Introduction
After the discussion of physics, chemistry, mathematics in approx 200 posts. Now after a long break, we have planed to discuss physics. Now a day, a more simple question is trending on the internet that is what is the main relation or co-relation between force and potential energy. So the answer is available but not in simple word. So here we have establish a relation between force and potential energy in simplest word. even a 10th student can understand it well.
So before any wasting of time lets start. But before we start, lets take an overview on the concept of force and potential energy.
We have already studied that when we talk about the term potential energy ie, we are actually talking about conservative forces. Because we have already studied in our previous classes that in case of Non-conservative forces, no storage of energy is possible. Full work done or whole amount of work done is concerned in terms of energy loss.
Whenever we talk about potential or stored form of energy that actually means we are talking about conservative forces. And in case of conservative forces, work done is always form of potential energy. Here what is work done means? The answer is work done is nothing but a change in potential energy.
Please not that the whole concept of this relation of force and potential energy is only that ''work done is equal to the change in potential energy".
Work done = Change in potential energy
Δ W = Δ U2 − Δ U1
∴ Δ W = Δ U
From the above short discussion, we have concluded that the relation between force and potential energy is work done with change in potential energy.
Before we discuss the relation with examples and graph. lets first understand some terms which is used here. like what is the meaning of force, conservative force, Potential energy etc.
What is Force in terms of conservative force?
If we not see the concept of conservative. Then we can define force as simple, a push or a pull of an object is called force. In other words, if a body experience some changes in its state wether it is in motion or rest. Then there must be something acted upon the body called force.
Now, let's define force in terms of conservative force. So, a conservative force is like a gravitational force that acted between two bodies having some masses. For example, gravitational force always acting between the earth and any other mass bodies.
For better understanding in conservative force. Always remember, those forces which acted on potential energy know as conservative force.
Conservative force always determined only when, when there will be the final displacement of the object in potential energy.
In simple words, conservative force helps in work done only when, there will be some changes in the position of the object.
Means the work done by conservative force not dependent on path.
Mathematical concepts of conservative force
The term Conservative force comes from Stored energy concept. Means conservative force comes from the concept of mechanical energy.
The most common conservative force are gravity and spring forces (also known as stored energy force).
What is potential energy in terms of conservative force?
Potential energy is a type of stored energy. If we decine potential energy in terms of conservative force. Then definition will be like, When work is done by the displacement of an object from initial to final position due to the conservative force known as potential energy.
You can consider gravitation force to understand the concept of potential energy.
Now, take a look for what you are expected from this post. i.e, the relationship between force and potential energy.
Establish a relation between force and potential energy.
Let's take an example to establish a relation and co relation between force (or conservative force) and potential energy.
Take a conservative field, and divide it into two parts (may or may not be equal). Mark a point A in the left part and point B in the right part. As shown in fig.
Suppose when a body is in a side of point A, then its energy is U¹ and when it is in side of point B, it's energy is U². Please note that it is given (U¹ >U²). This shows that body A has more energy than body B.
We all know that every body has a tendency to achieve the minimum energy state. So, in this case body at point B has less energy. Therefore, body at point A transfer the energy to the body B. And when body A releases the energy towards body B, then the conservative field will exert a conservative force towards the direction of energy flow.
Now, when a work is done to being body A to the body B by the force (F), then change kar n work done will be equal to the change in potential energy.
I.e,
ΔW = UA - UB
ΔW = f. dx
Similarly,
In the above example, when we bought a body A towards the body B, then potential energy of body A decreases and due to this the potential energy of the body B increases.
That meas to do work done, change in potential energy takes place.
Therefore,
ΔW = - ΔU
Here, minus sign indicates that the energy is decreasing.
The same thing happens when being body A towards body B but in step by step. I.e, to cover less amount of distance. The result of work done will be as equal to happen in above case.
Let suppose A and B are located at on x axis. And potential energy of a body is given as two function of position at every point where potential energy is changing.
So, when the body is going from one point to another point say A to B.
We can state the above example as, in displacing the body by dx, let dU be the decreasing in P.E of the system. Then we can use
- dU = F. dx
Because by displacing from point A to B by dx. The work done by force (F) is F. dx.
We can write:-
dU = - F. dx
Or,
F = -dU / dx
This relation is well at for undirectional varient in P.E.
Please note that :- The minus sign always shows that the force acting in the direction where potential energy is decreasing.
Here, we have discussed only about one dimensional system. But when potential energy acts on two dimension or three dimensions (or free space). Then what will happen. Let's take a look.
In free space or three dimensions variation of P.E. we use,
F = - ΔU = - gradient of U.
Here, delta is gradient oprator.
Hence, the required relation between force and potential energy is
F = - ΔU.
Relation Between Force and Potential Energy
In physics, the relationship between force and potential energy is fundamental to understanding the behavior of systems in fields such as mechanics, electromagnetism, and gravitation. The force acting on an object is related to the potential energy by the following mathematical relationship:
Where:
- is the force acting on the object,
- is the potential energy as a function of position (or configuration),
- is the position or configuration variable (like displacement).
The negative sign indicates that the force is directed in the opposite direction to the increase in potential energy. This can be understood as a system naturally moving in the direction where the potential energy is minimized.
Explanation
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Potential Energy (): It represents the energy stored in a system due to its configuration. It could depend on position, displacement, or other variables (depending on the type of force involved, such as gravitational, elastic, or electric potential energy).
-
Force (): The force is the negative gradient (or rate of change) of the potential energy with respect to position. This is because the force tends to move an object towards a position where its potential energy is lower (for stable systems).
- If the potential energy increases with displacement (), the force will act in the opposite direction to reduce the potential energy ().
- Conversely, if the potential energy decreases with displacement (), the force will act in the direction of increasing displacement ().
Examples of Force and Potential Energy Relationships
1. Gravitational Potential Energy
For a mass in a gravitational field near the Earth's surface, the gravitational potential energy is given by:
Where:
- is the mass of the object,
- is the acceleration due to gravity,
- is the height above the ground.
The gravitational force acting on the object is the negative derivative of potential energy with respect to height:
This negative sign indicates that the force acts downward, opposing the increase in height (potential energy).
2. Elastic Potential Energy (Hooke’s Law for Springs)
For a spring with spring constant , the elastic potential energy stored in the spring when displaced by from its equilibrium position is:
The force exerted by the spring is the negative derivative of the potential energy with respect to displacement:
This is Hooke’s law, which shows that the restoring force is proportional to the displacement and acts in the opposite direction.
3. Electric Potential Energy (Coulomb Force)
In the case of two charges and separated by a distance , the electric potential energy is given by:
Where is Coulomb’s constant. The force between the charges is the negative derivative of the potential energy with respect to the separation distance :
This shows that the force between the charges follows an inverse square law, and the force is attractive for opposite charges and repulsive for like charges.
Steps to Calculate Force from Potential Energy:
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Obtain the Potential Energy Function (): You need the potential energy function that describes the system. This function may depend on position, configuration, or other relevant variables.
-
Differentiate the Potential Energy: Differentiate with respect to the position (or another relevant variable) to obtain the rate of change of potential energy.
-
Apply the Force Formula: The force is the negative of the derivative of the potential energy:
-
Interpret the Force: The sign of the force indicates the direction in which the system will move. The force will tend to move the object toward regions of lower potential energy.
Key Takeaways
- The force on an object is related to the rate of change of its potential energy with respect to position.
- The direction of the force is opposite to the direction in which the potential energy increases.
- In systems where potential energy is well-defined, this relationship provides a powerful way to determine the behavior of forces within the system.