Max distance a truck can travel without box falling off

[I_Try_Math]

Homework Statement

A box rests on the (horizontal) back of a truck. The coefficient of static friction between the box and the surface on which it rests is 0.24. What maximum distance can the truck travel (starting from rest and moving horizontally with constant acceleration) in 3.0 s without having the box slide?

Relevant Equations

F = m*a
x = x_0 + v_0*t + (1/2)*a*t^2

The equation at the bottom is me attempting to solve for the distance. Without knowing the mass of the box and truck my approach to this problem isn't possible?

 

[erobz]

Draw a FBD of the box, then examine Newtons Second Law for it.

 

[Delta2]

Hint 1: Just focus on the box, the truck is part of the problem but is not the "core" part of the solution...

The no slide condition means that the acceleration of the truck must be always the same as the acceleration of the box.

Hint 2: Forget the no slide condition momentarily and think: what is the maximum acceleration that the box can have? beware not maximum velocity but maximum acceleration.

 

[I_Try_Math]

Hint 1: Just focus on the box, the truck is part of the problem but is not the "core" part of the solution...

The no slide condition means that the acceleration of the truck must be always the same as the acceleration of the box.

Hint 2: Forget the no slide condition momentarily and think: what is the maximum acceleration that the box can have? beware not maximum velocity but maximum acceleration.

I was able to find the answer (~10.6 meters) with your hints, thank you

 

[kuruman]

The question posed by this problem makes no sense to me. If the acceleration of the truck is constant, all the forces acting on the box are constant. Why would the box start sliding at ~10.6 m and not at 8 m or 2 m? What does distance traveled have to do with the imbalance of forces that is needed to cause the box to (all of a sudden) start moving relative to the bed of the truck?

Judging from the answer, the correct formulation of the question should be something like "What distance will the truck cover in 3 seconds while the box is on the verge of starting to slide?" Then the "correct" answer will make sense.

 

[Delta2]

Judging from the answer, the correct formulation of the question should be something like

You are very right, the formulation of what is asked to find is not what it should have been. Something wrong with the exact wording of question, but this is very common in these forums ...

 

[erobz]

The question posed by this problem makes no sense to me. If the acceleration of the truck is constant, all the forces acting on the box are constant. Why would the box start sliding at ~10.6 m and not at 8 m or 2 m? What does distance traveled have to do with the imbalance of forces that is needed to cause the box to (all of a sudden) start moving relative to the bed of the truck?

Judging from the answer, the correct formulation of the question should be something like "What distance will the truck cover in 3 seconds while the box is on the verge of starting to slide?" Then the "correct" answer will make sense.

We are to find the maximum distance the truck can travel in 3 seconds without having the box slide. It could for instance go 1 meter in 3 seconds (under constant acceleration) without slippage - that's fine- but its not the farthest it could travel. It cannot go farther than 10.6 meters in 3 seconds (under constant acceleration). 10.6 meters is the maximum distance the truck could travel under constant acceleration from rest, such that the box does not slip.

I’m not seeing your issue with the problem statement.

 

[kuruman]

It cannot go farther than 10.6 meters in 3 seconds (under constant acceleration). 10.6 meters is the maximum distance the truck could travel under constant acceleration from rest, such that the box does not slip.

Why? If the box is not slipping at 10 m and starts slipping at 10.6 m what has changed to cause it to slip? In the non-inertial frame of the truck all the forces on the box are balanced and it is at rest. What force imbalance causes the box to start moving all of a sudden? The acceleration is constant and does not depend on distance traveled.

 

[erobz]

Why? If the box is not slipping at 10 m and starts slipping at 10.6 m what has changed to cause it to slip? In the non-inertial frame of the truck all the forces on the box are balanced and it is at rest. What force imbalance causes the box to start moving all of a sudden? The acceleration is constant and does not depend on distance traveled.

The box doesn't start slipping at 10.6 m, it can go on indefinitely without slipping. 10.6 m is just the farthest distance the truck could travel in the 3 second time interval of constant acceleration while conscientious of the no-slip criteria of the package it carries (i.e. for the truck(and package) $a_{max} = \mu_s g$).

I've highlighted what I think you are missing.

What maximum distance can the truck travel (starting from rest and moving horizontally with constant acceleration) in 3.0 s

 

[kuruman]

The box doesn't start slipping at 10.6 m, it can go on indefinitely without slipping.

That's exactly my point. It's the placement and use of the word "maximum" that I find objectionable. I picture in my mind a truck accelerating at the constant threshold acceleration. If the distance traveled in 3 s is maximum, so must be the distance traveled in any number of seconds. It's the acceleration that is maximum not the distance.

 

[haruspex]

If the distance traveled in 3 s is maximum, so must be the distance traveled in any number of seconds. It's the acceleration that is maximum not the distance.

Yes, the choice of 3 seconds is arbitrary. So? I see no problem with the question wording. Except… I would omit that the acceleration is constant. That should be for the student to figure out.