Physics Notes Class 11
CHAPTER 6
WORK,
ENERGY AND POWER
Physics
Notes Class 11 CHAPTER 6 WORK,
ENERGY AND POWER
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, Ki
= initial kinetic energy
and Kf
= final kinetic energy.
Regarding the
work-energy theorem it is worth noting that
(i)
If Wnet is positive,
then Kf – Ki = positive, i.e., Kf > Ki
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
noninertial 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 ho and it strikes the ground with velocity vo
and after inelastic collision it rebounds with velocity v1 and rises
to a height h1, then
If after n
collisions with the ground, the body rebounds with a velocity vn and
rises to a height hn then en = vn / vo
= √hn / ho
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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