Physics - Work and Energy

Introduction

When acting (applying force), there is a displacement of the point of application in the direction of the force, is known as work.
The term work was first introduced by the French mathematician GaspardGustave Coriolis in 1826.

Work done by a force acting on the object is equal to the magnitude of the force multiplied by the distance moved in the direction of the force and it is calculated as −
Work done (W) = Force (F) × Displacement (s)
Work has the only magnitude and no direction.
The SI unit of work is the joule (J).

Energy

Energy can be converted in form, but cannot be created or destroyed. For example, producing electric energy from solar energy, etc.
The object which does the work loses energy and the object on which the work is done gains energy.
Further, the object that possesses energy can exert a force on another object to transfer energy from former to later.
The energy possessed by an object is therefore measured in terms of its capacity of doing work.
The SI unit of energy is joule (J).

Forms of Energy

Following are the major forms of the energy −
- Potential energy
- Kinetic energy
- Heat energy
- Chemical energy
- Electrical energy
- Light energy
Let’s discuss each one in brief

Potential Energy

The energy, possessed by a body by virtue of its position relative to others, is known as Potential energy.
So, potential energy is the stored energy in an object. For example, gravitational potential energy, elastic potential energy, electrical potential energy, etc.
The SI unit of potential energy is joule (J).
The term potential energy was introduced by Scottish engineer and physicist William Rankine.

Kinetic Energy

The energy that an object possesses because of its motion, is known as kinetic energy.

The running/moving body maintains its kinetic energy unless its speed changes (increases or decreases).
The SI unit of kinetic energy is joule (J).

Heat Energy

Heat is a form of energy transferred spontaneously from a hotter to a colder body.

Chemical Energy

The potential of a chemical substance to experience a transformation through a chemical reaction and transform other chemical substances is known as chemical energy. E.g. Breaking or making of chemical bonds, batteries, etc.
The chemical energy of a (chemical) substance can be converted to other forms of energy by a chemical reaction. E.g., green plants convert solar energy to chemical energy (commonly of oxygen) by the process of photosynthesis.

Electrical Energy

The energy, derived from electric potential energy or kinetic energy, is known as electrical energy.
Electricity is normally produced by electromechanical generators at a power station.
The electromechanical generators primarily are driven by heat engines fueled by the kinetic energy of flowing water and wind.
The electromechanical generators are also driven by heat engines fueled by chemical combustion or nuclear fission.

Light Energy

Light is a form of electromagnetic radiation.
Light energy most likely is the only form of energy that we can really see.

Light is transferring energy through the space in a natural way. E.g. solar energy.

Law of Conservation of Energy

Law of conservation of energy states that energy can neither be created nor be destroyed; however, it can be only transformed from one form to another.
According to the law of conservation of energy, the total energy before and after the transformation remains the same.
The law of conservation of energy remains valid in all conditions and locations and for all kinds of transformations.

Work

When a force acts on an object and the object actually moves in the direction of force, then the work is said to be done by the force.

Work done by the force is equal to the product of the force and the displacement of the object in the direction of force.

If under a constant force F the object displaced through a distance s, then work done by the force

W = F * s = F s cos θ

where a is the smaller angle between F and s.

Work is a scalar quantity, Its S1 unit is joule and CGS unit is erg.

∴ 1 joule = 10⁷ erg

Its dimensional formula is [ML²T^-2].

Work done by a force is zero, if

(a) body is not displaced actually, i.e., s = 0

(b) body is displaced perpendicular to the direction of force, i.e.,

θ = 90°

Work done by a force is positive if angle between F and s is acute angle.

Work done by a force is negative if angle between F and s is obtuse angle.

Work done by a constant force depends only on the initial and final Positions and not on the actual path followed between initial and final positions.

Work done in different conditions

(i) Work done by a variable force is given by

W = ∫ F * ds

It is equal to the area under the force-displacement graph along with proper sign.

Work done = Area ABCDA

(ii) Work done in displacing any body under the action of a number of forces is equal to the work done by the resultant force.

