Force & Motion Homework: Force Diagrams, Velocities & Accelerations

[Nicholas Egebak]

Homework Statement

A small box (mass m1) is placed upon a large box (mass m2); the large box i placed on a horizontal table. The boxes begin a rest on the table. The lower box is hit by a horizontal blow at the time t=0 resulting in a initial velocity v0. The frictioncoefficient between all surfaces are µk

Homework Equations

a) Draw force-diagrams for both boxes as the move relative to each other. Indicate all Newtons-Third-Law-Pairs.

b) Determine all velocities and accelerations of the boxes as the move relative to each other. At what time T do the boxes not move relative to each other anymore?

c) Determine the acceleration of the boxes when they do not move relative to each other. What length L do the boxes move in this fase? The initial velocity can be concidered know as v₁

The Attempt at a Solution

I started drawing the force diagrams. No problem there. It was easy to see that the normal force in both cases were pairing up with the gravitational force.

For question b) I came up with this equation:

v_box1=v₀-a*t​

Applied on the second box (the one underneath the other):

The acceleration must be the kinetic friction force divided by the mass:

v_box1=v₀-(f_k)/m*t​

Naturally I got:

v_box1=v₀-μ_k*g*t​

For the box on top:

v_box2=-μ_k*g*t​

For the acceleration I differentiate both:

a_box1=(v_box2)'=-g*µ_k
a_box2=(v_box2)'=-g*µ_k

It is now that I am lost. I do not know how to find the time by which the boxes are not moving relatively to each other anymore...
Picture of scenario:

Force Diagrams:

 

[mfb]

I started drawing the force diagrams. No problem there. It was easy to see that the normal force in both cases were pairing up with the gravitational force.

What are f_k1 and f_k2? The friction forces need more thought here.

Applied on the second box (the one underneath the other):

If you call it box 1, why do you say "second box"? Its acceleration is wrong because you are missing a force acting on it.

For the box on top, be careful with the sign. In which direction will it accelerate?

 

[Doc Al]

No problem there. It was easy to see that the normal force in both cases were pairing up with the gravitational force.

Careful: The normal force and the gravitational force are not 3rd-law pairs.

 

[Nicholas Egebak]

Careful: The normal force and the gravitational force are not 3rd-law pairs.

Oh, yeah! That's right! They are just equal and opposite...

What are f_k1 and f_k2? The friction forces need more thought here.If you call it box 1, why do you say "second box"? Its acceleration is wrong because you are missing a force acting on it.

For the box on top, be careful with the sign. In which direction will it accelerate?

Do you mean that box 2 has a velocity of 0 at t=0 and it therefore because of the friction will accelerate in the same direction as the velocity of box 1?

I understand that it is a bit confusing when referred to as both box 1 and 'second box' but they are the same

f_k1 and f_k2 should be the same - I realize that now

 

[mfb]

Do you mean that box 2 has a velocity of 0 at t=0 and it therefore because of the friction will accelerate in the same direction as the velocity of box 1?

For example, yes.

f_k1 and f_k2 should be the same - I realize that now

They are not, no matter what they represent (which is still undefined).