In order to derive a general expression for change in potential energy, we refer to the vertical motion of a ball, projected upward. As the ball moves up, initial kinetic energy of the ball decreases by the amount of negative work done by the gravity. In the meantime, however, separation between Earth and ball also increases. Consequently, the potential energy of "Earth - ball" system increases by the same amount. In effect, gravity transfers energy "from" the kinetic energy of the ball "to" the potential energy of the "Earth - ball" system.

Figure 1: Gravity transfers energy between object and system.

Ball thrown vertically

In the reverse motion i.e. when ball starts moving downward, gravity transfers energy "from" potential energy of the "Earth - ball" system "to" the kinetic energy of the ball. The transfer of energy in "up" and "down" motion is same in quantity. This means that exchange of energy between "ball" and "Earth - ball" system is replicated in reverse motion.

A positive work on the object means that energy is being transferred from the system to the object. Consequently, the potential energy of the system decrease. As such, final potential energy of the system is lesser than initial potential energy. On the other hand, a negative work on the object means that energy is being transferred from the object to the system. Consequently, the potential energy of the system increases. As such, final potential energy of the system is greater than initial potential energy.

Between any two instants, the change in potential energy of the "Earth - ball" system is negative of the work done by the gravity on the "ball".

Δ U = - W Δ U = - W

The result so obtained for gravitational force can be extended to any conservative force. The change in potential energy of a system, therefore, is defined "as the negative of the work done by conservative force". Note the wordings here. We have defined potential energy of a system - not to an individual entity. In practice, however, we refer potential energy to a particle with an implicit understanding that we mean a system - not an individual entity.

We must note following important aspects of potential energy as defined above :

1: We have derived this expression in the context of gravitational force, but is applicable to a class of forces (called conservative forces), which transfer energy in above manner. It is important to understand that there are non-conservative forces, which transfer energy to the system in other energy forms like heat etc. - not in the form of potential energy.

2: This is the expression for change in potential energy, which assumes different forms in different contexts (for different conservative forces).

3: We should realize that we deal here with the "change" in potential energy - not with "unique" value of potential energy corresponding to a specific configuration. This can be handled with the help of a known reference potential energy. The measurement of potential energy with respect to a chosen reference value enables us to assign a unique value to potential energy corresponding to a specific system configuration. We shall further elaborate this aspect while discussing specific potential energies.

4: The change in potential energy differs to kinetic energy in one aspect that it can assume negative value as well. We must remember that kinetic energy is always positive. A positive change in potential energy means that final potential energy is greater than initial potential energy. On the other hand, a negative value of change in potential energy means that final potential energy is lesser than initial potential energy.

5: Potential energy is defined only in the context of conservative force. For the sake of clarity, we should also realize that the conservative force doing work is internal to the system, whose potential energy is determined. We can rewrite the defining relation of potential energy that work refers to work by conservative as :

Δ U = - W C Δ U = - W C

We shall evaluate expression for potential energy for the vertical motion of a ball, which is thrown up with a certain speed. The gravity "-mg" works against the motion. Importantly, gravity is a constant force and as such we can use the corresponding form for the change in potential energy. For a displacement from a height y 1 y 1 to y 2 y 2 , the change in potential energy for the "Earth - object" system is :

Figure 2: Gravity transfers energy between object and system.

Ball thrown vertically

⇒ Δ U = - ( - m g ) ( y 2 - y 1 ) cos 0 0 ⇒ Δ U = m g ( y 2 - y 1 ) ⇒ Δ U = - ( - m g ) ( y 2 - y 1 ) cos 0 0 ⇒ Δ U = m g ( y 2 - y 1 )

The change in potential energy is, thus, directly proportional to the vertical displacement. It is clear that potential energy of "Earth - ball" system increases with the separation between the elements of the system . Evaluation of change in gravitational potential energy requires evaluation of change in displacement.

Motion of an object on Earth involves "Earth - object" as the system. Thus, we may drop reference to the system and may assign potential energy to the object itself as the one element of the system i.e. Earth is common to all systems that we deal with on Earth. This is how we say that an object (not system) placed on a table has potential energy of 10 J. We must, however, be aware of the actual context, when referring potential energy to an object.

We should also understand here that we have obtained this relation of potential energy for the case of vertical motion. What would be the potential energy if the motion is not vertical - like for motion on a smooth incline? The basic form of expression for the potential energy is :

⇒ Δ U = - F r cos θ ⇒ Δ U = - F r cos θ

Here, we realize that the right hand expression yields to zero for θ = 90°. Now, any linear displacement, which is not vertical, can be treated as having a vertical and horizontal component. The enclosed angle between gravity, which is always directed vertically downward, and the horizontal component of displacement is 90°. As such, the expression of change in potential energy evaluates to zero for horizontal component of displacement. In other words, gravitational potential energy is independent of horizontal component of displacement and only depends on vertical component of displacement.

This is an important result as it renders the calculation of gravitational potential lot easier and independent of actual path of motion. The change in gravitational potential energy is directly proportional to the change in vertical altitude. As a matter of fact, this result follows from the fact that the work done by the gravitational force is directly proportional to the change in vertical displacement.