(iii) In equilibrium (static or dynamic), the resultant force is zero therefore resultant work done is zero.

(iv) If work done by a force during a rough trip of a system is zero, then the force is conservative, otherwise it is called non-conservative force.

Gravitational force, electrostatic force, magnetic force, etc are conservative forces. All the central forces are conservative forces.
Frictional force, viscous force, etc are non-conservative forces.

(v) Work done by the force of gravity on a particle of mass m is given by W = mgh

where g is acceleration due to gravity and h is height through particle one displaced.

(vi) Work done in compressing or stretching a spring is given by

W = 1 / 2 kx²

where k is spring constant and x is displacement from mean position.

(vii) When on end of a spring is attached to a fixed vertical support and a block attached to the free end moves on a horizontal

table from x = x₁ to x = x₂ then W = 1 / 2 k (x²x₂ – x²x₁)

(viii) Work done by the couple for an angular displacement θ is given by W = i * θ

where i is the torque of the couple.

power

The time rate of work done by a body is called its power.

Power = Rate of doing work = Work done / Time taken

If under a constant force F a body is displaced through a distance s in time t, the power

p = W / t = F * s / t

But s / t = v ; uniform velocity with which body is displaced.

∴ P = F * v = F v cos θ

where θ is the smaller angle between F and v.

power is a scalar quantity. Its S1 unit is watt and its dimensional formula is [ML²T^-3].

Its other units are kilowatt and horse power,

1 kilowatt = 1000 watt

1 horse power = 746 watt

Energy

Energy of a body is its capacity of doing work.

It is a scalar quantity.

Its S1 unit is joule and CGS unit is erg. Its dimensional formula is [ML³T^-3].

There are several types of energies, such as mechanical energy (kinetic energy and potential energy), chemical energy, light energy, heat energy, sound energy, nuclear energy, electric energy etc.

Mechanical Energy

The sum of kinetic and potential energies at any point remains constant throughout the motion. It does not depend upon time. This is known as law of conservation of mechanical energy.

Mechanical energy is of two types:

1. Kinetic Energy

The energy possessed by any object by virtue of its motion is called its kinetic energy.

Kinetic energy of an object is given by

k = 1 / 2 mv² = p² / 2m

where m = mass of the object, U = velocity of the object and p = mv = momentum of the object.

2. Potential Energy

The energy possessed by any object by virtue of its position or configuration is called its potential energy.

There are three important types of potential energies:

(i) Gravitational Potential Energy If a body of mass m is raised through a height h against gravity, then its gravitational potential energy = mgh,

(ii) Elastic Potential Energy If a spring of spring constant k is stretched through a distance x. then elastic potential energy of the spring = 1 . 2 kx²

The variation of potential energy with distance is shown in figure.

Potential energy is defined only for conservative forces. It does not exist for non-conservative forces.

Potential energy depends upon frame of reference.

(iii) Electric Potential Energy The electric potential energy of two point charges ql and q’l. separated by a distance r in vacuum is given by

U = 1 / 4πΣ₀ * q₁q₂ / r

Here 1 / 4πΣ₀ = 9.0 * 10¹⁰ N-m² / C² constant.

Work-Energy Theorem

Work done by a force in displacing a body is equal to change in its kinetic energy.

where, K_i = initial kinetic energy

and K_f = final kinetic energy.

Regarding the work-energy theorem it is worth noting that

(i) If W_net is positive, then K_f – K_i = positive, i.e., K_f > K_i or kinetic energy will increase and vice-versa.

(ii) This theorem can be applied to non-inertial frames also. In a non-inertial frame it can be written as:

Work done by all the forces (including the Pseudo force) = change in kinetic energy in non-inertial frame.

Mass-Energy Equivalence

According to Einstein, the mass can be transformed into energy and vice – versa.

When Δm. mass disappears, then produced energy

E = Δmc²

where c is the speed of light in vacuum.

Principle of Conservation of Energy

The sum of all kinds of energies in an isolated system remains constant at all times.

