticky surface. If μ was low (eg ice) you would have to calculate if the static friction limited the amount of torque that could be applied at the wheels. In that case the max velocity would be slower or even zero... or even negative..