Reference potential energy

In order to assign unique value of potential energy for a specific configuration, we require to identify reference gravitational potential energy, which has predefined value - preferably zero value. Let us consider the expression of change in potential energy again,

⇒ Δ U = m g ( y 2 - y 1 ) ⇒ Δ U = m g ( y 2 - y 1 )

The important aspect of this relation is that change in displacement is not dependent on the coordinate system. Whatever be the reference, the difference in displacement ( y 2 - y 1 ) ( y 2 - y 1 ) remains same. Thus, we are at liberty as far as choosing a gravitational potential reference (and preferably assign zero value to it). Let us consider here the motion of ball which rises from y 1 = 2 m y 1 = 2 m to y 2 = 5 m y 2 = 5 m with reference to ground. Here,

Figure 3: Change in potential energy is independent of choice of reference.

Ball thrown vertically

⇒ Δ U = m g ( y 2 - y 1 ) ⇒ Δ U = m g ( 5 - 2 ) = 3 m g ⇒ Δ U = m g ( y 2 - y 1 ) ⇒ Δ U = m g ( 5 - 2 ) = 3 m g

Let us, now consider if we measure displacement from a reference, which is 10 m above the ground. Then,

y 1 = - ( 10 - 2 ) = - 8 m y 2 = - ( 10 - 5 ) = - 5 m y 1 = - ( 10 - 2 ) = - 8 m y 2 = - ( 10 - 5 ) = - 5 m

and

⇒ Δ U = m g ( y 2 - y 1 ) ⇒ Δ U = m g { - 5 - ( - 8 ) } = 3 m g ⇒ Δ U = m g ( y 2 - y 1 ) ⇒ Δ U = m g { - 5 - ( - 8 ) } = 3 m g

If we choose ground as zero gravitational potential, then

y 1 = 0 , U 1 = 0 , y 2 = h (say) , and U 2 = U y 1 = 0 , U 1 = 0 , y 2 = h (say) , and U 2 = U

Gravitational potential energy of an object (i.e. "Earth - ball" system) at a vertical height "h" is :

⇒ U 2 - U 1 = m g ( y 2 - y 1 ) ⇒ U - 0 = m g ( h - 0 ) ⇒ U = m g h ⇒ U 2 - U 1 = m g ( y 2 - y 1 ) ⇒ U - 0 = m g ( h - 0 ) ⇒ U = m g h

We must be aware that choice of reference does not affect the change in potential energy, but changes the value of potential energy corresponding to a particular configuration. For example, potential energy of the object at ground in ground reference is :

U g = m g h = m g x 0 = 0 U g = m g h = m g x 0 = 0

The potential energy of the object at ground in reference 10 m above the ground is :

U' g = m g h = m g x - 10 = - 10 m g U' g = m g h = m g x - 10 = - 10 m g

Thus, we must be consistent with the choice of reference potential, when analyzing a situation.

Spring force transfers energy like gravitational force through the system of potential energy. In this sense, spring force is also a conservative force. When we stretch or compress a spring, the coils of the spring are accordingly extended or compressed, reflecting a change in the arrangement of the elements within the spring and the block attached to it. The spring and the block attached to the spring together form the system here as we consider that there is no friction involved between block and the surface. The relative change in the spatial arrangement of this system indicates a change in the potential energy of the system.

Figure 4: Energy is exchanged between block and "spring - block" system

Spring block system

When we give a jerk to the block right ways, the spring is stretched. The coils of the spring tend to restore the un-stretched condition. In the process of this restoration effort on the part of the spring elements, spring force is applied on the block. This force tries to pull back the spring to the origin or the position corresponding to relaxed condition. The spring force is a variable force. Therefore, we use the corresponding expression of potential energy :

⇒ Δ U = - ∫ F ( x ) ⅆ x ⇒ Δ U = - ∫ F ( x ) ⅆ x

⇒ Δ U = - ∫ ( - k x ) ⅆ x ⇒ Δ U = - ∫ ( - k x ) ⅆ x

Integrating between two positions from the origin of the spring,

Figure 5: Energy is exchanged between block and "spring - block" system

Spring block system

⇒ Δ U = ∫ x i x f k x ⅆ x ⇒ Δ U = k [ x 2 2 ] x i x f ⇒ ⇒ Δ U = 1 2 k ( x f 2 - x i 2 ) ⇒ Δ U = ∫ x i x f k x ⅆ x ⇒ Δ U = k [ x 2 2 ] x i x f ⇒ ⇒ Δ U = 1 2 k ( x f 2 - x i 2 )

Considering zero potential reference at the origin, x i = 0 , U i = 0 , x f = x (say) , U f = U (say) , x i = 0 , U i = 0 , x f = x (say) , U f = U (say) , we have :

⇒ U = 1 2 k x 2 ⇒ U = 1 2 k x 2

The important aspect of this relation is that potential energy of the block (i.e. "spring - block" system) is positive for both extended and compressed position as it involves squared terms of "x". Also, note that this relation is similar like that of work except that negative sign is dropped.