`➢` The spring force is an example of a variable force which is conservative. Fig. 6.7 shows a block attached to a spring and resting on a smooth horizontal surface.
`➢` The other end of the spring is attached to a rigid wall. The spring is light and may be treated as massless.

`➢` In an ideal spring, the spring force `F_s` is proportional to `x` where x is the displacement of the block from the equilibrium position. The displacement could be either positive [Fig. 6.7(b)] or negative [Fig. 6.7(c)]. This force law for the spring is called Hooke’s law and is mathematically stated as

`F_s=-kx`

`=>` The constant `k` is called the spring constant. Its unit is `N m^(-1)`. The spring is said to be stiff if k is large and soft if `k` is small.

`➢` Suppose that we pull the block outwards as in Fig. 6.7(b). If the extension is `x_m`, the work done by the spring force is

`color{purple} {W_s = int_0^(x_m) F_s dx=- int_0^(x_m) kx dx}`

`color {blue}{=- (kx_m^2)/2}`

............(6.15)

`➢` This expression may also be obtained by considering the area of the triangle as in Fig. 6.7(d). Note that the work done by the external pulling force F is positive since it overcomes the spring force

`color {blue}{W= + (kx_m^2)/2}`..........(6.16)

`➢` The same is true when the spring is compressed with a displacement `x_c (< 0)`. The spring force does work `W_s=-kx_c^2//2` while the external force `F` does work `+ kx_c^2//2` I
`=>` f the block is moved from an initial displacement `x_i` to a final displacement `x_f` , the work done by the spring force `W_s` is

`color {blue}{W_s= - int_(x_i)^(x_f) kx dx = (kx_i^2)/2 -(k x_f^2)/2}` ...........(6.17)

`➢` Thus the work done by the spring force depends only on the end points. Specifically, if the block is pulled from xi and allowed to return to `x_i` ;

`color {blue}{W_s=- int_(x_i)^(x_f) kx dx= (kx_i^2)/2 - (kx_i^2)/2=0}`

.........(6.18)


`➢` The work done by the spring force in a cyclic process is zero. We have explicitly demonstrated that the spring force
(i) is position dependent only as first stated by Hooke, `(F_s = − kx)`;
(ii) does work which only depends on the initial and final positions, e.g. Eq. (6.17).

`color{red}☞` Thus, the spring force is a conservative force We define the potential energy `V(x)` of the spring to be zero when block and spring system is in the equilibrium position. For an extension (or compression) `x` the above analysis suggests that

`color {blue}{V(x) =(kx^2)/2}`..............(6.19)

`➢` You may easily verify that `− dV/dx = −k x`, the spring force. If the block of mass m in Fig. 6.7 is extended to `x_m` and released from rest, then its total mechanical energy at any arbitrary point x, where `x` lies between `– x_m` and `+ x_m`, will be given by

`color{orange}{1/2 kx_m^2 =1/2 kx^2 + 1/2 mv^2}`

`=>` where we have invoked the conservation of mechanical energy. This suggests that the speed and the kinetic energy will be maximum at the equilibrium position, `x = 0`, i.e.,

`1/2 kx_m^2+ 1/2 mv^2`

`➢` where we have invoked the conservation of mechanical energy. This suggests that the speed and the kinetic energy will be maximum at the equilibrium position, `x = 0`, i.e.,

`1/2 mv_m^2 = 1/2 kx_m^2` where `v_m` is the maximum speed.

or `v_m = sqrt(k/m) x_m`

`color{red}☞` Note that `k//m` has the dimensions of `[T^(-2)]` and our equation is dimensionally correct. The kinetic energy gets converted to potential energy

and vice versa, however, the total mechanical energy remains constant. This is graphically depicted in Fig. 6.8.

`color{red}☞` We conclude this section by making a few remarks on conservative forces.
(i) In the example considered above, we can calculate the compression, but not the time over which the compression occurs. A solution of Newton’s Second Law for this system is required for temporal information.
(ii) Not all forces are conservative. Friction, for example, is a non-conservative force. The principle of conservation of energy will have to be modified in this case. This is illustrated in Example 6.9.
(iii) The zero of the potential energy is arbitrary. It is set according to convenience.
`=>` For the spring force we took V(x) = 0, at x = 0, i.e. the unstretched spring had zero potential energy. For the constant gravitational force `mg,` we took `V = 0` on the earth’s surface.
`➢` In a later chapter we shall see that for the force due to the universal law of gravitation, the zero is best defined at an infinite distance from the gravitational source. However, once the zero of the potential energy is fixed in a given discussion, it must be consistently adhered to throughout the discussion. You cannot change horses in midstream.

`➢ "We have seen that the total mechanical energy of the system is conserved if the forces doing work on it are conservative."`

`➢` If some of the forces involved are non-conservative, part of the mechanical energy may get transformed into other forms such as heat, light and sound.
`➢` However, the total energy of an isolated system does not change, as long as one accounts for all forms of energy. Energy may be transformed from one form to another but the total energy of an isolated system remains constant.
`➢` `"Energy can neither be created, nor destroyed."`
`➢` Since the universe as a whole may be viewed as an isolated system, the total energy of the universe is constant. If one part of the universe loses energy, another part must gain an equal amount of energy.

`➢` The principle of conservation of energy cannot be proved. However, no violation of this principle has been observed.
`➢` The concept of conservation and transformation of energy into various forms links together various branches of physics, chemistry and life sciences. It provides a unifying, enduring element in our scientific pursuits. From engineering point of view all electronic, communication and mechanical devices rely on some forms of energy transformation.

