# Only external forces can accelerate the center of mass of a system- paradox

We know that only external forces can accelerate a system, not internal forces. But I don't know why, I draw a contradictory conclusion from the following examples:

1. Let's say I consider a person standing at rest on a ground (man is the system). He wants to jump upwards.

Just like in the first case shown in picture- Jumping upwards requires the man to lift his heels (with an acceleration "a") which in turn bends his knees, but, the toes are still in touch. the normal force increases from N= mg to N=mg+ma.

BUT, work done by normal is zero as the point of application's displacement (the toes) is zero.

Thus the whole work of jump is done by internal forces- which are the muscles. So the muscle force ends up accelerating the system and thus the centre of mass accelerates upwards. Hence internal forces accelerate the system. But this is wrong...

1. Another example to explain what i am asking-

Fundamentals of Physics by Halliday, Resnick, and Walker mentions the following about the work done in internal energy transfers:

An external force can change the kinetic energy or potential energy of an object without doing work on the object- that is, without transferring energy to the object. Instead, the force is responsible for transfers of energy from one type to another inside the object.

It states an example:

An initially stationary ice-skater pushes away from a railing and then slides over the ice. Her kinetic energy increases because of an external force F on her from the rail. However, that force does not transfer energy from the rail to her. Thus, the force does no work on her. Rather, her kinetic energy increases as a result of internal transfers from the biochemical energy in her muscles.

Here, the normal force from the railing is that force that does no work. So in this case also, muscular force of skater, being internal (skater is the system), will end up accelerating the skater and the centre of mass of skater.

Thus work is done by internal forces- which are the muscles. So the chemical energy in the muscles gets converted in to kinetic energy which ends up accelerating the system and thus the centre of mass accelerates upwards. Hence, internal forces accelerate the system. But that is wrong...

Please explain where do I go wrong?

Edit: How is it that the internal force, the muscular force ends up accelerating the center of mass?

• In both your examples you are mixing up work and acceleration. A force does not need to do work to cause acceleration. For instance, a homogeneous magnetic field does no work on a charged particle but causes it to accelerate in circular motion. Commented Sep 28, 2023 at 19:41
• @MariusLadegårdMeyer to understand this more intuitively, does this have to do with momentum conservation? Commented Sep 29, 2023 at 8:12
• See BioPhysicist answer here physics.stackexchange.com/questions/563889/… Commented Sep 29, 2023 at 19:23

As already pointed out by @Marius Ladergard Meyer 6 a force doesn’t need to do work to cause acceleration. Another example is the static friction force by the ground acting forward on a car wheel causing the car to accelerate. The energy comes from the fuel not the ground.

What’s more if the person and the skater are the system then the reaction force of the ground and railing is an external force accelerating the COM of the person and skater.

On the other hand if you include the Earth in the system the forces are internal and the acceleration of the COM of the system is zero.

Hope this helps.

• The static friction force on a car (system) due to the ground is an external force acting on an accelerating car. To an observer on the ground that static frictional force is moving and doing work on the car which increases the car's kinetic energy. Commented Sep 28, 2023 at 23:26
• @Farcher do you agree that the static friction force applied by the ground does not transfer energy to the car ? Commented Sep 28, 2023 at 23:36
• If that static friction force was not there the car would not accelerate, ie there would not be a transfer of chemical energy (fuel) to kinetic energy (car). It is the "agent" which enables the transfer to occur. Commented Sep 29, 2023 at 6:28
• @Farcher when you say that, "To an observer on the ground that static frictional force is moving and doing work on the car which increases the car's kinetic energy." I don't think that's the case, since there is no displacement of the point on the tyre on which static friction is acting in general rotation of the tyre (when the car is not skidding). It doesn't matter if the force is moving or not, all that matters is the point on which the force acts, and here it is not moving in the case of tyre for the time duration for which static friction acts on the tyre. Commented Sep 29, 2023 at 8:01
• @Farcher I realize static friction “enables” the car to accelerate. I’m asking if you agree that the static friction force applied by the ground does not transfer energy to the car? Commented Sep 29, 2023 at 8:03

BUT, work done by normal is zero as the point of application's displacement (the toes) is zero.

This is correct. Remember that there are two definitions of work. One is that work is force times displacement. As you have correctly analyzed, this is zero.

Work is also a transfer of energy. So since work is zero that means zero energy is transferred. That makes sense, the energy comes from the muscles, not from the floor.

The work is zero because no energy is transferred. Internal energy is converted to kinetic energy, but not transferred.

