# Force on table/chair rolling back

1. Sep 6, 2015

### Entr0py

1. The problem statement, all variables and given/known data
Today I was sitting in a chair (with wheels on bottom) and decided to apply a force to a large desk. While I applied this force with my hand, the chair and I rolled backwards. My question is why is this so? I was thinking that if my hand is object "a" then it exerts a force on the desk (object "b"), F(ab) while the desk exerts a force F(ba) on my hand. Now if these action-reaction forces are equal and opposite then why do I accelerate backwards (or away from my desk)? What I think happens is that I DO accelerate the table but why can't I see its acceleration?

2. Relevant equations
Newton's third law and ΣF=ma where F and a are vectors

3. The attempt at a solution

2. Sep 6, 2015

### Titan97

This is a common misconception. Even if actions reaction forces are equal and opposite, they are acting on two different bodies. The force you exert acts on the table. The table exerts an equal and opposite force on you. Hence for you, there is an unbalanced force acting on your body

3. Sep 6, 2015

### Phynos

I believe the reason you accelerate while the desk appears not to is due to a series of action-reaction forces taking place. Someone else can chime in if something I say sounds off.

You accelerate because there is nothing for the force the table exerts on your hand to transfer to beyond you and your chair, with the exception of the floor (or wheel barrings), but the floor (or the wheel barrings) is not able to exert a force on the chair great enough to cancel it out, thus you accelerate. Basically static friction is overcome.

The table does not appear to accelerate because the floor and possibly the wall (depending on how your desk is set up) are able to exert a force on the desk equal and opposite to the force you are exerting on the desk. The force moves to the objects holding the table in place, and so on. That set of objects holding one another up reacts to your push by accelerating. So the table does accelerate, but along with the floor, and the wall, etc, and since the mass of all those objects far exceeds your own, the acceleration they experience is much smaller.

I was going to say the wall and floor could deform instead of the whole system moving but at first glance that seems to go against conservation of momentum? The momentum you gain must be matched by that gained by the object you are pushing on, only in the opposite direction. Comments on this part are welcome, I'm not sure of this.

4. Sep 6, 2015

### Entr0py

I understand that. I understand that the action reaction forces act on different bodies and so I accelerate because of the net force from the chair. So does the table accelerate at all? Or what forces are causing it to stand still relative to me?

5. Sep 6, 2015

### Entr0py

I think the conservation of momentum applies to the isolated system of me, the chair,the ground and table. So in a given time interval, let's say 5 s, the table loses momentum p1 and I (and chair) gain momentum -(p1).

6. Sep 6, 2015

### Titan97

Table may accelerate depending on friction. If the floor is very rough and table is very heavy, then it won't move. If the system is kept on ice, then both the table and chair will slide.

7. Sep 6, 2015

### Titan97

Also, momentum can't be conserved if the floor has friction.

8. Sep 6, 2015

### Entr0py

So since the table lays on top of a wooden floor then the coefficient of static friction is higher than on ice and so it doesn't accelerate. Therefore net force acting on table is 0 but for me and the chair it isn't. So the force from the table onto me is able to overcome static friction of the floor and I accelerate away from the table?

9. Sep 6, 2015

### Entr0py

If that because friction is a non conservative force?

10. Sep 6, 2015

### Phynos

I would agree that the momentum of the chair-table system is not conserved, but that's an open system. For momentum to be conserved we have to consider what is holding the table in place. That momentum should be transferred to the object exerting the friction, the floor.

Imagine we have a cannon stationary on a wooden platform fin space. If the cannon fires perpendicular to the surface we know the momentum of the cannon ball is the same as the momentum of the cannon and floor, in opposite directions of course.

Now instead the cannon fires at an angle from the floor so it scrapes across the floor. Friction between the cannon and the floor played a role, but in the transfer of momentum from the cannon to the floor. The sum of the momentum for each object should still be zero, or else it would appear that a closed system gained momentum from nowhere.

Maybe I am mistaken, but I thought momentum was always conserved.

