# Maximum acceleration uphill of a truck - not sure what the equations mean

I'm trying to solve a problem that reads:

The coefficient of static friction between a truck's tires and the road is 0.850. What is the maximum acceleration uphill that the truck can have if the road is tilted 12 degrees to the horizontal?

I drew a free body diagram and came up with the following equations:

X direction: $f=ma+mgsin\theta = \mu_sF_n$

Y direction: $F_n = mgcos\theta$

Substituting $mgcos\theta$ into the x direction equation and solving for $a$ gives:

$a=g[\mu_s cos\theta-sin\theta]$

At this point I'm not sure what this last equation means or how to find the maximum acceleration from it. It's saying that the acceleration depends only on the force of gravity, the angle of the incline, and the coefficient of static friction. I'm confused because shouldn't the acceleration depend on things like the horsepower of the engine as well as the variables in the equation? Does this equation give the value of some kind of hard cap as to how much friction can contribute to movement given a certain coefficient of friction?

Thanks!

• Imagine that the truck has access to a, say, 25,000 hp engine, and that you simply attach that engine to the wheel and fire it up. What happens to the wheels? – dmckee Mar 20 '15 at 0:17

Its true that the truck's acceleration does depend on the engine specifications.

However, the maximum acceleration the truck can have is the same as the maximum friction of the wheels from the force of static friction. We don't want the truck wheels to slip. If it accelerates above the static friction force then the wheels will begin to slip. Without this force of friction the truck would not move at all.

$$a=g[\mu_s\cos \theta − \sin \theta]$$ is correct; plugging the values in will give you the maximum acceleration of the truck.

As an additional afterthought to the problem. The fact that the maximum acceleration does not depend on mass implies that it can be said that the maximum acceleration of the truck does not depend on the number of wheels that it has. Assume a truck with 8 wheels (an APC weighing m1=4 tons), versus a truck with 4 wheels (a SUV weighing m2=1 ton), versus a bike with 2 wheels (m3=20 lbs); the number of wheels reduces the mass parameter used in Newton's second law by a fraction of the total. The m parameter in your equations could be written $$m = m1/8$$, or $$m = m2/4$$, or $$m = m3/2$$ This mass parameter would still cancel out in the end, the APC, the SUV, and the Bike all would have the same maximum acceleration.

• I think I follow now. Like you said, the truck is accelerated by friction, and the maximum force of static friction that the ground can deliver is $\mu_s F_n$, which is what I set the friction in the FBD equations to equal. Maximum friction $\rightarrow$ maximum acceleration. Thanks! – user47989 Mar 20 '15 at 19:25

"Does this equation give the value of some kind of hard cap as to how much friction can contribute to movement given a certain coefficient of friction?"

Yes. If it tries to accelerate any faster, it will skid.

Imagine trying to move as fast as you can on ice. You can walk on it, but if you try to move too fast you'll slide. Rubber on asphalt has much more friction, so a truck can accelerate a lot faster, but the same principle applies.