# Maximum Acceleration and Coefficient of Friction

1. Oct 21, 2014

### Jazz

I'm teaching myself Physics (I really want to study Physics in college), and this is one of the few places where I can ask for help. I hope my questions aren't so silly.

1. The problem statement, all variables and given/known data

(a) If half of the weight of a small $1.00×10^3kg$ utility truck is supported by its two drive wheels, what is the magnitude of the maximum acceleration it can achieve on dry concrete?

(b) Will a metal cabinet lying on the wooden bed of the truck slip if it accelerates at this rate?

Given/known data:

$m = 1.00×10^3\ kg$
$\mu_{s(rubber-dry\ concrete)} = 1.0$
$\mu_ {s(metal-wood)} = 0.5$

2. Relevant equations

$F_{net} = ma$
$f_{s(max)} = \mu_sN$

3. The attempt at a solution

To solve (a), I understand than even when the wheels are rotating they are stationary relative to the ground. So as long as they are not slipping (by hitting the brakes hardly or by skiing on a wet surface) the following, I think, should hold:

$F_{net} = f_{s(max)}$
$F_{net} = \mu_{s(rubber-dry\ concrete)}N(0.5)$
$ma = \mu_{s(rubber-dry\ concrete)}mg(0.5)$
$a = \mu_{s(rubber-dry\ concrete)}g(0.5)$
$a = (1.0)(9.8\ m/s^2)(0.5)$
$a = 4.90\ m/s^2$

In the case of (b), I think the slipperiness occurs because of Newton’s First Law. The cabinet will remain at rest while the truck will be moving at $4.90\ m/s^2$. But this would be the same as the cabinet accelerating at $4.90\ m/s^2$ in the opposite direction while the truck remains at rest; but only if:

$m_{cabinet}a > \mu_ {s(metal-wood)}m_{cabinet}g$
$a >\mu_ {s(metal-wood)}g$
$4.90\ m/s^2 > (0.5)(9.8\ m/s^2)$
$4.90\ m/s^2 > 4.90\ m/s^2$

Since the inequality doesn't hold, the cabinet will not slide.

Doubts:

As you can see, in (a) and (b) mass cancels. Does it mean that the maximum acceleration only depends on the $\mu_s$ of the surfaces (and on the planet I’m driving)?

if my bicycle’s wheels and the wheels of a racing car have the same $\mu_s$ (on the same road with the same conditions), does it mean that maximum acceleration of both is the same? The difference is in how quickly the car and I can reach that acceleration, right?

Probably I'm neglecting other things that come into play (like drag force), but is this assumption theoretically correct?

Thanks!!

2. Oct 21, 2014

### haruspex

Yes, and this can be predicted by dimensional analysis.
Yes, except that the weight may be distributed differently between front and back. As your calculation has shown, the greater the share of load on the driving wheels the better the acceleration (but the worse the steering).
There need be no delay n reaching maximum acceleration. It may not feel like it, but when you put your full weight on the forward pedal, in horizontal crank position, in bottom gear, you are at maximum acceleration immediately.

3. Oct 21, 2014

### Jazz

So, as an example, this can be considered one of the reason why some tractors have a sort of movable axle to use when transporting heavy load.

Then, the difference between the two is in how much times that acceleration can be mantained, right?

4. Oct 21, 2014

### haruspex

It is likely that human legs and a car engine have different performance characteristics, but I'm no expert on such matters.