Contents
Physics Topics can be both theoretical and experimental, with scientists using a range of tools and techniques to understand the phenomena they investigate.
What are the Two Types of Mechanical Energy? What is the Simple Definition of Potential Energy?
Only mechanical energy is discussed in detail in the follow-ing sections.
Mechanical Energy
The ability of a body to do work due to its motion, position or configuration, is called its mechanical energy.
Mechanical energy is of two types:
- kinetic energy and
- potential energy.
Kinetic Energy
Definition: The ability of a body to do work due to its motion is called its kinetic energy.
An external force is needed to stop a moving body. Work done by the body against the applied force till it comes to rest, is equal to its initial kinetic energy.
Examples of kinetic energy:
i) A bullet fired from a gun can penetrate through a target, whereas a stationary bullet cannot. A high speed bullet is able to do some work due to the kinetic energy acquired from its motion. Most of this kinetic energy is converted to heat energy.
ii) Wind carries a boat forward when the sails of the boat are hoisted. Wind has the ability of doing work due to its motion.
iii) An athlete takes a long run to acquire kinetic energy, before a long jump. Using this energy, the athlete does work against air friction and gravity, and can cover a longer distance during the jump.
iv) While hammering a nail into a wall, the hammer should be brought swiftly to the nail to apply an impulsive force. The hammer acquires kinetic energy due to its rapid motion. This energy is used for doing work against the resistance of the wall on the nail, as it penetrates through the wall.
Kinetic energy of water and wind is used for different purposes, including production of electrical energy, wind mill operations, etc.
Expression of linear kinetic energy
Suppose a body of mass m moves in a straight line with velocity u. Let a constant force F be applied against the motion of the body. Hence, a retardation a is produced and after a further displacement s the body comes to rest.
So, work done against the force F until the body stops
= Fs = mas …… (1)
As the final velocity of the body is zero,
o = u2 – 2as [using the formula, v2 = u2 + 2as]
or, as = \(\frac{u^2}{2}\) …. (2)
or, mas = \(\frac{m u^2}{2}\)
∴ Work done = \(\frac{1}{2}\)mu2
This expression, \(\frac{1}{2}\)mu2, is regarded as the measure of the kinetic energy (K) of a body of mass m , moving with a velocity u. It is equal to the work done to stop the motion of the body.
Hence, K = \(\frac{1}{2}\)mu2 …….. (3)
or, linear kinetic energy = \(\frac{1}{2}\) × mass × (linear velocity)2
If due to application of a force, a body accelerates, its kinetic energy increases, and the opposite happens when the body decelerates.
Relationship between momentum and kinetic energy: A body of mass m, moving with a velocity u, has a momentum, p = mu.
∴ Its kinetic energy, K = \(\frac{1}{2} m u^2\) = \(\frac{1}{2} \cdot \frac{m^2 u^2}{m}\) = \(\frac{p^2}{2 m}\)
∴ p2 = 2mK or, p = \(\sqrt{2 m K}\)
Work-energy theorem:
Let m = mass of a particle, \(\vec{F}\) = force or resultant of forces, acting on the particle, \(\vec{s}\) = displacement of the particle in time t, \(\vec{v}\) = velocity of the particle at time t, and its initial and final values are vA and vB.
We shall consider the most general case [Fig.]
- The path ACB of the particle is a curved path,
- \(\vec{F}\) is a variable force, changing with time both in magnitude and in direction,
- the angle, between \(\vec{F}\) and the infinitesimal displace-ments d\(\vec{s}\) along the path, also varies with time.
Then, the work done for the entire motion along the path ACB is,
W = \(\int_A^B \vec{F} \cdot d \vec{s}\) = \(\int_A^B m \vec{a} \cdot d \vec{s}\) = \(m \int_A^B \vec{a} \cdot d \vec{s}\)
where, \(\vec{a}\) = acceleration of the particle = \(\frac{d \vec{v}}{d t}\) we also know, \(\vec{v}\) = \(\frac{d \vec{s}}{d t}\).
i.e., work done = final K.E. – initial K.E. = change in K.E.
