In the last few chapters, we have discussed,
- Different ways of describing the motion of objects
- The cause of motion and gravitation
Another important concept that encourages us to understand and explain many natural phenomena is ‘work’, which is closely related to energy and power.
We require energy for other activities such as playing, singing, reading, speaking, writing, thinking, jumping, cycling, running, etc.
You are working hard to push a large rock. Let us say the rock does not move an inch despite all the effort put in by you. You get exhausted fully. But, you have not performed any work on the rock as there is no displacement of the rock.
No displacement
If we use the scientific definition, the above two activities involve a lot of work. Activities such as playing on the ground, interacting with friends, humming a tune, watching a movie, attending a function are sometimes not considered work.
Example 1 - Carrying a box:
Carrying a box
In the case of carrying a box, you are using a vertical force to lift a box while you are moving horizontally in the distance. This indicates zero work is being done as the force and distance are not applied in the same direction.
Example 2 - A book falls off a table (free falls to the ground):
Book fall from table
This is an example of work. There is a force (gravity) that acts on the book, which causes it to be displaced in a downward direction (i.e., "fall").
A closer look at the above scenarios shows that two conditions require to be satisfied for work to be done:
- A force should act on an object, and
- The object must be relocated or displaced
Let us consider,
Work done
F → Constant force acting on an object
s → Displacement of the object in the direction of the force
W → Work done
Work is calculated as the product of the force and displacement
\(Workdone\ =\ Force\ \times\ Displacement\)
\(W\ =\ F\ \times\ s\)
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 Nm.
Consider the scenario in which an object moves with a uniform velocity along a particular direction. Retarding force, F, is applied in the opposite direction to the object's movement. That is, the angle between the two directions is 180º. Let the object stops after a displacement.
In such cases, the work done by the force, F, is taken as negative and denoted by the minus sign.
\(Workdone\ =\ Force\ \times\ Displacement\)
\(W\ =\ - F\ \times\ s\)
From the above discussion, the work done by a force can be either positive or negative.
Positive work:
When force and displacement are in the same direction, the work performed on an object is said to be positive work.
Negative work:
Negative work is performed if the displacement is opposite to the direction of the force applied.
Energy:
Life is impossible without energy. The energy demand is ever increasing.
The Sun is the most significant natural source of energy for us. Many of our sources of energy are derived from the Sun. We can also take energy from the nuclei of atoms, the inside of the earth, and the tides.
An object having the ability 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.
Therefore, the unit of energy is the same as that of work: \(joule (J)\).
\(1\ J\) is the energy required to do \(1\ joule\ of\ work\). Sometimes a larger unit of energy called kilojoule (kJ) is used. \(1\ kJ\) equals \(1000\ J\).
Energy can exist in two basic mechanical forms,
- Kinetic Energy - which is the energy of motion, and
- Potential Energy - which is the stored energy due to position or state
In this section, we will derive the formula to find the kinetic energy of an object.
Kinetic energy:
Kinetic energy is the energy contained 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 obtain that velocity.
Let us now represent the kinetic energy of an object in the form of an equation.
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.
From the equation of work done, We know that,
\(Work\ done\ =\ F\ \times s\)
The work done on the object will cause a change in its velocity. Let its velocity change from initial velocity (u) to final velocity (v).
Let a be the acceleration produced.
We studied three equations of motion. The relation connecting the initial velocity (u) and final velocity (v) of an object moving with a uniform acceleration (a), and the displacement (s) is,
\(v^2\ -\ u^2\ =\ 2as\)
Rearranging it to gets, \(s\ =\ \frac{v^2\ -\ u^2}{2a}\)
We also know that,
\(Force\ (F)\ =\ Mass\ (m)\ \times\ Acceleration(a)\)
Substitute the equation s and F in work done,
\(Work\ done\ =\ F\ \times s\)
\(=\ F\ \times \frac{v^2\ -\ u^2}{2a}\)
\(=\ F\ \times \frac{v^2\ -\ u^2}{2a}\)
Simplifying it weget,= \(\frac{1}{2}\ \times\ m\ \times\ (v^2\ -\ u^2)\)
If the object starts from its stationary position, that is, \(u\ =\ 0\), then,
\( Work\ done (W)\ =\ \frac{1}{2}\ \times\ m\ \times\ (v^2)\)
It is clear that the work done is equal to the change in the kinetic energy of an object.
If \(u\ =\ 0\), the work done will be \(\frac{1}{2}\ \times\ m\ \times\ (v^2)\).
Thus, the kinetic energy possessed by an object of mass (m) and moving with a uniform velocity (v) is,
\(E_k\ =\ \frac{1}{2}\ \times\ m\ \times\ (v^2)\)
Objects in motion contain energy. We define this energy as kinetic energy.
Potential Energy:
Let us try to get a basic knowledge of potential energy through the following activity.
- Take a bamboo stick and create a bow, as shown in the below figure.
- Locate an arrow made of a light stick on it with one end supported by the stretched string.
- Now stretch the string and release the arrow.
- Observe the arrow flying off the bow.
- Notice the change in the shape of the bow.
Discussion and Conclusions of the activity:
- The potential energy stored in the bow due to the change of shape is thus used as kinetic energy in throwing off the arrow.
- When the arrow is released from the bow, it acquires its original configuration as the elastic potential energy of the bow and stretched string converts into kinetic energy of the arrow.
Potential energy is energy that is stored or conserved in an object or substance. This stored energy is based on the position, arrangement or state of the object or substance.
Potential energy is mainly classified into two types:
- Gravitational potential energy
- Elastic potential energy
Gravitational potential energy is energy in an object that is held in a vertical position.
Elastic potential energy is energy stored in objects that can be stretched or compressed.
The potential energy possessed by the object is the energy present in it by virtue of its position or configuration.
Gravitational Potential energy:
The potential energy of an object at a particular height depends on the ground level or the zero level you select. An object in a given position can have specific potential energy corresponding to one level and a different value of potential energy with respect to another level.
It is beneficial to note that the work done by gravity depends on the difference in vertical heights of the initial (Ground level) and final positions of the object and not on the path along which the object is moved.
Let the work done on the object against gravity be W. That is,
\(Workdone\ (W)\ =\ Force(F)\ \times\ Displacement(s)\)
We know that, \(Force\ (F)\ =\ Mass(m)\ \times\ Acceleration\ due\ to\ gravity(g)\)
\(Displacement\ (s)\ =\ h\)
Subtituting the above equations in Workdone,
\(W\ =\ (m\ \times\ g)\ \times\ h\)
\(W\ =\ mgh\)
Since,
\(Workdone\ =\ mgh\),
Energy equal to mgh units is gained by the object. mgh is the potential energy of the object.
The power is represneted as rate of doing work, \(P\ = \frac{E}{T}\)
Total mechanical energy:
The sum of kinetic energy and potential energy of an object is its total mechanical energy. We find that during the free fall of the object, the decrease in potential energy appears as an equal amount of increase in kinetic energy at any point in its path. (Here, the effect of air resistance on the motion of the object has been ignored) There is thus a continual transformation of gravitational potential energy into kinetic energy.