# Relationship Between Potential Energy And Distance:Detailed Facts

In this article, we are going to discuss the relationship between potential energy and distance.

The potential energy may increase or decrease with distance depending upon the types of forces acting on the system. For repulsive forces, the potential energy will reduce and for attractive forces, the potential energy will escalate with increasing distance.

## Does Potential Energy Increase with Distance between Particles

The potential energy of any system veritably depends upon the distance depending upon the types of forces acting on the objects. The potential energy undeniably depends upon the distance and is inversely proportional to each other. If the force between the two is attractive then the potential energy will increase with increasing distance between them both, and if the force is repulsive then the potential energy will increase will decreasing the distance between them.

Let us elaborate our discussion on how does the change in distance will affect the potential energy.

Gravity:

A weak gravitational pull is felt on all the objects present near and on the surface of the Earth due to the gravity of the Earth. The potential energy of any object on the surface of the Earth is always equal to zero. The potential energy related to any object depends upon the weight of the object due to gravity and the height of the object from the ground. Thus the potential energy is denoted as:-

V=mgh

This shows that the potential energy is directly proportional to the height of the object above the ground. But, as the gravitational force of the Earth is exerted on the objects surrounding it, the object is pulled back towards the Earth’s surface and during that time the potential energy of the object is utilized to return back to the ground converting the potential energy into kinetic energy that accelerates the object to the ground. This is the universal fact that all the objects tend to occupy the least potential energy level.

Gravitation Potential Energy between two bodies in space:

The gravitation force exerted on the two bodies in space is inversely proportional to the square of the distance between them both. It is represented by the formula

F=G*(m1m2)/r2

Where G is a gravitational constant.

The potential energy of the masses due to the gravitational force is the integral value of the attractive force experience between the two masses and hence we can write

Potential energy –

From the above equation, we can say that the potential energy of the two bodies depends upon the distance between the two joined in a line. The negative sign indicates that the work done is negative and hence the force is an attractive force. Therefore, if the distance between the two increases then less will be the work done and the potential energy will increase.

This clearly shows that, if the distance between the heavenly bodies is large, this means that the potential energy associated with the heavenly object is also greater.

Electric potential energy:

Electric potential is the amount of potential per unit charge, whereas electric potential energy is the amount of energy required to bring the charged particle from the distance to that point.

Consider a charge particle q1 having a positive charge of 1C kept at a distance ‘r’ from the point charge.

Then the potential of a charge q1will be

When the point charge is replaced by a charge q2 having the same charge as that of q1 then the force experience on charge q1 due to q2 and on charge q2 due to q1 is

Since potential energy associated with the particle is the integral result of all the forces acting on the particle. Therefore, the potential energy will be equal to

In this case, as the separation between the two charges will increase, the potential energy will decrease with respect to distance. As two equal charges will repel away from each other, large potential energy is required to bring the two charges closer to one another.

Now, if the charge kept at the source was negatively charged, then the electrostatic force exerted equally on each particle would be

The negative sign of the potential energy indicates that the force between the two oppositely charge particle is attractive. In this case, as the distance between the two charge particles increases the potential energy generated by the particles to attract each other will also increase.

Potential energy due to spring:

Consider a wooden block of mass ‘m’ attached to the one end of the spring having spring constant value ‘k’ and another end of the spring attached to the rigid wall. The force is applied in the direction as shown in the figure to displace the wooden block to the distance ‘x’ from the initial position. The force required to pull the wooden block attached to the string is kx. A block will gain enough potential energy which when released is converted into kinetic energy and due to the elastic property of the spring, a block will move slightly towards the rigid wall.

The potential energy of the spring is equal to the work done by the spring and is given as the integral of the force acting on the mass due to spring. Therefore,

If the mass attached to the string is pulled further, the potential energy will also increase by the square of the displacement.

## How Mass and Distance of an Object Affect the Potential Energy

As per the mass-energy equivalence relation described by Albert Einstein and formulated by the equation E=mc2; the mass of the object is correlated to the energy it is associated with. The internal energy stored in the object is its potential. To do any work this potential energy is utilized by transforming this energy to some other form of energy.

In most of the cases as seen above, the potential energy decreases with distance because the potential energy is inversely proportional to the distance between the object and the source or between the two objects. But we have also noticed that when the force between the two bodies is attractive then the potential energy increases as the distance separating both increases.

## Why Does Potential Energy Increase with Distance

If the distance separating the two objects increases then the potential energy will also rise if the force acting between the two objects is an attractive force. This is because; less and lesser force will be required to keep the objects apart from each other if we keep on increasing the distance between them both since the force decreases to the square of the distance between them increased.

Work done by the system resembles the potential energy required to do the work. If the two bodies attractive to each other are placed closer to one another, then very less potential energy will be required to bring the two objects closer. If the bodies are far away then more potential will be required to do the same work. More and more potential energy will be required if the distance between the two objects becomes larger.

## Find out the potential energy of the string having a spring constant 25Nm-1 and displacement 20cm.

Given:

k=25Nm-1

x=20cm=0.2m

The potential energy stored in a string stretched at a distance of 20 cm is 0.5 J.

## Why does the potential energy of the attractive force is negative?

In attractive forces, the particles exert a force of attraction to pull them closer to each other. Work has to be done to keep them both at a finite distance.

Force applied to do so is in the direction opposing the force of attraction between the two particles and hence the work done in a direction to oppose the attractive force. Since the potential energy applied is in corresponds to the work done, therefore it is negative.