Introduction
In the previous chapter, we have learnt about "Laws of Motion". In this chapter, we shall study notion of work and energy and relation between work and energy by the help of work energy theorem. The terms ‘work’, ‘energy’ and ‘power’ are frequently used in everyday language. A farmer ploughing the field, a construction worker carrying bricks, a student studying for a competitive examination, an artist painting a beautiful landscape, all are said to be working.
In physics, we use the term work to describe the process of transferring energy from object or system to another or converting energy from one form to another.
The Scalar or Dot Product of Two Vectors
The scalar product or dot product of any two vectors `\vec{A}` and `\vec{B}`, denoted as `\vec{A}` . `\vec{B}` (read as `\vec{A}` dot `\vec{B}`) is defined as
`\vec{A}` . `\vec{B}` = AB cosθ
Where A & B are magnitudes of vectors `\vec{A}` and `\vec{B}` respectively and θ is the smaller angle between them. Dot product is called scalar product as A, B and cosθ are scalars. Both vectors have a direction but their scalar product does not have a direction.
Properties
Dot product is commutative
A . B = B . ADot product is distributive
A . (B + C) = A . B + A . C
Dot product of a vector with itself gives square of its magnitude
A . A = AA cosθ = A
A . (λB) = λ(A . B)
where λ is a real number
- `\hat{i}` . `\hat{j}` = `\hat{j}` . `\hat{k}` = `\hat{k}` . `\hat{i}` = 0
- `\hat{i}` . `\hat{i}` = `\hat{j}` . `\hat{j}` = `\hat{k}` . `\hat{k}` = 1
Work Done by a Force
The work done by the force is defined to be the product of component of the force in the direction of the displacement and the magnitude of this displacement.
W = (F cosθ) d = F . d
We see that if there is no displacement, there is no work done even if the force is large. Thus, when you push hard against a rigid brick wall, the force you exert on the wall does no work.
Recommended Books
- NCERT Textbook For Class 11 Physics Part 1 & 2
- CBSE All In One Physics Class 11 2022-23 Edition
- Oswaal CBSE Chapterwise Question Bank Class 11 Physics Book
- Modern's abc Plus of Physics for Class-11 (Part I & II)
Read also: System of Particles and Rotational Motion Class 11 Physics Notes Chapter 7
Unit of Work
- SI unit of work is joule (J),
- CGS unit is erg.
- 1 J = 107 erg
- Gravitational unit of work in SI is kg m and
- Gravitational unit of work in CGS is g cm
No work is done if
the displacement is zero. A weightlifter holding a 150 kg mass steadily on his shoulder for 30 s does no work on the load during this time.
the force is zero. A block moving on a smooth horizontal table is not acted upon by a horizontal force, but may undergo a large displacement.
the force and displacement are mutually perpendicular. since, for θ = π/2 rad (= 90o), cos (π/2) = 0.
Read also: Thermodynamics Class 11 Notes Chemistry Chapter 6
Energy
Energy is a physical quantity which enables a system to perform work. There are so many form of energy like heat energy, sound energy, nuclear energy which will be discussed in their respective topics. Here we will discuss mechanical energy.
Mechanical Energy
Mechanical energy is the energy that is possessed by an object due to its motion or due to its position. Mechanical energy of an object is the sum of its kinetic and potential energy.
Mechanical energy = KE + PE
Kinetic Energy
kinetic energy K of an object of mass m is given as
K = 1 / 2 mv . v = 1 / 2 mv2
It is a scalar quantity. It is a measure of the work an object can do because of its motion. Sailing ships use KE of the wind. KE of a fast-flowing stream has been used for grinding corn.
Relation between KE and linear momentum
Let mass of body be m and velocity of body be v.
p = mv = linear momentum
`KE=\frac{1}{2}mv^2`
`KE=\frac{1}{2m}(m^2v^2)`
`KE=\frac{p^2}{2m}`
`p=\sqrt{2KE.m}`
Read also: Conceptual Questions for Class 11 Physics Chapter 6 Work, Power and Energy
Potential Energy
Potential energy is the energy that an object has because of its position relative to other objects. It is usually defined in equations by the capital letter U or sometimes by PE.
Gravitational Potential Energy
Gravitational potential energy as a function of the height h, is denoted by U(h) and it is the negative of work done by the gravitational force in raising the body to that height.
U(h) = mgh
where, m = mass of a body
The SI unit of potential energy is Joule, the same as kinetic energy or work. Commercial unit of energy is kWh, 1 kWh = 3.6 x 106 J.
