The work done by a force on a particle is defined as the product of the component of the force in the direction of the displacement and the magnitude of the displacement. Suppose a constant force F acts on a block to produce a displacement s in it. The angle contained between the vectors F and s is θ . Then the work done by the force is:
W = Fs cos θ
Work is the scalar product of force and displacement vectors, which means work itself is a scalar and has no direction. If the angle between F and s is an acute angle (< 90°), work is positive. If the angle between F and s is an obtuse angle (> 90°), work is negative. If the angle between F and s is 90° , work done is zero. When a block moves on a horizontal floor, the force of gravity exerted by the earth and the normal force exerted by the floor do not do any work on the block because these forces act at right angles to the motion of the block. The dimensions of work are
[ML2T-2], and its SI unit is joule (J).
To calculate the work done by a varying force, we need a more rigorous, calculus-based approach. Let us consider the case of a block attached to a spring. Suppose one end of a spring is attached to a fixed wall and the other end is attached to a block resting on a horizontal, frictionless floor. Initially, the spring is at its natural length. Now if an external force is applied to the block to move it slowly without acceleration further away from the wall, the spring stretches and exerts a variable restoring force opposite to the block's motion. The magnitude of this spring force is directly proportional to the displacement of the block.
The constant of proportionality is known as the force constant of the spring. If the spring is stretched to take the block from initial position x = 0 to final position x = x, it can be shown by integration that the work done by the external force is:
Wext= 1/2 kx2
The corresponding work done by the spring force is:
Wspr= -1/2 kx2
Here k is the force constant of the spring. Notice that the external force does positive work because it has the same direction as the block's displacement. The spring force does negative work (of the same absolute value) because it is directed opposite to the block's displacement.
The energy of a body can be loosely defined as its capacity of doing work, measured by the total amount of work the body can do. Energy can be of different types, e.g. mechanical, thermal, light, electrical, magnetic, sound, chemical and nuclear. In classical mechanics, we shall mostly deal with mechanical energy which is further classified into two types: kinetic energy associated with the motion of a body, and potential energy associated with the configuration of a system such as gravitational, elastic and electrical potential energy. All types of energy have the same dimensions and units as those of work.
The kinetic energy K of a particle of mass m moving with speed v is defined as:
K= 1/2 MV2
This is a scalar quantity which is always positive. According to the work-energy theorem, the net work done on a particle is equal to the change in its kinetic energy. In mathematical terms:
Wnet = 1/2 mv22 - 1/2 mv12 = K2 - K1 = ΔK
where v1 and v2 are the initial and final speeds, K1 and K2 are the initial and final kinetic energies, Δ K is the change in kinetic energy of the particle.
The power supplied by a force is defined as the rate at which the force does work. It is the scalar product of force and velocity vectors. That is:
P = F.v = Fv cos θ
where θ is the angle contained between F and v. The dimensions of power are
[ML2T-3], and its SI unit is watt (W).
The potential energy of a system is the energy possessed by it by virtue of its configuration. It is measured by the work the system can do if it returns from its present configuration to some standard configuration usually associated with zero potential energy.
Forces in nature can be divided into two groups: conservative force and non-conservative force. A force is conservative if the total work done by it on a particle moving in a closed path is zero. In other words, the work done by a conservative force on a particle is independent of the path it takes to move from one point to another. Examples: force of gravity, spring force.
The energy diagram is the graph that shows the variation in the potential energy U(x) of a system with the displacement x of a particle forming part of the system. Energy diagram helps us to determine whether a body is in stable equilibrium, unstable equilibrium or neutral equilibrium.
According to the principle of conservation of mechanical energy, the total mechanical energy of an isolated system of particles which interact only through conservative forces is constant. If K1 , U1 are the initial values and K2 , U2 are the final values of the kinetic and potential energies of the system then:
K1 + U1 = K2 + U2
The principle can be applied to solve a host of mechanical problems. The method is often referred to as the energy method of solving a problem. It is usually more elegant and easier than force-based dynamical method of solving a problem.
Another conservation principle with broader scope is the principle of conservation of energy. It states that energy can neither be created nor destroyed. It may be transformed from one form to another, but the total energy of an isolated system is always constant. From a universal point of view, the total energy of the universe is constant.
Any discussion on energy conservation remains incomplete without reference to Einstein's famous formula on equivalence of mass and energy:
E0 = m0 c2
In this expression, m0 is the rest mass of a particle, E0 is the rest energy which is energy equivalent of m0 , and c is the speed of light in vacuum.