Maximum Acceleration and Coefficient of Friction

[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. Homework Statement

(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:

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

Homework Equations

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

The Attempt at a Solution

[/B]
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!

 

[haruspex]

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)?

Yes, and this can be predicted by dimensional analysis.

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?

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).

The difference is in how quickly the car and I can reach that acceleration, right?

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.

 

[Jazz]

Yes, and this can be predicted by dimensional analysis.

Thanks for answering.

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).

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.

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.

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

 

[haruspex]

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

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

 

[HalfLostAndSearching]

First of all, great job on your attempt at solving the problem! Your understanding of Newton's First Law and the equations for net force and maximum static friction are all correct.

To answer your doubts, yes, the maximum acceleration does indeed depend on the coefficient of friction and the mass of the object being accelerated. This is because the maximum static friction is directly proportional to the normal force (which is equal to the weight of the object in this case), and the net force is equal to the mass times acceleration. Therefore, the mass does not cancel out in the equation, but rather plays a crucial role in determining the maximum acceleration.

In the case of your bicycle and a racing car, even though they may have the same coefficient of friction on the same road, their mass and other factors such as engine power and aerodynamics will affect their maximum acceleration. So, the car may be able to reach a higher maximum acceleration due to its larger mass and other factors.

Lastly, you are correct in assuming that other factors such as air resistance (drag force) may come into play in real-world situations. These factors may affect the maximum acceleration achieved by an object, but they are not included in this problem and are more complex to calculate.

Overall, your understanding of the concepts is solid and you are thinking critically about the problem. Keep up the good work and good luck with your studies in physics!