Internal and external forces

When a person lifts a book while standing on the earth's surface. Why is his force considered external to the book-earth system? Isn't his force internal like the case of the forces that cause the exploson of a bomb. I think the forces the person exerts on both the book and the earth are internal since they are equal but opposite therefore resulting in the position of the centre of mass of the system being maintained and if i'm right is the mechanical energy of the system constant? Am i wrong? Please correct me.

• hint-the categorization of internal/external is dependent on the 'system' taken -in the book -earth system the agency is external and it does some positive work such that the total mechanical energy of the book changes Aug 11, 2018 at 22:34

I think the forces the person exerts on both the book and the earth are internal since they are equal but opposite therefore resulting in the position of the centre of mass of the system being maintained and if I'm right is the mechanical energy of the system constant? Am I wrong? Please correct me.

The importance of categorizing a force as being either internal or external is related to the ability of that type of force to change an object's total mechanical energy when it does work upon an object.

When net work is done upon an object by an external force, the total mechanical energy (KE + PE) of that object is changed.

If the work is positive work, then the object will gain energy. If the work is negative work, then the object will lose energy.

The gain or loss in energy can be in the form of potential energy, kinetic energy, or both. Under such circumstances, the work that is done will be equal to the change in mechanical energy of the object. Because external forces are capable of changing the total mechanical energy of an object, they are sometimes referred to as non-conservative forces.

When the only type of force doing net work upon an object is an internal force (for example, gravitational and spring forces), the total mechanical energy (KE + PE) of that object remains constant. In such cases, the object's energy changes form.

For example, as an object is "forced" from a high elevation to a lower elevation by gravity, some of the potential energy of that object is transformed into kinetic energy. Yet, the sum of the kinetic and potential energies remains constant.

When the only forces doing work are internal forces, energy changes forms - from kinetic to potential (or vice versa); yet the total amount of mechanical energy is conserved. Because internal forces are capable of changing the form of energy without changing the total amount of mechanical energy, they are sometimes referred to as conservative forces.

Another classic example of this is that you cannot grab yourself by the hair and lift yourself up off the ground. That is because your hand is part of your body. So you cannot really create a system where your hand is external to the rest of your body. Of course, you could define the system to be your body minus your hand and say your hand is external to this system.

But when your hand pulls on your hair, your hair will pull back on the hand. And since your whole body is connected, ultimately, there will be no acceleration of the centre of mass of the hand-body system. But someone else could grab you by the hair and pull you up off the ground.

ref.-

• @ drvm: ok, but from your link it says that external forces also cause an acceleration: i am assuming that the accelaration being refered to is that of the system or in other words ; the centre of mass of the system. If we imagine a person lifting the book it seems like the centre of mass of the system does not change position since the person displaces the earth with his feet and the book with his hands there by apply eaual and opposite forces at both ends like an interaction. Therefore the system does not accelerate. Is my reasoning ok or am i lost? Aug 12, 2018 at 6:46
• I think if the agency i.e. the man- book-earth all three are taken as a system, then no forces external to the system can be visualized. Aug 12, 2018 at 7:13

It all depends on what you define the system to be. If your system only consists of the book and the earth, then any force you supply is external. If you include yourself as part of the system, then your force is internal. If your system is just you and the book, then gravity is an external force.

Forces being internal or external has nothing to do with what they physically are and everything to do with how the system is subjectively defined (see drvrm's answer to understand the reasoning one would consider in this subjective decision).

• But does the earth-book system accelerate if the man is excluded from the system? Aug 12, 2018 at 7:31
• @Energy No matter what the labels, all forces are still present. If the man lifts the book, both the book and the earth feel a force. They are both pushed away from each other. The book just moves way more than the earth due to its small mass. Aug 12, 2018 at 12:38

Your confusion might well arise because of certain (erroneous) simplification which are made when these ideas are introduced in elementary Physics courses.
The system is assumed (although not necessarily stated) to be the book and so the force that your hand is exerting on the book is external to the system.
The work done by that force is then equated to the gain in gravitational potential energy of the book.
This type of analysis provides the “correct” result because the mass of the Earth is so much greater than the mass of the book.
It also ignores the important fact that gravitational potential energy is stored in the book and Earth system.

The mechanical energy of the system is the sum of kinetic energies and potential energies.
Any forces associated with potential energy must be conservative.

An external force originates from outside a system whereas an internal force originates from within a system and must always have a Newton third law pair which is equal in magnitude and opposite in direction.

To simplify the discussion let us assume that the book is going to be raised by an initially compressed, massless (ideal) spring and initially everything is at rest.

Let the system be the book, the spring and the Earth and consider what happens if the spring is allowed to expand.
In such an instance spring potential energy will be converted into kinetic energy of the book and the Earth and gravitational potential energy of the system.
The mechanical energy of the system will remain unchanged as the decrease in spring potential energy will equal the increase in kinetic energy and gravitational potential energy.
All relevant forces are internal and conservative.

If the system is only the book and the Earth then the forces exerted on the book and the Earth by the spring are external forces.
Those external forces do work on the system wrist the spring expands and the mechanical energy of the system increases because the gravitational potential energy (and the kinetic energy) increase.