The force from the floor transfers momentum, not energy. Momentum and energy are separately conserved quantities. And they are separately transferred quantities. A force that transfers momentum may not transfer energy, as in this case

How is it that the internal force, the muscular force ends up accelerating the center of mass?

Internal forces cannot accelerate the center of mass, that requires an external force. The acceleration of the center of mass is a change in momentum. It is the external force that provides the change in momentum.

• Alright, so i get that the energy is conserved here. 1. Could you also elaborate on how does the normal transfer momentum in these cases? impulse. Does gravity create an impulse, possibly lowers the net impulse in the jumpers case but not in the skaters case? 2. To compare the amount of impulse imparted in these cases to let’s say fission of any particle/bomb? Commented Sep 29, 2023 at 7:35
• 3. In Skater and jumper case, impulse due to the forces- normal and gravity, considering all of this happens in less time, thus imparts more force – in jumpers case- it’s sufficient to overcome gravity with an acceleration, in skaters case possibly the static friction initially. (but force is not as much as collision/fission?) 4. Am I right in saying that all this happens in less time? But I guess time is more compared to a collision/ or a bomb internally breaking into fragments- hence less force is imparted to the jumper in comparison to what would possibly be imparted to fragments in bomb? Commented Sep 29, 2023 at 7:36
• 5. Are time and force the only factor or is there anything else at play in sometimes less, sometimes more force being imparted? Sorry for bombarding you with questions, but if you could please explain these 5 points. Commented Sep 29, 2023 at 7:36
• @Sirat That is too much for comments. Please ask those as actual questions instead
– Dale
Commented Sep 29, 2023 at 11:03
• Internal [chemical energy] energy is converted to kinetic energy. That can only happen in the jumpers case because the body is not rigid and different parts accelerate at differing rates. Commented Sep 29, 2023 at 16:24

You get something wrong, it is not an internal force which makes the jump possible, but the normal force from the earth. but one uses like in the iceskating internal energy, so the jumping and the iceskating are the same kind of acceleration, both need external force and inner energy. if the man would be floating in space without support he could not move forward.

Please, don't mess things up.

## Dynamics - some results

The second principle of dynamics reads, assuming constant mass of the system, $$$$m \mathbf{a}_G = \mathbf{F}^{ext} \ .$$$$

The theorem of kinetic energy reads $$$$\dfrac{d K}{dt} = P^{tot} = P^{ext} + P^{in} \ ,$$$$ i.e. the time derivative of the kinetic energy of a system equals the power of all the forces acting on the system, both external and external.

In an incremental form (resulting from time integration), says that the difference in the kinetic energy of the system equals the work done by all the forces acting on the system. $$$$\Delta K = L^{tot} = L^{ext} + L^{in} \ .$$$$

The center of mass accelerates because of the force exchanged with the ground.

The kinetic energy increases because of the power of the forces of your muscles that stretch and contract (so displacement) to transmit stresses (so forces),

$$$$\text{stress} \times \text{strain} \times muscle \ volumes \approx \text{force} \times \text{displacement} \approx \text{Inner muscle power} \ .$$$$

Please explain where do I go wrong?

You are not dealing with rigid bodies.

Imagine a horizontal massless compressed spring which is attached to a box, which is on a horizontal frictionless surface, on its left side and a wall on its right side.

The spring exerts a force $$F$$ on the box to the left and a force $$F$$ on the wall to the right.

The wall exerts a force $$F$$ to the left on the spring and that force does not move so it does no work on the spring.
The force $$F$$ on the wall due to the spring does not move so it does no work on the wall.

The force $$F$$ on the box does move and so does work on the box and so there is a transfer of elastic potential energy (spring) to kinetic energy (box).

In the jumping scenario the feet do not move up because of the work done by the normal force on the feet due to the ground rather they are pulled up by the forces exerted on the feet by the rest of the body which is moving upward due to the muscles ("springs") exerting forces which move and are doing work.

• when you say we're not dealing with rigid bodies, what does that imply? Commented Sep 29, 2023 at 9:15
• Think of the jumper and how the feet get off the ground.It is not due to the normal force on the feet due to the ground it is due to a force due to the rest of the body acting on the feet, pulling the feet up.The rest of the body acts like a piece of elastic.You can think of the centre of mass of the rest of the body moving upwards whist the feet are at rest.What happens eventually?The stretched elastic pulls the feet up whilst also pulling the centre of mass of the rest of the body down.All this movement between the rest of the body of the body and the feet implies that the body is not rigid. Commented Sep 29, 2023 at 9:26