Inertial frame of reference question for stacked boxes

In summary: Both blocks would be moving relative to the frictionless ground at the same rate as each other, which is why static friction does not prevent motion.
  • #1
khakitat
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0
Hello,

I was wondering about a question and how it would be reconciled within Newton's laws of motion. Take a case where two boxes are stacked on top of each other, and the bottom box rests on a frictionless surface. Now, imagine a rope is attached to the top box, and tension is applied to the rope. Also, imagine that there is static friction on the surface where the top box meets the bottom box.

Imagine the tension applied is less than the max value of static friction that can be achieved at that interface. From this case, it would seem that static friction would match tension when viewed at rest from the ground (the inertial reference frame) but the system of boxes would still be seen to accelerate from the ground since the bottom box is on a frictionless surface and is pushed by the top boxes static frictional force.

It seems that one from the ground would conclude that the forces on the top box were balanced despite the fact that both of the boxes as a system would accelerate. I feel my misunderstanding has something to do with reference frames, but am not sure how to answer this myself.

Any help is appreciated.

Thanks,
 
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  • #2
khakitat said:
it would seem that static friction would match tension
Why?
 
  • #3
That is the nature of how static friction at that interface would work. As tension was applied to the top box, the static friction would match the tension as long as tension was at or below the max static friction.
 
  • #4
khakitat said:
That is the nature of how static friction at that interface would work.
Says who?
 
  • #5
fsta.gif

This diagram is from hyper physics. Read the boxed information in the middle of the graph.
 
  • #6
Static friction does not prevent motion, it only prevents slipping. It assumes any value necessary to prevent slipping up to the max.
 
  • #7
khakitat

This assumes that the acceleration of the body is zero. This is not the case here.
But indeed this is a case worth considering.
In the ground reference frame the difference between T and the friction force will be equal to the mass of the top body times its acceleration.
 
  • #8
khakitat said:
This diagram is from hyper physics. Read the boxed information in the middle of the graph.
What nasu said.
 
  • #9
nasu said:
khakitat

This assumes that the acceleration of the body is zero. This is not the case here.
But indeed this is a case worth considering.
In the ground reference frame the difference between T and the friction force will be equal to the mass of the top body times its acceleration.
So, what you are saying is that static friction must actually be less than tension in that case, and that the amount less depends on the acceleration?

I think the reason this explanation gives me a problem is because if this is the case where tension is greater than static friction, it would seem that the top block should begin to move relative to the bottom block.

I think what does make this correct in my head though, is despite the fact that it would seem the top block would begin to move relative to the lower block, the lower block would be moving relative to the frictionless ground at the same rate as the top block is moving relative to the bottom block therefore causing both blocks to remain static relative to each other. Does this seem like a proper interpretation, or have I missed something?

Thanks,
 
  • #10
khakitat said:
it would seem that the top block should begin to move relative to the bottom block.
This can happen, if the static friction is not sufficient to accelerate the lower block at the same rate as the upper one accelerates.
 
  • #11
Yes. This is a correct interpretation. But you can get a much better understanding if you use free body diagrams. Draw a free body diagram for each of the boxes, showing the external forces acting on each. Now write a force balance on each of the boxes, based on the free body diagram. In these force balances, set the acceleration of each of the boxes to the same acceleration a. What do you get?

Chet
 
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Likes khakitat
  • #12
khakitat said:
So, what you are saying is that static friction must actually be less than tension in that case, and that the amount less depends on the acceleration?

I think the reason this explanation gives me a problem is because if this is the case where tension is greater than static friction, it would seem that the top block should begin to move relative to the bottom block.

I think what does make this correct in my head though, is despite the fact that it would seem the top block would begin to move relative to the lower block, the lower block would be moving relative to the frictionless ground at the same rate as the top block is moving relative to the bottom block therefore causing both blocks to remain static relative to each other. Does this seem like a proper interpretation, or have I missed something?

Thanks,
Both situations can happen. They move together (same acceleration) for small values of tension, up to a threshold value.
After that, the top body starts to slide over the bottom one (which moves too, but with a different acceleration).
To see all these, do the math, starting with the free body diagrams, as was suggested already.
 
