Two boxes; one pushing the other

[Karajovic]

Homework Statement

Two blocks, of mass M and mass m, are in contact on a horizontal frictionless table (with the block of mass M on the left and the block of mass m on the right). A force F1 is applied to the block of mass M and the two blocks move together to the right.

a. draw a free body diagram

b. Suppose the larger block M exerts a force F2 on the smaller mass m. By Newton’s third law, the smaller block m exerts a force F2 on the larger block M. Argue whether F1 = F2 or not. Justify your reasoning.

c. Derive an expression for the acceleration of the system.

d. Derive an expression for the magnitude of the force F2 that the larger block exerts on the smaller block.

e. Choose different values of M and m (e.g. M = 2m, M = 5m, including the case M = m) and compare the magnitudes of F1 and F2.

Homework Equations

The Attempt at a Solution

a. For box of mass M:
F_N
|
F2<--o--->F1
|
Fg

For box of mass m:
F_N
|
o--->F2
|
Fg
b. Using Newton's Third Law, for every force there is an equal reaction force. Since the box with mass M is being pushed by the force F1, doesn't that mean that it is pushing the smaller box of mass m with the same force? But this doesn't seem right because they don't have the same masses... and if F1=F2 that means the boxes wouldn't be moving right?

c. a = F_net/m
a = (m_net*a_net)/(M+m)?

d. F2 = Ma

e. F1 = M*a, while F2 = 2m*a (I don't really understand this part)...

 

[tiny-tim]

Welcome to PF!

Hi Karajovic! Welcome to PF!

The important point is that both boxes have the same acceleration, a.

Write out the F = ma equation for:
i] both boxes
ii] the left box
iii] the right box.

That will give you all the equations you need relating m M F₁ and F₂ …

what do you get? ​

 

[Karajovic]

Write out the F = ma equation for:
i] both boxes
ii] the left box
iii] the right box.

Okay.. so I would get:

i] F = F₁ + F₂ (?)
ii] F₁ = Ma
iii] F₂ = ma

I don't know if that's it
Thanks for your time.

 

[tiny-tim]

Hi Karajovic!

i] F = F₁ + F₂ (?)
ii] F₁ = Ma
iii] F₂ = ma

iii] is correct.

for ii], you need to use both forces on the M block (there was only one force on the m block, so iii] was ok).

i] isn't an F = ma equation, is it?

 

[Karajovic]

Hi Karajovic!


iii] is correct.

for ii], you need to use both forces on the M block (there was only one force on the m block, so iii] was ok).

i] isn't an F = ma equation, is it?

Haha, you're right Then how would I use two forces in one equation? Because for ii] you have F1 = Ma (acting on the M box) and you have F2 = ma (also acting on the M box)...

 

[tiny-tim]

Because for ii] you have F1 = Ma (acting on the M box) and you have F2 = ma (also acting on the M box)...

Nope … ii] is for the M box.

On the LHS, you have all the forces on the M box (ie, the net force).

On the RHS, you have only the mass you're looking at.

(and I'm off to bed :zzz: … see you tomorrow!)

 

[Karajovic]

Nope … ii] is for the M box.

On the LHS, you have all the forces on the M box (ie, the net force).

On the RHS, you have only the mass you're looking at.

(and I'm off to bed :zzz: … see you tomorrow!)

Okay so my conclusion:
b) F1 does not equal F2. First of all F1 = Ma while F2 = ma, also, if the forces were equal, the boxes would not be moving

c) anet = (f1-f2)/(M+m)

d) F2 = ma

now for e) I'm not sure.. F1 = Ma while F2 = ma... Now how do I use different values of M and m? I don't seem to understand what it is asking me to find...

thanks in advance!

 

[tiny-tim]

No, F₁ is not equal to Ma … there are two forces on M.

And c) is wrong … there is only one force on M+m.

Try again.

 

[Karajovic]

No, F₁ is not equal to Ma … there are two forces on M.

And c) is wrong … there is only one force on M+m.

Try again.

F₁ has to equal Ma ? That is the force exerted on the big box, of mass M.

and for c) its asking for the net acceleration, right? So isn't that equal to Fnet/mnet, Fnet equals (F1 - F2 (because F2 is the force hitting the little box by the big, but then an equal force hits the big box)) while mnet is (M+m)

I attached a FBD diagram.

Thanks for the help!

 

[tiny-tim]

There is only one external force on M+m (F₂ is an internal force … they don't count).

(and what do you mean by "net acceleration" … there's no such thing )

 

[Karajovic]

There is only one external force on M+m (F₂ is an internal force … they don't count).

yes, but f1 does equal Ma. but for c) doesn't the F2 force sort of act like a frictional force to the F1 force? Like its a reaction force, but it pushes against the box with f1, so the net is f1 - f2, or not?

(and what do you mean by "net acceleration" … there's no such thing )

oh, sorry then :/ what i meant is the acceleration of the system... so a = F1/(M + m) ? then my above theory is wrong :(

Thanks for your time!

 

[tiny-tim]

yes, but f1 does equal Ma. but for c) doesn't the F2 force sort of act like a frictional force to the F1 force? Like its a reaction force, but it pushes against the box with f1, so the net is f1 - f2, or not?

c) is the whole system. Internal forces don't count.

... so a = F1/(M + m) ?

Yes!

 

[Karajovic]

c) is the whole system. Internal forces don't count.

Okay, thank you! But may I ask, the net force is just f1 then? You do not subtract f2 from it and is my b) statement still okay?

 

[tiny-tim]

Okay, thank you! But may I ask, the net force is just f1 then? You do not subtract f2 from it

The net force on M+m is F₁.

The net force on m is F₂.

The net force on M is F₁ - F₂.

Write out the three F = ma equations (for each of the three systems), to see how they work together …

you should find that 2 of them add to make the third one. ​

… and is my b) statement still okay?

which b)?

do you mean

b) F1 does not equal F2. First of all F1 = Ma while F2 = ma, also, if the forces were equal, the boxes would not be moving

that's ok except for F1 = Ma.

 

[Karajovic]

The net force on M+m is F₁.

The net force on m is F₂.

The net force on M is F₁ - F₂.

Write out the three F = ma equations (for each of the three systems), to see how they work together …

you should find that 2 of them add to make the third one. ​

okay, well i get f1 = (M+m)a while F2 = ma and F1-F2 = Ma + ma - ma = Ma

and f1 does not equal f2 because f1 = (M+m)a while F2 = ma, if they equal, no movement would occur!

How's that it makes sense now!

The total net force of the system is F1, therefore the acceleration of the system is a = f1/(M+m) ??

and what do they mean for e?

thanks again hey, i asked my teach and he said that f1 does equal f2 ? Is he wrong?