Understanding relative friction

[Chenkel]

TL;DR Summary

In the book I'm reading "Essential calculus-based physics by Chris McMullen": A problem is given of a box of bananas on a truck. The coefficient of friction is $\frac 1 5$. What is the maximum acceleration that the truck can have without the box of bananas sliding backward relative to the truck.

Hello everyone!

I'm reading this book and trying to get a more concrete understanding of friction and it's relation to Newton's third law. So in the solution he writes$$D - f - F_R = m_ta_x$$Where D is the driving force of the truck, f is the frictional force of the box on the truck, and $F_R$ is the resistant force of the truck, I don't really feel that last variable $F_R$ is necessary, but I might be wrong.) $m_t$ is the mass of the truck, and $m_a$ is the acceleration of the truck and the box. This equation confuses me a little, I think it's talking about f as the maximum friction force that is required for the box to slide.. The box is carried forward by the friction, so there's a frictional force on the truck, which acts opposite the truck's direction of motion, and an equal and opposite force on the box, which acts in the direction of the box and the truck. I'm wondering if anyone can clearify the relationship friction has to the box/truck.

Let me know what you guys think, thank you!

 

[Doc Al]

Keep it simple. What's the maximum static friction between truck and box? That should tell you the maximum acceleration of the box (and truck) without the box sliding.

 

[Chenkel]

Keep it simple. What's the maximum static friction between truck and box? That should tell you the maximum acceleration of the box (and truck) without the box sliding.

We have$f=\frac 1 5 m_bg = m_ba_x$ so let g = 10 and we have $a_x = \frac {10} 5 = 2$ I know it's a simple solution, but for some reason I have trouble wrapping my brain around it, how does friction carry the box forward?

Thank you for the response.

 

[Doc Al]

We have$f=\frac 1 5 m_bg = m_ba_x$ so let g = 10 and we have$a_x = \frac 10 5 = 2$ I know it's a simple solution but for some reason I have trouble wrapping my brain around it, how does friction carries the box forward?

The only force accelerating the box is the friction from the truck.

Max friction force = μmg; thus, μmg = ma, giving a = μg.

 

[Chenkel]

The only force accelerating the box is the friction from the truck.

Max friction force = μmg; thus, μmg = ma, giving a = μg.

For some reason simple things elude me, I'll meditate on your answer and see if I can make some sense of things. Thanks again.

 

[Chenkel]

The only force accelerating the box is the friction from the truck.

Max friction force = μmg; thus, μmg = ma, giving a = μg.

I wonder why friction is an accelerating force that speeds up an object, I'm used to it acting against the force moving the object, so it's hard for me to see the analogy with the box/truck example since both things are going in the same direction.

 

[Doc Al]

I wonder why friction is an accelerating force that speeds up an object, I'm used to it acting against the force moving the object, so it's hard for me to see the analogy with the box/truck example since both things are going in the same direction.

The only horizontal force acting on the box is the friction force. So that's the only force determining the acceleration of the box.

If there were no friction, the box wouldn't accelerate no matter what the truck did. But with friction, the box can accelerate along with the truck -- up to a point. What determines that point is the maximum friction force available to accelerate the box.

 

[Doc Al]

But I understand your point. In many static friction problems, the static friction is opposing some applied force. This is a bit different.

 

[Dale]

I wonder why friction is an accelerating force that speeds up an object, I'm used to it acting against the force moving the object, so it's hard for me to see the analogy with the box/truck example since both things are going in the same direction.

Friction prevents slipping. You cannot blindly assume that means that it slows things down. Sometimes things have to accelerate to prevent slipping.

 

[Chenkel]

But I understand your point. In many static friction problems, the static friction is opposing some applied force. This is a bit different.

In this example the friction is the applied force (that differs from classical problems where the friction force opposes the applied force), and that frictional force is being applied by the truck to the box, and the box to the truck. If you consider the truck/box system, the friction is no longer an external force as we usually treat it in classical problems, now the frictional force is internal to the truck/box system. I'm still trying to understand how friction accelerates an object, that simple thing eludes me. I appreciate your feedback.

 

[Chenkel]

Friction prevents slipping. You cannot blindly assume that means that it slows things down. Sometimes things have to accelerate to prevent slipping.

This makes me think of analyzing the problem in terms of a reference frame that represents the bed of the truck, while the object is not slipping there is some force A opposing the frictional force B, while the object is not slipping A + B = 0. I'm still trying to come to terms of how friction can carry an object forward. I appreciate your feedback.

 

[Doc Al]

If you consider the truck/box system, the friction is no longer an external force as we usually treat it in classical problems, now the frictional force is internal to the truck/box system.

And for some purposes, it's perfectly fine to treat the truck + box as a single system.

I'm still trying to understand how friction accelerates an object, that simple thing eludes me. I appreciate your feedback.

Think about that truck and how its tires grip the road. When you step on the gas, what ends up accelerating the truck is the friction between tires and ground.

 

[Doc Al]

This makes me think of analyzing the problem in terms of a reference frame that represents the bed of the truck, while the object is not slipping there is some force A opposing the frictional force B, while the object is not slipping A + B = 0.

You could, but realize that the reference frame of the truck is accelerating. You'll have to add "fictional" (inertial) forces to make use of Newton's laws in that frame. Do that after you understand how friction can accelerate something.

 

[Chenkel]

Friction prevents slipping. You cannot blindly assume that means that it slows things down. Sometimes things have to accelerate to prevent slipping.

