Let us consider the example of an ideal simple pendulum (frictionless). We can see that the mechanical energy of this system is a combination of its kinetic energy and gravitational potential energy. As the pendulum swings back and forth, a constant exchange between the kinetic energy and potential energy takes place. When the bob attains its maximum height, the potential energy of the system is the highest whereas the kinetic energy is zero. At the mean position, the kinetic energy is the highest and the potential energy is zero. Between these two extreme points, we see that the system possesses both kinetic and potential energy, the sum of which is constant. These observations tell us a lot about the conservation of mechanical energy. But how can we prove it for every other system? In this section, we shall learn more about the conservation of mechanical energy using a suitable example.
Conservation of mechanical energy
According to the principle of conservation of mechanical energy, the total mechanical energy of a system is conserved i.e., the energy can neither be created nor be destroyed; it can only be internally converted from one form to another, if the forces doing work on the system are conservative in nature.
In order to understand this statement more clearly, let us consider an example of a one-dimensional motion of a system. Here a body, under the action of a conservative force F, gets displaced by Δx, then from the work-energy theorem we can say that the net work done by all the forces acting on a system is equal to the change in the kinetic energy of the system.
Mathematically, ΔKE = F(x) Δx
Where, ΔK is the change in kinetic energy of the system.Considering only conservative forces are acting on the system Wnet = Wc.
Thus Wc = ΔKE
Also, If conservative forces do the work in a system, the system loses potential energy equal to the work done. Hence, Wc = -PE.
Which implies that, the total kinetic energy and potential energy of a system remains constant if the process involves only conservative forces.
KE + PE = constant
KEi+ PEi = KEf+ PEf
Where, denotes the initial values and f denotes the final values of KE and PE.
This law applies only to the extent that the forces are conservative in nature. Mechanical energy of the system is defined as the total kinetic energy plus the total potential energy. In a system that comprises of only conservative forces, each force is associated with a form of potential energy and the energy only changes between the kinetic energy and different types of potential energy, such that, the total energy remains constant.
Let us understand this principle more clearly with the following example. Let us say, a ball of mass m is dropped from a cliff of height H, as shown above.
At height H:
Potential energy (PE) = m×g×H
Kinetic energy (K.E.) = 0
Total mechanical energy = mgH
At height h:
Potential energy(PE) = m×g×h
Kinetic energy (K.E.) =1/2(mv^2)
Using the equations of motion, the velocity v1 at a height h for an object of mass m falling from a height H can be written as
Hence the kinetic energy can be given as,
Total mechanical energy = (mgH – mgh) – mgh = mgH
At height zero:
Potential energy: 0
Kinetic energy: 1/2(mv^2)
Using the equations of motion we can see that velocity v at the bottom of the cliff, just before touching the ground is
Hence, the kinetic energy can be given as,
Total mechanical energy: mgH
We saw the total mechanical energy of the system is constant throughout.
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