Now assume that the spring is immersed in treacle ie when the spring expands non-conservative frictional force are acting between the spring and the treacle.

With the book, spring and treacle and Earth as the system there will be a decrease in the mechanical energy of the system as the decrease in spring potential energy will not equal the increase in gravitational potential energy.
There will be a conversion of some of the mechanical energy of the system into heat.

If the system is just the book and the Earth then the mechanical energy of the system increases.

Now this second example with the spring in treacle can be thought of as being equivalent to the bomb that you mentioned with the spring potential energy being replaced by chemical potential energy stored in the explosive (or you lifting the book with chemical potential energy stored in you) and the action of the treacle being mirrored by the production of heat during the chemical reactions.

• All the forces internal to the system come pairwise so the net force is always zero which means that no net work is done internally inside the system and the system's energy stays constant. The mechanical energy (KE+PE) also remains constant. The internal forces can be econservative and nonconservative. I guess the change between PE and KE, while the mechanical energy stays constant, is only possible if the internal forces are conservative. I am not sure how the work done by conservative forces can change PE in KE and vice versa since the net work by conservative forces is zero... Dec 6, 2020 at 22:02

http://physics.bu.edu/~duffy/HTML5/field_potential3.html

• When an external force is exerted on an object in a system, positive work is done on that system in the form of gain in kinetic energy • When a system exerts a force on an object outside of that system, negative work is done by that system in the form of loss in kinetic energy

What is gradient? Every continuous graph has its own gradient, which shows how the value of y changes when x increases by 1. For example, when x increases by 1 and y increases by 1, the gradient is 1 and when x increases by 1 and y increases by 2, the gradient is 2. When x increases by 1 and y decreases by 1, the gradient is -1 and when x increases by 1 and y decreases by 0.5, we say the gradient is -0.5. When x increases by 1 and y does not change at all, we say that the gradient is 0.

When the net force (a.k.a. resultant force) acting on an object in a system is zero, there is no change (gain/loss) of kinetic energy in that system. So when there exists a forward internal force on an object in a system and we introduce a backwards external force on that object against the forward force that is equal in magnitude to the forward force, there is no net force or resultant force since the signs are opposite, thus the object does not accelerate (hence no change in kinetic energy of the system) since only resultant force or net force results in acceleration. However, we are still moving at constant velocity. In this case, work is still being done on the system because the object is still being displaced (displacement = constant velocity × time). This work done is not in the form of gain in kinetic energy of the system, but in the form of gain in potential energy of the system due to the (equal and opposite) forward force, a conservative, non-dissipative force.

All charges 'feel' a force from other charges. We want to move a positive test charge a certain displacement against an electric force from another positive charge. So we begin at infinity because it is the universal reference point. When we start at r= infinity and begin to move q positive test charge a certain displacement in the negative r direction, we need to introduce our own backwards external force on the test charge to balance out the forward internal force exerted on it. The type of work done on the system is considered 'positive' work in the form of gain in kinetic energy as both the backwards external force and the displacement are in the negative r (i.e., same) and this gain in kinetic energy is converted into gain in potential energy (potential difference) and stored. On the other hand, the forward internal force is also doing work - however the work done is 'negative' as the forward force is in the positive r direction while the displacement is in the negative r direction. This 'negative' work, unlike the work done by the force we exert on the test charge, is not done on the system because it is done by an 'internal' force, which simply converts energy from kinetic to potential, or potential back to kinetic.

Note the shape of the graph. As r moves from infinity inwards (in the negative r direction), for every decrease in r by a fixed value, the increase in V (potential energy) is getting larger and larger. This means that the backwards force required to be put in to balance out the forward force so that the velocity stays constant is ever increasing.

Finally we have reached our destination at r= 0. We have put in the maximum amount of backwards force and we have our maximum amount of potential energy. Now we let go of that so that there is no more backwards force. Now there is a resultant (net) force in the form of the forward force, and the forward force causes acceleration (hence change in kinetic energy) as it converts potential to kinetic in fulfilling its 'role' as an internal force. So now the potential energy stored will gradually decrease from maximum at r= 0 to 0 at r= infinity as all of it is converted into kinetic energy bit by bit (conservation of energy) by the forward force (fulfilling its 'role' as an internal force). The role of the internal force (positive force) is to convert all the potential to kinetic bit by bit. As r moves from r= 0 outwards (in the positive direction), for every increase in r by a fixed value, the decrease in potential energy which is equal to the increase in kinetic energy is getting smaller and smaller. This is because you when put in a lot of work, you get a lot of kinetic energy back and when you only put in a bit of work, you only get a bit of energy back.

What we were talking about is potential gradient. The potential gradient is ever increasing because as r increases (moves outwards), the decrease in potential energy (hence increase in kinetic) is getting less steep because the forward force is getting lesser and lesser. The equation

Increase in potential energy = backward force × decrease in r (negative displacement)

can be rearranged into

Backward force = increase in potential energy / negative displacement

which is same as potential gradient (decrease in potential energy / positive displacement). In fact, forward force is negative potential gradient, thus the electric field strength (forward force per unit positive charge) is -(potential gradient). The values of V are always +ve when the 2 charges are of the same sign and always -ve when the 2 charges are of different signs.