If the box is set on the bed of the truck, and the truck accelerates, the box wants to stay with the truck to prevent slipping. The frictional force keeping the box from sliding relative to the truck is proportional to the acceleration of the truck, but it's maximum value is proportional to the acceleration of gravity because $a=\mu g$, hopefully I'm making sense.

 

[Chenkel]

In the example of a frictional force counteracting an applied force, it makes sense to me that the frictional force reaches some maximum.

And I suppose I also understand how a force is used to accelerate an object to prevent slipping, that seems almost non negotiable for any object with non zero static friction coefficient, from an intuitive and mathematical perspective, but one problem is for me to understand why that force reaches a maximum in that particular case.

These are two different examples, one where the frictional force counteracts the applied force, and the other where the frictional force prevents slipping but reaches a maximum, the latter is a little confusing to me.

Thank you everyone for your guidance, I feel I'm slowly putting the puzzle together

 

[erobz]

I think why there is a maximum starts to come more into focus when you consider microscopically what is going on:

This is what this mating surfaces of the box and truck look like. There is a certain amount of acceleration (inertial forces) or force that must be applied so the blue surface (box surface) can climb over the peaks and valleys of the truck bed (black line). Thats how someone explained it to me once, I'm not saying its fully developed theory (and I'm waiting to be lambasted for presenting it), but I think it answers the question you have.

 

[Chenkel]

I think why there is a maximum starts to come more into focus when you consider microscopically what is going on:

View attachment 304028

This is what this mating surfaces of the box and truck look like. There is a certain amount of acceleration (inertial forces) or force that must be applied so the blue surface (box surface) can climb over the peaks and valleys of the truck bed (black line). Thats how someone explained it to me once, I'm not saying its fully developed theory (and I'm waiting to be lambasted for presenting it), but I think it answers the question you have.

I believe I have some degree of a working understanding of it. The box on top has the same acceleration as the truck, so we know a 'frictional force' is applied to the box from the truck, and is $f = F_b = m_ba$, this is non negotiable since the box has the same acceleration as the truck, the truck has an equal and opposite force onto it, so we can write $D - f = m_ta$ where D is the driving force of the truck, we know that f will be growing as the driving force increases, but have a maximum value of $f = m_ba = \mu g m_b$ so we divide by $m_b$ and find the acceleration of the truck when the frictional force is at it's max, and that acceleration is $\mu g$. After the acceleration of the truck is larger than, or equal to $\mu g$ the acceleration of the truck will increase for larger driving force, but the box's acceleration will remain constant. Of course the acceleration of the box will be different than the trucks once the frictional force is greater than or equal to $\mu g$ but I thought it would be more simple to write the equations with one acceleration variable.

 

[PeroK]

What if you replace "static friction" with a little glue? What happens then?

 

[Chenkel]

What if you replace "static friction" with a little glue? What happens then?

I'm guessing you mean what happens if there's glue between the two objects, I imagine the frictional constant would be greater than 1 most likely.

 

[PeroK]

I'm guessing you mean what happens if there's glue between the two objects, I imagine the frictional constant would be greater than 1 most likely.

Not necessarily. With a little bit of glue perhaps the box could be accelerated only so fast before the glue fails? Is that really so different from static friction?

 

[Chenkel]

Not necessarily. With a little bit of glue perhaps the box could be accelerated only so fast before the glue fails? Is that really so different from static friction?

I believe this problem can probably be modeled with static friction, so I don't believe it's very different from static friction

 

[PeroK]

I believe this problem can probably be modeled with static friction, so I don't believe it's very different from static friction

It's not clear what you think the box should do when the truck accelerates? Always slide backwards relative to the truck? Even then, from the ground frame, kinetic friction is accelerating the block. That leaves the only option that the box remains at rest relative to the ground and slides smoothly across the truck bed.

Does that mean the box could be nailed down or glued to the truck, but cannot be held in place by friction?

 

[Chenkel]

It's not clear what you think the box should do when the truck accelerates? Always slide backwards relative to the truck? Even then, from the ground frame, kinetic friction is accelerating the block. That leaves the only option that the box remains at rest relative to the ground and slides smoothly across the truck bed.

Does that mean the box could be nailed down or glued to the truck, but cannot be held in place by friction?

In my understanding I see the more rough the surface between the box and the truck the higher $\mu$ so the force of the truck onto the box has a higher maximum for a rougher surface. As soon as the force pushing the block along with the truck riches a maximum value the box will start sliding in the bed of the truck. I'm still a little confused about some aspects, but I have more of a feel for the physics than earlier. I'm wondering what happens at the microscope level when the friction force reaction it's maximum value.

 

[anorlunda]

I'm wondering what happens at the microscope level when the friction force reaction it's maximum value.

At the microscopic level, things become very messy. Each atom might be experiencing more/less/zero force at any particular instant. Forces on boundary atom can cause the crystal lattice of the solid to deform in many ways.

Friction is an average, and the averaging wipes out the microscopic differences. So when you study atomic forces with each atom modeled separately, there is no such thing as friction.

The same applies with temperature. Temperature is defined only for the average behavior of many particles. Look at just one particle and there is no such thing as temperature.

 

[PeroK]

I'm wondering what happens at the microscope level when the friction force reaction it's maximum value.

At that point the loose bonds between the surfaces break and you transform to a kinetic friction model.

 

[Dale]

At the microscopic level, things become very messy. Each atom might be experiencing more/less/zero force at any particular instant. Forces on boundary atom can cause the crystal lattice of the solid to deform in many ways.

And this microscopic messiness is exactly why friction produces heat.