This is known as the work-energy theorem: the work done by the net force on a particle is equal to the change in its kinetic energy. Or, alternatively, the work done on a particle transforms into an equal amount of kinetic energy of the particle.
Kinetic energy due to an explosion: In a bomb explosion, the body splits into many fragments and these scatter in different directions. Hence, the explosion imparts kinetic energy to the fragments.
Let us assume that, a body of mass (M + m) was initially at rest. So, initial momentum = 0. After explosion, two fragments of masses in and M scatter off in mutually opposite directions. Let their velocities be v and -V, respectively.
Then, final momentum = mv – MV.
From the law of conservation of momentum,
0 = mv – MV or, v = \(\frac{M V}{m}\).
If velocitiy V of mass M is in the direction opposite to v, then
Hence, kinetic energy is inversely proportional to the mass.
K1 > K2 if M > m.
This means that, after an explosion, the smaller fragments have larger amounts of K.E. If a fragment is too heavy, it remains almost stationary.
Also, \(\frac{K_1}{K_1+K_2}\) = \(\frac{M}{M+m}\)
or, K1 = \(\frac{M\left(K_1+K_2\right)}{M+m}\) or, K1 = \(\frac{M K}{M+m}\)
where K = K1 + K2 = total kinetic energy of the two fragments.
Similarly, K2 = K1ᐧ\(\frac{m}{M}\) = \(\frac{m K}{M+m}\)
where K = K1 + K2 = total kinetic energy of the two fragments.
Similarly, K2 = K1ᐧ\(\frac{m}{M}\) = \(\frac{m K}{M+m}\).
Potential Energy
Definition: The ability of a body to do work due to its special position or configuration, is called potential energy of the body.
An equilibrium position or configuration is primarily taken as the standard or reference; any other position or configuration is then termed as special position. The reference position or configuration is also called the zero state of the body.
Some work has to be done on a body to take it from its reference state to any special state. This work is stored as potential energy in the body. When it comes back to its reference state, it can do some work using its stored potential energy.
Potential energy due to change We position: Work has to be done against gravity to lift a body above the surface of the earth. This work gets stored in the body in the form of gravitational potential energy. While returning to its reference position, i.e., the earth’s surface, the body can do work using this stored energy. As the body does work, its potential energy decreases and finally on reaching the reference position, its potential energy reduces to zero. For example:
i) A nail is half inserted into the ground. If a hammer is just held on the head of the nail, no further penetration takes place. But if the hammer is raised and allowed to fall on the nail head, the nail penetrates further into the earth, doing some work against the resistance of the earth’s surface. The raised hammer, acquires the ability to do this work. This capacity to do work is its potential energy.
ii) Hydroelectric power is generated mainly using the gravitational potential energy of stored water. River water is stored by constructing a dam on it. The water in the reservoir has a large amount of potential energy due to its height. While flowing down through pipes this potential energy changes to kinetic energy which is used to rotate the turbines of an electric generator.
Potential due to change in configuration: Let us consider an elastic body. The relative positions between the different parts of that body may be altered and the shape of the body changed. The work done to alter its shape gets stored as elastic potential energy. The body spends this energy by doing work, to regain its original shape, and the process continues until the potential energy reduces to zero. For example:
i) When the hairspring of a watch is wound, the work done is stored as potential energy. This energy is used to unwind the spring slowly and thus the watch works. However, when the spring unwinds completely and returns to its original shape, the stored potential energy becomes zero and the watch stops.
ii) In order to shoot an arrow from a bow, the bow-string is pulled to change the natural shape of the bow. This stored potential energy in the bow transforms into the kinetic energy of the arrow.
Hence, work has to be done to change the shape and size of an object; this work is stored as potential energy of the object in its special configuration.