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

Question

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...

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2. 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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

Please, don't mess things up.

Dynamics - some results

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

The theorem of kinetic energy reads \begin{equation} \dfrac{d K}{dt} = P^{tot} = P^{ext} + P^{in} \ , \end{equation} 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. \begin{equation} \Delta K = L^{tot} = L^{ext} + L^{in} \ . \end{equation}

Your example

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),

\begin{equation} \text{stress} \times \text{strain} \times muscle \ volumes \approx \text{force} \times \text{displacement} \approx \text{Inner muscle power} \ . \end{equation}

Answer 5

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.