Principle of Conservation of Mechanical Energy

For conservative forces the sum of kinetic and potential energies of any object remains constant throughout the motion.

According to the quantum physics, mass and energy are not conserved separately but are conserved as a single entity called ‘mass-energy’.

Collisions

Collision between two or more particles is the interaction for a short interval of time in which they apply relatively strong forces on each other.

In a collision physical contact of two bodies is not necessary. rrhere are two types of collisions:

1. Elastic collision

The collision in which both the momentum and the kinetic energy of the system remains conserved are called elastic collisions.

In an elastic collision all the involved forces are conservative forces.

Total energy remains conserved.

2. Inelastic collision

The collision in which only the momentum remains conserved but kinetic energy does not remain conserved are called inelastic collisions.

In an inelastic collision some or all the involved forces are non-conservative forces.

Total energy of the system remains conserved.

If after the collision two bodies stick to each other, then the collision is said to be perfectly inelastic.

Coefficient of Restitution or Resilience

The ratio of relative velocity of separation after collision to the velocity of approach before collision is called coefficient of restitution resilience.

It is represented by e and it depends upon the material of the collidingI bodies.

For a perfectly elastic collision, e = 1

For a perfectly inelastic collision, e = 0

For all other collisions, 0 < e < 1

One Dimensional or Head-on Collision

If the initial and final velocities of colliding bodies lie along the same line, then the collision is called one dimensional or head-on collision.

Inelastic One Dimensional Collision

Applying Newton’s experimental law, we have

Velocities after collision

v₁ = (m₁ – m₂) u₁ + 2m₂u₂ / (m₁ + m₂)

and v₂ = (m₂ – m₁) u₂ + 2m₁u₁ / (m₁ + m₂)

When masses of two colliding bodies are equal, then after the collision, the bodies exchange their velocities.

v₁ = u₂ and v₂ = u₁

If second body of same mass (m₁ = m₂) is at rest, then after collision first body comes to rest and second body starts moving with the
initial velocity of first body.

v₁ = 0 and v₂ = u₁

If a light body of mass m₁ collides with a very heavy body of mass m₂ at rest, then after collision.

v₁ = – u₁ and v₂ = 0

It means light body will rebound with its own velocity and heavy body will continue to be at rest.

If a very heavy body of mass m₁ collides with a light body of mass m₂(m₁ > > m₂₁) at rest, then after collision

v₁ = u₁ and v₂ = 2u₁

In Inelastic One Dimensional Collision

Loss of kinetic energy

ΔE = m₁m₂ / 2(m₁ + m₂) (u₁ – u₂)² (1 – e²)

In Perfectly Inelastic One Dimensional Collision

Velocity of separation after collision = 0.

Loss of kinetic energy = m₁m₂ (u₁ – u₂)² / 2(m₁ + m₂)

If a body is dropped from a height h_o and it strikes the ground with velocity v_o and after inelastic collision it rebounds with velocity v₁and rises to a height h₁, then

If after n collisions with the ground, the body rebounds with a velocity v_n and rises to a height h_n then

eⁿ = v_n / v_o = √hⁿ / h^o

Two Dimensional or Oblique Collision

If the initial and final velocities of colliding bodies do not lie along the same line, then the collision is called two dimensional or oblique Collision.

In horizontal direction,

m₁u₁ cos α₁+ m₂u₂ cos α₂= m₁v₁ cos β₁ + m₂v₂ cos β₂

In vertical direction.

m₁u₁ sin α₁ – m₂u₂ sin α₂ = m₁u₁ sin β₁ – m₂u₂ sin β₂

If m₁ = m₂ and α₁ + α₂ = 90°

then β₁ + β₂ = 90°

If a particle A of mass m₁ moving along z-axis with a speed u makes an elastic collision with another stationary body B of mass m₂

From conservation law of momentum

m₁u = m₁v₁ cos α + m₂v₂ cos β

O = m₁v₁ sin α – m₂v₂ sin β

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WORK POWER AND ENERGY