`➢` In the previous section we have discussed mechanical energy. We have seen that it can be classified into two distinct categories : one based on motion, namely kinetic energy; the other on configuration (position), namely potential energy. Energy comes in many a forms which transform into one another in ways which may not often be clear to us.

`color{red} ► color {blue} bbul" Heat"`

`➢` We have seen that the frictional force is excluded from the category of conservative forces. However, work is associated with the force of friction.

`➢` A block of mass `m` sliding on a rough horizontal surface with speed `v_0` comes to a halt over a distance `x_0`.

`➢` The work done by the force of kinetic friction `f` over `x_0` is `–f x_0`.
`➢` By the work-energy theorem `m v_0^2 //2 = f x_0` . If we confine our scope to mechanics, we would say that the kinetic energy of the block is llost due to the frictional force.

`➢` On examination of the block and the table we would detect a slight increase in their temperatures.
`➢` Thus the work done by friction is this raises the internal energy of the block and the table. In winter, in order to feel warm, we generate heat by vigorously rubbing our palms together.
`➢` We shall see later that the internal energy is associated with the ceaseless, often random, motion of molecules. A quantitative idea of the transfer of heat energy is obtained by noting that `1 kg` of water releases about `42000 J` of energy when it cools by `10 °C`.

`color{red} ► color {blue} bbul" Chemical Energy"`

`➢` As we learnt to rub two flint stones together (mechanical energy), got them to heat up and to ignite a heap of dry leaves (chemical energy), which then provided sustained warmth.
`➢` A matchstick ignites into a bright flame when struck against a specially prepared chemical surface. The lighted matchstick, when applied to a firecracker, results in a spectacular display of sound and light.


`➢` Chemical energy arises from the fact that the molecules participating in the chemical reaction have different binding energies. A stable chemical compound has less energy than the separated parts.
`➢` A chemical reaction is basically a rearrangement of atoms. If the total energy of the reactants is more than the products of the reaction, heat is released and the reaction is said to be an exothermic reaction.
`➢` If the reverse is true, heat is absorbed and the reaction is endothermic. Coal consists of carbon and a kilogram of it when burnt releases `3 xx 10^7 J` of energy.

`color{red}☞` Chemical energy is associated with the forces that give rise to the stability of substances. These forces bind atoms into molecules, molecules into polymeric chains, etc. The chemical energy arising from the combustion of coal, cooking gas, wood and petroleum is indispensable to our daily existence.

`color{red} ► color {blue} bbul " Electrical Energy"`

`➢` The flow of electrical current causes bulbs to glow, fans to rotate and bells to ring. There are laws governing the attraction and repulsion of charges and currents, which we shall learn later. Energy is associated with an electric current. An urban Indian household consumes about 200 J of energy per second on an average.

`color{red} ► color {blue} " The Equivalence of Mass and Energy"`

`➢` Till the end of the nineteenth century, physicists believed that in every physical and chemical process, the mass of an isolated system is conserved. Matter might change its phase, e.g. glacial ice could melt into a gushing stream, but matter is neither created nor destroyed; Albert Einstein (1879-1955) however, showed that mass and energy are equivalent and are related by the relation

`E=mc^2`...............(6.20)

where `c`, the speed of light in vacuum is approximately `3 xx 10^8 m s^(–1)`. Thus, a staggering amount of energy is associated with a mere kilogram of matter

`E= 1 xx (3 xx 10^8)^2J = 9 xx 10^16 J`

`=>` This is equivalent to the annual electrical output of a large `(3000 MW)` power generating station.

`color{red} ► color {blue} bbul " Nuclear Energy"`

`=>` The most destructive weapons made by man, the fission and fusion bombs are manifestations of the above equivalence of mass and energy [Eq. (6.20)].

`➢` Often it is interesting to know not only the work done on an object, but also the rate at which this work is done.
`➢ \ \ "Power"` is defined as the time rate at which work is done or energy is transferred.
`➢` The average power of a force is defined as the ratio of the work, `W`, to the total time `t` taken

`P_(av)= W/t`

`➢` The instantaneous power is defined as the limiting value of the average power as time interval approaches zero,

`color {blue}{P=(dW)/(dt)}`..................(6.21)

`➢` The work `dW` done by a force F for a displacement `dr` is `dW = F*dr`. The instantaneous power can also be expressed as

`P= F * (dr)/(dt)`

`color {blue}{=F*v}`...............(6.22)

`=>` where `v` is the instantaneous velocity when the force is `F`.

`color{red} ►` Power, like work and energy, is a scalar quantity. Its dimensions are `[ML^2T^(–3)]`.
`color{red} ►` In the SI, its unit is called a watt (W). The watt is 1 `J s^(–1)`. The unit of power is named after James Watt, one of the innovators of the steam engine in the eighteenth century. There is another unit of power, namely the horse-power (`hp`)

`1` hp = `746` W

`➢` This unit is still used to describe the output of automobiles, motorbikes, etc. We encounter the unit watt when we buy electrical goods such as bulbs, heaters and refrigerators. A `100` watt bulb which is on for `10` hours uses 1 kilowatt hour (kWh) of energy.

`100` (watt) × `10` (hour)`

`= 1000` watt hour `=1 ` "kilowatt hour (kWh)"`

`= 10^3 (W) xx 3600 (s)`

`color {blue}{= 3.6 xx 10^6 J}`

Our electricity bills carry the energy consumption in units of kWh. Note that kWh is a unit of energy and not of power.