# Finding the time it takes for two stacked blocks to travel

## Homework Statement

The coefficient of static friction is 0.604 between the two blocks shown. The coefficient of kinetic friction between the lower block and the floor is 0.104. Force F causes both blocks to cross a distance of 3.44m, starting from rest. What is the least amount of time in which the motion can be completed without the top block sliding on the lower block, if the mass of the lower block is 1.13kg and the mass of the upper block is 2.25kg?

## The Attempt at a Solution

I looked at the attached file that my teacher gave us
I found the normal of the top block which was 22.0 N
Since it equals one of the normals of the bottom block, I substituted in the same value in for calculating the other normal. This "newfound" normal was used to calculate the acceleration in the x direction using Fnet=ma ---> ma = friction static - friction kinetic

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nrqed
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## Homework Statement

The coefficient of static friction is 0.604 between the two blocks shown. The coefficient of kinetic friction between the lower block and the floor is 0.104. Force F causes both blocks to cross a distance of 3.44m, starting from rest. What is the least amount of time in which the motion can be completed without the top block sliding on the lower block, if the mass of the lower block is 1.13kg and the mass of the upper block is 2.25kg?

## The Attempt at a Solution

I looked at the attached file that my teacher gave us
I found the normal of the top block which was 22.0 N
Since it equals one of the normals of the bottom block, I substituted in the same value in for calculating the other normal. This "newfound" normal was used to calculate the acceleration in the x direction using Fnet=ma ---> ma = friction static - friction kinetic

Let's call the block under "A" et the block on top "B". The applied force is only on A? And it is horizontal?

What you have to do is to determine the maximum horizontal force on block B (top) such that it does not slide. This is given by the max static friction force. Calculate this first, and then set that equal to $m_B a$. That will give you the acceleration, which will be the acceleration of both blocks since they move as one.

Let's call the block under "A" et the block on top "B". The applied force is only on A? And it is horizontal?

What you have to do is to determine the maximum horizontal force on block B (top) such that it does not slide. This is given by the max static friction force. Calculate this first, and then set that equal to $m_B a$. That will give you the acceleration, which will be the acceleration of both blocks since they move as one.
does this maximum static friction force equal the force being applied to pull the boxes? In the FBD it indicates so but do i just calculate the mac static friction force alone?

nrqed
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does this maximum static friction force equal the force being applied to pull the boxes? In the FBD it indicates so but do i just calculate the mac static friction force alone?
No, it is not equal to the force applied. You need to use the equation for the maximum static friction force on an object (the one that contains $\mu_s$).

No, it is not equal to the force applied. You need to use the equation for the maximum static friction force on an object (the one that contains $\mu_s$).
So i get:

Fs = (coefficient of static friction) x (normal force of top block)
Fs = 13.3

13.3 = mb(a)
13.3= 2.25kg (a)
a = 5.91 m/s^2

d = Vit + 1/2at^2 (Vit vanishes because theres no initial velocity)
d = 1/2at^2
3.44 = 1/2 (5.91)t^2
3.44/2.955 = t^2 //sqrt both sides
1.08 s = t

this is apparently wrong :(

nrqed
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So i get:

Fs = (coefficient of static friction) x (normal force of top block)
Fs = 13.3

13.3 = mb(a)
13.3= 2.25kg (a)
a = 5.91 m/s^2

d = Vit + 1/2at^2 (Vit vanishes because theres no initial velocity)
d = 1/2at^2
3.44 = 1/2 (5.91)t^2
3.44/2.955 = t^2 //sqrt both sides
1.08 s = t

this is apparently wrong :(
I don't see any mistake. The force is only acting on block A, right?

I don't see any mistake. The force is only acting on block A, right?
yeah it is.

But what i wish i could show you is the diagram. I should probably mention that the top block has a rope attached to it and this is where that question i asked you about calculating static friction on its own came from. Knowing this, would it affect the outcome?

nrqed
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yeah it is.