Elastic Potential Energy of Spring
The energy associated with the state of compression or expansion of a spring is known as elastic potential energy. Elastic potential energy is given by
U= 1/2 kx2
where, k = force constant of given spring
The Law of Conservation of Energy
According to the law of conservation of energy, the total energy of an isolated system does not change. Energy may be transformed from one form to another but the total energy of an isolated system remains constant.
Besides mechanical energy, the energy may manifest itself in many other forms. Some of these forms are: thermal energy, electrical energy, chemical energy, visual light energy, nuclear energy etc. According to Einstein, mass and energy are inter-convertible. That is, mass can be converted into energy and energy can be converted into mass.
The Conservation of Mechanical Energy
According to this, "The total mechanical energy of a system is conserved if the forces doing work on it, are conservative."
Suppose that a body undergoes displacement Δx under the action of a conservative force F. From the work-energy theorem,
`\triangle K=\int F(x)dx`
If the force is conservative, the potential energy function U(x) can be defined such that
`\triangle U=-\int F(x)dx`
Adding the above two equations, we get
`\triangle K+\triangle U=0`
`\triangle (K+U)=0`
K + U = constant
The Work-Energy Theorem
The work-energy theorem states that the work done by all forces acting on a particle is equal to the change in the particle’s kinetic energy.
`W_{Total}=\triangle KE`
`W_{Total}=\frac{1}{2}mv_{f}^{2}-\frac{1}{2}mv_{i}^{2}`
Confining to one dimension, rate of change of kinetic energy with time is
`\frac{dK}{dt}=\frac{d}{dt}(\frac{1}{2}mv^{2})`
`\frac{dK}{dt}=\frac{1}{2}m\frac{d}{dt}(v^{2})`
`\frac{dK}{dt}=\frac{1}{2}m(2v\frac{dv}{dt})`
`\frac{dK}{dt}=m\frac{dv}{dt}v`
`\frac{dK}{dt}=Fv` ...`[\because F=ma]`
`\frac{dK}{dt}=F\frac{dx}{dt}`
dK = Fdx
`\int_{K_{i}}^{K_{f}} dK=\int_{x_{i}}^{x_{f}}Fdx`
Where Ki and Kf are the initial and final kinetic energies corresponding to xi and xf respectively.
`K_{f}-K_{i}=W`
Power
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}=\frac{W}{t}`
The instantaneous power is defined as the limiting value of the average power as time interval approaches zero.
`P=\frac{dW}{dt}`
The work dW done by force `\vec{F}` for a displacement d`\vec{r}` is
dW=`\vec{F}` . d`\vec{r}`
The instantaneous power can also be expressed as
`P=\vec{F}.\frac{d\vec{r}}{dt}=\vec{F}.\vec{v}`
where `\vec{v}` is the instantaneous velocity when the force is `\vec{F}`. Power is a scalar quantity like work and energy. Its dimensions are [ML2T–3]. Its SI unit is Watt (W).
[∵ 1 hp = 746 W ]
Collisions
If two object collide during their motion then this event is called Collision. Generally, collisions are two type-
(i) Elastic Collision
If, in a particular collision, there is no dissipation of energy, the total kinetic energy of the objects before collision is equal to total kinetic energy of the objects after collision. Such a collision is termed as elastic collision. As, Collision of two steel's ball.
(ii) Inelastic Collision
When there is a loss of kinetic energy, in any collision Then this tyoe of collision is called perfectly inelastic collision. As, When a soft mud ball is thrown against the wall
Summary
The work done by the force is defined as the product of component of the force in the direction of the displacement and the magnitude of this displacement.
SI unit of work is joule (J),
CGS unit is erg.
1 J = 107 erg
Gravitational unit of work in SI is kg m and
Gravitational unit of work in CGS is g cm
1 kg m = 9.8 J,
1 g cm = 980 erg,
1 kg m = 105 g cm
Energy is defined as the capacity to do work.
Kinetic energy of a body is defined as the energy possessed due to its motion.
Potential energy of a body is defined as the energy possessed due to its position or configuration.
The total mechanical energy of a system is conserved if the forces, doing work on it, are conservative (This is the principle of conservation of mechanical energy).
The work-energy theorem states that the change in kinetic energy of a body is the work done by the net force on the body. Kf – Ki = Wnet.
Power is defined as the time rate at which work is done or energy is transferred.
An elastic collision is the one in which the initial kinetic energy is equal to the final kinetic energy. A completely inelastic collision is the one in which the two particles move together after the collision.
- An inelastic collision is the one in which deformation is partly relieved and some of the initial kinetic energy is lost.
During a collision,
- (i) The total linear momentum is conserved at each instant of the collision.
- (ii) In elastic collision kinetic energy before collision is equal to kinetic energy after collision.