  • #13
Chestermiller said:
Yes. This is a correct interpretation. But you can get a much better understanding if you use free body diagrams. Draw a free body diagram for each of the boxes, showing the external forces acting on each. Now write a force balance on each of the boxes, based on the free body diagram. In these force balances, set the acceleration of each of the boxes to the same acceleration a. What do you get?

Chet[/QUOT
Chestermiller said:
Yes. This is a correct interpretation. But you can get a much better understanding if you use free body diagrams. Draw a free body diagram for each of the boxes, showing the external forces acting on each. Now write a force balance on each of the boxes, based on the free body diagram. In these force balances, set the acceleration of each of the boxes to the same acceleration a. What do you get?

Chet
When I do that, it shows that the tension is the net force for the system of boxes, and it shows that when the system is accelerating uniformly, the tension varies from greater than 1 to less than or equal to 2 times the value of static friction currently acting depending on the ratio of masses between the boxes. As the mass of the top box gets larger in relation to the mass of the bottom box, the tension can increase. Therefore, it appears that if they are boxes of equal mass, the max tension can be twice that of max static friction and still cause uniform acceleration for the boxes as a system.

This appears to mean that uniform acceleration of the system will occur until the Tension applied is greater than twice that of the max static friction possible. It also appears to mean that in this case, static friction will assume its value between 1/2 and up to 1 (depending on mass) times the amount of tension. When the system would accelerate uniformly, it would seem that static friction would assume half of tension's value when the masses were equal, and could approach but not be equal to tension as the ratio of masses got smaller.

This is an interesting result to me. Have I made some mistake in this interpretation?

Thank you very much for your time,
 
  • #14
khakitat said:
When I do that, it shows that the tension is the net force for the system of boxes, and it shows that when the system is accelerating uniformly, the tension varies from greater than 1 to less than or equal to 2 times the value of static friction currently acting depending on the ratio of masses between the boxes. As the mass of the top box gets larger in relation to the mass of the bottom box, the tension can increase. Therefore, it appears that if they are boxes of equal mass, the max tension can be twice that of max static friction and still cause uniform acceleration for the boxes as a system.

This appears to mean that uniform acceleration of the system will occur until the Tension applied is greater than twice that of the max static friction possible. It also appears to mean that in this case, static friction will assume its value between 1/2 and up to 1 (depending on mass) times the amount of tension. When the system would accelerate uniformly, it would seem that static friction would assume half of tension's value when the masses were equal, and could approach but not be equal to tension as the ratio of masses got smaller.

This is an interesting result to me. Have I made some mistake in this interpretation?

Thank you very much for your time,
It's hard to tell without our seeing your actual equations. I don't want to take the time to do the problem myself, but I can tell from seeing your equations if what you say is correct.

Chet
 

1. What is an inertial frame of reference?

An inertial frame of reference is a coordinate system in which Newton's first law of motion holds true. This means that an object at rest will remain at rest and an object in motion will continue in a straight line at a constant speed, unless acted upon by an external force.

2. How does an inertial frame of reference relate to stacked boxes?

In the context of stacked boxes, an inertial frame of reference is important because it allows us to understand the forces acting on the boxes and how they will behave. It helps us determine if the boxes will remain at rest or if they will start to move due to an external force.

3. What happens to the boxes if they are in an inertial frame of reference?

If the boxes are in an inertial frame of reference, they will remain at rest if they were initially at rest. If they were initially in motion, they will continue to move at a constant speed in a straight line, unless acted upon by an external force.

4. How is an inertial frame of reference different from a non-inertial frame of reference?

An inertial frame of reference follows the principles of Newton's laws of motion, while a non-inertial frame of reference does not. In a non-inertial frame, objects may appear to accelerate or experience fictitious forces due to the frame's acceleration or rotation.

5. How is the concept of inertia related to an inertial frame of reference?

Inertia is the property of matter that causes objects to resist changes in their state of motion. In an inertial frame of reference, objects will exhibit this resistance to changes in motion as described by Newton's first law of motion.

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