I can see how it might be argued that it is not conserved. Energy is lost through friction as heat. However momentum does not have units of energy, it is different. Kinetic energy would be lost through friction, but that relies on velocity just like momentum, so if that changes momentum should as well.

I have confused myself. I would like to see someone with more understanding on the subject clear this up.

EDIT: Isn't the 'loss of momentum to friction via heat' just the transfer of the momentum from the large scale object to the individual constituents of the object?

11. Sep 6, 2015

### Titan97

Yes.
But I would like a PF mentor to make things clear like @Phynos said.
Since you are sitting on a chair with wheels, the situation will be more complex.
Let's say the table exerts a force N on you and the hinge exerts a force F on the centre of wheel along horizontal. So for wheel to roll,
$F-f=ma$
$fR=I\alpha$ side hinge force acts along centre, it won't exert Torque.
And for pure rolling, $a=R\alpha$
$a=\frac{FR^2}{mR^2+I}$
Where I is the moment of inertia.

12. Sep 6, 2015

### Entr0py

I'm confused where you got that equation. Is moment of inertia times angular acceleration=net torque (I haven't learned about torque yet. I'm still trying to finish up the chapter on Newton's laws and then go onto Applications of Newton's laws).

13. Sep 7, 2015

### Titan97

Whatever equation I gave was for a single wheel. Let's just avoid wheels and chair. Imagine you are pushing a wall. The wall will exert an equal and opposite force on your hands. So a net external force acts along the horizontal direction on your body. You will see that sometimes, your leg slips. This is because of lesser coefficient of static friction.

14. Sep 7, 2015

### Kilgour22

When you push against the table and your chair rolls backwards but not the table, then somewhere there has to be friction. The coefficient of static friction is NEVER less than the coefficient of kinetic friction or rolling friction between two surfaces. As a result, if not enough force is applied to overcome the coefficient of static friction, then that object (in this case, the table) will never move.

The reason why the chair rolls back is because the coefficient of static friction between the floor and the wheels is less than the coefficient of static friction between the floor and the table legs. Therefore when you apply enough force to overcome the wheel and floor coefficient of static friction, the wheels move backwards! Now instead of static friction, we are dealing with rolling friction which is a very small coefficient for a lot of surfaces against each other. This is why when you push against the table and you move backwards, you seem to move backwards with a considerable amount of velocity relative to how much force you applied on the table.

But where does the force you applied on the table go? Obviously it has to go somewhere! This is where work comes in to play. Our system loses mechanical energy, but gains thermal energy, as it must due to the law of conservation of energy. Because we have a non-zero change in mechanical energy, there must be non-conservative work being done to the system. Our non-conservative work is made up of dissipative work and external work, which are done by the system on the environment and the environment on the system, respectively. Friction generates thermal energy, so we have negative work done on the system and positive work done on the environment. As a result, the energy you put into the system was turned into thermal energy, which ended up being transferred into the environment (in this case, the floor). The same process is happening to the chair, but at a slower rate due to a low coefficient of rolling friction. This is why the chair slows down, because the kinetic energy is being constantly turned into thermal energy, which is non-conservative work, which is work done on the floor!

15. Sep 7, 2015

### Entr0py

So the force from the wall on me is greater than the force due to static friction. Wouldn't this mean that net force in horizontal direction=(force from wall) + (static friction) which would be Fwall-Fstaticfriction?

16. Sep 7, 2015

### Entr0py

Very awesome reply.I now have a good understanding for why my chair and I roll backwards and the table doesn't. Thank you all for your help (this is making me love physics even more)!

17. Sep 7, 2015

### Kilgour22

Not a problem. The more you learn and understand physics, the more fascinating and intriguing it becomes. :)

18. Sep 7, 2015

### Entr0py

You are 100% correct!

19. Sep 8, 2015

### Titan97

No. If you start moving, net force will be Fwall-Fkinetic friction
But Fwall should be greater than Fstatic friction to start motion.

20. Sep 9, 2015

### Entr0py

I don't understand why? If I start to move wouldn't the net force be F (from wall) - F(from static friction). Wouldn't I still be accelerating even if I move just a bit since my velocity has changed?