Work and Energy
Class 09 ScienceTwo conditions need to be satisfied for work to be done: (i) a force should act on an object (ii) the object must be displaced. If any one of the above conditions does not exist, work is not done.
Work Done By Constant Force
Let a constant force, F act on an object. Let the object be displaced through a distance, s in the direction of the force. Let W be the work done. We define work to be equal to the product of the force and displacement.
Work done = force × displacement
W = F s
Work done by a force acting on an object is equal to the magnitude of the force multiplied by the distance moved in the direction of the force. Work has only magnitude and no direction.
If F = 1 N and s = 1 m then the work done by the force will be 1 N m. The unit of work is newton metre (N m) or joule (J). 1 J is the amount of work done on an object when a force of 1 N displaces it by 1 m along the line of action of the force.
Work can be Positive or Negative
Consider a situation in which an object is moving with a uniform velocity along a particular direction. Now a retarding force, F, is applied in the opposite direction. That is, the angle between the two directions is 180°. Let the object stop after a displacement s. In such a situation, the work done by the force, F is taken as negative and denoted by the minus sign.
The work done by a force can be either positive or negative. Work done is negative when the force acts opposite to the direction of displacement. Work done is positive when the force is in the direction of displacement.
Energy
An object having a capability to do work is said to possess energy. The object which does the work loses energy and the object on which the work is done gains energy.
An object that possesses energy can exert a force on another object. When this happens, energy is transferred from the former to the latter. The second object may move as it receives energy and therefore do some work. Thus, the first object had a capacity to do work. This implies that any object that possesses energy can do work.
The energy possessed by an object is measured in terms of its capacity of doing work. Therefore, the unit of energy is the same as that of work, that is, joule (J). 1 J is the energy required to do 1 joule of work. Sometimes a larger unit of energy called kilo joule (kJ) is used. 1 kJ equals 1000 J.
Forms of Energy
The various forms include mechanical energy (potential energy + kinetic energy), heat energy, chemical energy, electrical energy and light energy.
Kinetic Energy
A moving object can do work. An object moving faster can do more work than an identical object moving relatively slow. A moving bullet, blowing wind, a rotating wheel, a speeding stone can do work.
Objects in motion possess energy. This energy is called kinetic energy. Kinetic energy is the energy possessed by an object due to its motion. The kinetic energy of an object increases with its speed.
The kinetic energy of a body moving with a certain velocity is equal to the work done on it to make it acquire that velocity. Consider an object of mass, m moving with a uniform velocity, u. Let it now be displaced through a distance s when a constant force, F acts on it in the direction of its displacement.
W = F s
The work done on the object will cause a change in its velocity. Let its velocity change from u to v. Let a be the acceleration produced.
$$ v^2 - u^2 = 2as $$
$$ s = \frac{v^2 - u^2}{2a} $$
$$ F = ma $$
$$ W = F \cdot s $$
$$ W = (ma) \cdot \left( \frac{v^2 - u^2}{2a} \right) $$
$$ W = \frac{1}{2}m (v^2 - u^2) $$
If the object is starting from its stationary position, that is, u = 0, then
$$ W = \frac{1}{2}m v^2 $$
The work done is equal to the change in the kinetic energy of an object.
The kinetic energy possessed by an object of mass, m and moving with a uniform velocity, v is
$$ E_k = \frac{1}{2}m v^2 $$
Potential Energy
The potential energy possessed by the object is the energy present in it by virtue of its position or configuration.
Potential Energy as Height
An object increases its energy when raised through a height. This is because work is done on it against gravity while it is being raised. The energy present in such an object is the gravitational potential energy.
The gravitational potential energy of an object at a point above the ground is defined as the work done in raising it from the ground to that point against gravity.
Consider an object of mass, m. Let it be raised through a height, h from the ground. A force is required to do this. The minimum force required to raise the object is equal to the weight of the object, mg. The object gains energy equal to the work done on it. Let the work done on the object against gravity be W.
W = force × displacement
W = mg × h
W = mgh
This is the potential energy of the object.
$$ E_p = mgh $$
The work done by gravity depends on the difference in vertical heights of the initial and final positions of the object and not on the path along which the object is moved.
Law of Conservation of Energy
Whenever energy gets transformed, the total energy remains unchanged. This is the law of conservation of energy.
According to this law, energy can only be converted from one form to another; it can neither be created or destroyed. The total energy before and after the transformation remains the same. The law of conservation of energy is valid in all situations and for all kinds of transformations.
For example, let an object of mass, m be made to fall freely from a height, h. At the start, the potential energy is mgh and kinetic energy is zero. The total energy of the object is mgh. As it falls, its potential energy will change into kinetic energy. If v is the velocity of the object at a given instant, the kinetic energy would be 1⁄2 mv2. As the fall of the object continues, the potential energy would decrease while the kinetic energy would increase. When the object is about to reach the ground, h = 0 and v will be the highest. Therefore, the kinetic energy would be the largest and potential energy the least. However, the sum of the potential energy and kinetic energy of the object would be the same at all points.
potential energy + kinetic energy = constant
The sum of kinetic energy and potential energy of an object is its total mechanical energy.
During the free fall of the object, the decrease in potential energy, at any point in its path, appears as an equal amount of increase in kinetic energy.
Rate of Doing Work
Power measures the speed of work done, that is, how fast or slow work is done. Power is defined as the rate of doing work or the rate of transfer of energy.
If an agent does a work W in time t, then power is given by
$$ \text{Power} = \frac{\text{Work}}{\text{Time}} $$
$$ P = \frac{W}{t} $$
The unit of power is watt having the symbol W. 1 watt is the power of an agent, which does work at the rate of 1 joule per second. We can also say that power is 1 W when the rate of consumption of energy is 1 J s–1.
1 watt = 1 joule/second
1 W = 1 J s–1
1 kW = 1000 W = 1000 J s–1
The power of an agent may vary with time. This means that the agent may be doing work at different rates at different intervals of time. Therefore the concept of average power is useful. We obtain average power by dividing the total energy consumed by the total time taken.