But what i wish i could show you is the diagram. I should probably mention that the top block has a rope attached to it and this is where that question i asked you about calculating static friction on its own came from. Knowing this, would it affect the outcome?
AHH! It changes everything. I was assuming there were no other force than friction on the top block

AHH! It changes everything. I was assuming there were no other force than friction on the top block
yes i am so sorry ;(! thats why I was getting confused because I thought that the tension in the rope = static friction of the top block. Im back to square one now, please help!!!

haruspex
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So i get:

Fs = (coefficient of static friction) x (normal force of top block)
Fs = 13.3

13.3 = mb(a)
13.3= 2.25kg (a)
a = 5.91 m/s^2

d = Vit + 1/2at^2 (Vit vanishes because theres no initial velocity)
d = 1/2at^2
3.44 = 1/2 (5.91)t^2
3.44/2.955 = t^2 //sqrt both sides
1.08 s = t

this is apparently wrong :(
Your method would be right if F were applied to the lower block, but it is applied to the top block. Clearly, the friction between the lower block and ground will be important, but you have not considered it.

nrqed
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yes i am so sorry ;(! thats why I was getting confused because I thought that the tension in the rope = static friction of the top block. Im back to square one now, please help!!!
No, *I* am sorry, I opened the file you had attached, but I did not notice there were two pages!! I just saw the first one, so I did not realize there was a rope there. I am sorry

nrqed
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Your method would be right if F were applied to the lower block, but it is applied to the top block. Clearly, the friction between the lower block and ground will be important, but you have not considered it.
It is my fault, I had not seen the FBD and I gave instructions thinking that there were no horizontal forces on the top block. My mistake.

It is my fault, I had not seen the FBD and I gave instructions thinking that there were no horizontal forces on the top block. My mistake.
So if i calculate the force of the normal from the top block, according to the diagram it is the same magnitude for the bottom block? Or do i reverse its signs?

nrqed
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So if i calculate the force of the normal from the top block, according to the diagram it is the same magnitude for the bottom block? Or do i reverse its signs?
Watch out, there are two normal forces on the bottom block, one due to the contact with the top block and one with the floor. Calculate these two. Then calculate the net horizontal force on the bottom block, using for the static friction force the maximum value, which you had calculated before, $f_s = \mu_s m_{top} g$. Then the net horizontal force on the bottom block is $f_s - f_k$. Set this to $m_{bot} a$ and then do as before.

Watch out, there are two normal forces on the bottom block, one due to the contact with the top block and one with the floor. Calculate these two. Then calculate the net horizontal force on the bottom block, using for the static friction force the maximum value, which you had calculated before, $f_s = \mu_s m_{top} g$. Then the net horizontal force on the bottom block is $f_s - f_k$. Set this to $m_{bot} a$ and then do as before.
so if i calculate the normal force of the top block to be 22.0 N, since there are two normal forces acting on the bottom block, the normal force of the top block will act as one of the normal forces of the bottom block. Should it be 22.0 N or -22.0 N?

nrqed
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so if i calculate the normal force of the top block to be 22.0 N, since there are two normal forces acting on the bottom block, the normal force of the top block will act as one of the normal forces of the bottom block. Should it be 22.0 N or -22.0 N?
The magnitude is 22 N while the y component is -22 N. But for the calculation of the static friction force, $f_s$, you need to use the magnitude. You will of course get the same result as before.

The magnitude is 22 N while the y component is -22 N. But for the calculation of the static friction force, $f_s$, you need to use the magnitude. You will of course get the same result as before.
im still lost, I don't know what to do :(, any more hints?

haruspex
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im still lost, I don't know what to do :(, any more hints?
You need to analyse the forces and acceleration of the lower block.

nrqed
First step: calculate the maximum static friction force $f_s$ (it is the same static friction force on th stop block an don the bottom block by the action-reaction principle ). Second step: calculate the force of kinetic friction on the bottom block (due to the friction with the ground). Do you see how to do those two steps?