Work:

Work Done by a Constant Force:

The term work is incorrectly utilized in everyday life as applied in any form of activity where we use muscular or mental effort. But in physics term work is utilized in the particular sense. So, in scientific sense work is done when the force moves its point of application along direction of its line of action.

For instance, in given figure if constant force F moves from point A to point B a distance of s in a constant direction, then work done by this force is stated as

Work = Force x distance moved by force

W = Fs........ Eq.1

If force applies at an angle θ to direction of motion of point of application of force as illustrated in Figure then work is stated as product of component of the force in direction of motion and the displacement in that direction. I.e.

W = (F cos Θ)s

Note that when Θ = 0, Cos Θ = 1 and so, W =Fs. This agrees with Eq.1. When Θ =90^o, Cos Θ = 0 and we see that F has no component in direction of motion and so, no work is done. This signifies that if we associate this to force of gravity, it is clear that for horizontal motion, no work is made by force of gravity.

Unit of Work:

The unit of work is the unit of force multiplied by unit of distance in any specific system of measurement. In SI system, unit of force is Newton and unit of distance is meter; so in this system unit of work is one Newton meter (I Nm). This is known as the joule (IJ).

In cgs system, unit of work is one dyne centimeter (1 dyn cm) and it is known as one erg. Note that as 1m = 100cm and 1N = 10⁵ dyn, then

1Nm = 10⁷dyn cm or 1J = 10⁷erg.

In engineering system, unit of work is one foot pound (1ft lb):

Note: IJ = 0.7376 ft 1b

And 1 ft lb = 1.356 J

When numerous forces act on the body, we resolve them in their components and determine algebraic sum of work done by effective component forces. This follows as work is a scalar quantity.

Work Done by Varying Force:

Work could be done by the force that different in magnitude or direction during displacement of body. For instance on stretching the spring slowly, force needed to do this increases gradually as spring lengthens. Also gravitational force pulling the upward vertically projected particle downward decreases inversely as square of distance from the centre of the earth.

We can determine work done by the varying force graphically as follows:

Suppose force is F when displacement is x, then for the further small displacement dx is Fdx (that is if we take dx to be so small that F is considered constant). If complete area under curve AB is divided in small narrow strips, total work done during the displacement S will be provided by area under curve AB that is Area OABC.

Energy:

The body is said to expend energy when it performs work on another body. For instance if body A performs work by applying a force on body B, then body A is said to lose energy. This energy lost by body A is equivalent in amount to work it done on body B. Therefore we can state energy as that which allows the body to perform work. So when it is said that one has some energy, it means that he is able to do some work.

Energy is estimated in joules just similar to work. Work done can be taken to be the measure of quantity of energy transferred between two bodies. If for instance, body P performs 10 joules of work on body Q then energy transfer from P to Q is 10 joules.

Power:

Power of equipment means the rate at which it performs work. This is the same as rate at which machine or appliance converts energy from one form to another. Unit of power is watt (W). When one joule of work is performed in one second it is called as watt or that energy expended is IW

Therefore 1W = 1js^-1

Kinetic Energy:

Kinetic energy is energy the body posses by virtue of its motion. For instance the moving hammer does work against resistance of wood in which the nail is being driven. We get expression of kinetic energy by calculating amount of work done by the body whereas body is being brought to rest. Consider the body of constant mass, m moving with velocity u. The constant force F acts on it to bring it to rest in the distance s

When it comes to rest, final velocity, v is zero. Then from equation of motion:

V² = u² + 2as

Where a is acceleration

Therefore O = u² + 2as

And a = -u²/2s

Negative sign in equation illustrates that acceleration is in opposite direction to motion of body therefore body decelerates. We expect acceleration in direction of force F to be + u²/2s.

Now, kinetic energy of body is equal to the work, W body does against F, thus,

Kinetic energy, K.E of body = W = Fs

But Fs = mass

:. K.E = mass

Putting a = u²/2s

K.E. = 1/2mu²

If work is done on the body gain of kinetic energy when velocity increases from zero to u can be illustrated also to be 1/2mu².

We now generalize. If the body of mass, m with the initial velocity of u moves when work is done on it by the force acting over a distance s and if its final velocity is v then work done Fs is given by

Fs = 1/2mv² - 1/2mu²

This equation is known as work- energy equation. It may be expressed in words as: Work done by forces (Acting on the body) = change in kinetic energy of body.

Potential Energy:

Potential energy of the system of bodies is energy the body possesses by virtue of the relative position of parts of the body of the system. Potential energy P.E arises when the body experiences the force in the region or field. The example is gravitational field of the earth. In this situation, body engages the position with respect to earth. P.E is then taken to be the joint property of body-earth system and not of either body separately. Therefore P.E is found by relative position of body and earth. It is observed that greater the separation, the greater the P.E. P.E of the body on surface of the earth is always taken to be zero. But for the body of mass m at a height h above ground level, P.E. is equal to the work which will be done against gravity, to elevate body to this height. This signifies that the force equal and opposite to mg is required to be applied to body to elevate it to required height. That is why we have assumed g to be constant near surface of earth. Therefore,

Work done by external force (against gravity)

= Force x displacement

= mgh

Therefore P.E = mgh

When body returns straight to ground level an equal amount of potential energy is lost.

Conservation of Energy:

Word conserve could be taken to mean preserve so that nothing is lost. For instance if body of mass m is projected vertically upwards and if its early velocity is u at point of projection A say, it will perform work against constant force of gravity,

Consider velocity of body at the higher point B is V and height obtained at this point is h. Now, by definition K.E lost between points A and B = work done by body against mg. Also by definition of P.E Gain of P.E between A and B = work done by body against mg. Thus, we have that

Loss of K.E. = gain of P.E

Therefore 1/2 mu² - 1/2 mv² = mgh

This is principle of conservation of mechanical energy. This principle is expressed as:

A total amount of mechanical energy (K.E + P.E) that bodies in the isolated system have is constant.

This is valid only to frictionless motion that is to conservative system. Also, gain in P.E will rely on path taken but it doesn't in the conservative system.

Work done against frictional forces is frequently accompanied by the temperature rise. Thus in energy account take this in consideration.

By so doing energy conservation principles will be extended to comprise non- conservative systems and it becomes loss of K.E = gain of P.E + gain of internal energy.

Energy may be transformed from one form to another, but it can't be created or destroyed, i.e. total energy of the system is constant.

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