Page 1
89
4.1 Introduction
Work is said to be done when a force applied on the body displaces the body
through a certain distance in the direction of force.
4.2 Work Done by a Constant Force
Let a constant force F be applied on the body such that it makes an angle
? with the horizontal and body is displaced through a distance s.
Then work done by the force in displacing the body through a distance s is
given by
W = (F cos ?) s = Fs cos ? ? W = (F cos ?) s = Fs cos ?
W =
4.3 Nature of Work Done
Positive work Negative work
Positive work means that force (or its Negative work means that force (or its
component) is parallel to displacement component) is opposite to displacement
0º = ? < 90º i.e., 90º < ? = 180º
The positive work sigmnes that the external The negative work sigmnes that the
external force favours the motion force opposes the motion of the body.
of the body.
Page 2
89
4.1 Introduction
Work is said to be done when a force applied on the body displaces the body
through a certain distance in the direction of force.
4.2 Work Done by a Constant Force
Let a constant force F be applied on the body such that it makes an angle
? with the horizontal and body is displaced through a distance s.
Then work done by the force in displacing the body through a distance s is
given by
W = (F cos ?) s = Fs cos ? ? W = (F cos ?) s = Fs cos ?
W =
4.3 Nature of Work Done
Positive work Negative work
Positive work means that force (or its Negative work means that force (or its
component) is parallel to displacement component) is opposite to displacement
0º = ? < 90º i.e., 90º < ? = 180º
The positive work sigmnes that the external The negative work sigmnes that the
external force favours the motion force opposes the motion of the body.
of the body.
4.4 Work Done by aVariable Force
When the magnitude and direction of a force varies with position, the work
done by such a force for an infinite simal displacement is given by
dW = .
The total work done in going from A to B is W = .
Area under force displacement curve with proper algebraic sign represents
work done by the force.
4.5 Work Depends on Frame of Reference
With change of frame of reference (inertial) force does not change while
displacement may change. So the work done by a force will be different in
different frames.
Examples : If a person is pushing a box inside a moving train, the work done
in the frame of train will while in the frame of earth will be
where is the displacement of the train relative to the ground.
4.6 Energy
The energy of a body is defined as its capacity for doing work.
(1) It is a scalar quantity.
(2) Dimension : [ML
2
T
2
] it is same as that of work or torque.
(3) Units : Joule [S.I.], erg [C.G.S.]
Practical units : electron volt (eV), Kilowatt hour (KWh), Calories (Cal)
Relation between different units :
1 Joule = 10
7
erg
1 eV = 1.6 × 10
–19
Joule
1 KWh = 3.6 × 10
6
Joule
1 Calorie = 4.18 Joule
(4) Mass energy equivalence : The relation between the mass of a particle
m and its equivalent energy is given as E = mc
2
where c = velocity of
light in vacuum.
Page 3
89
4.1 Introduction
Work is said to be done when a force applied on the body displaces the body
through a certain distance in the direction of force.
4.2 Work Done by a Constant Force
Let a constant force F be applied on the body such that it makes an angle
? with the horizontal and body is displaced through a distance s.
Then work done by the force in displacing the body through a distance s is
given by
W = (F cos ?) s = Fs cos ? ? W = (F cos ?) s = Fs cos ?
W =
4.3 Nature of Work Done
Positive work Negative work
Positive work means that force (or its Negative work means that force (or its
component) is parallel to displacement component) is opposite to displacement
0º = ? < 90º i.e., 90º < ? = 180º
The positive work sigmnes that the external The negative work sigmnes that the
external force favours the motion force opposes the motion of the body.
of the body.
4.4 Work Done by aVariable Force
When the magnitude and direction of a force varies with position, the work
done by such a force for an infinite simal displacement is given by
dW = .
The total work done in going from A to B is W = .
Area under force displacement curve with proper algebraic sign represents
work done by the force.
4.5 Work Depends on Frame of Reference
With change of frame of reference (inertial) force does not change while
displacement may change. So the work done by a force will be different in
different frames.
Examples : If a person is pushing a box inside a moving train, the work done
in the frame of train will while in the frame of earth will be
where is the displacement of the train relative to the ground.
4.6 Energy
The energy of a body is defined as its capacity for doing work.
(1) It is a scalar quantity.
(2) Dimension : [ML
2
T
2
] it is same as that of work or torque.
(3) Units : Joule [S.I.], erg [C.G.S.]
Practical units : electron volt (eV), Kilowatt hour (KWh), Calories (Cal)
Relation between different units :
1 Joule = 10
7
erg
1 eV = 1.6 × 10
–19
Joule
1 KWh = 3.6 × 10
6
Joule
1 Calorie = 4.18 Joule
(4) Mass energy equivalence : The relation between the mass of a particle
m and its equivalent energy is given as E = mc
2
where c = velocity of
light in vacuum.
4.7 Kinetic Energy
The energy possessed by a body by virtue of its motion is called kinetic
energy.
Let m = mass of the body, v = velocity of the body then K.E. = .
(1) Kinetic energy depends on frame of reference : The kinetic energy
of a person of mass m, sitting in a train moving with speed v, is zero in
the frame of train but in the frame of the earth.
(2) Work-energy theorem : It states that work done by a force acting on a
body is equal to the change produced in the kinetic energy of the body.
This theorem is valid for a system in presence of all types of forces
(external or internal, conservative or non-conservative).
(3) Relation of kinetic energy with linear momentum : As we know
E= p= 2mE
p2
2m
(4) Various graphs of kinetic energy
4.8 Potential Energy
4.9 Potential Energy
Potential energy is defined only for conservative forces. In the space occupied
by conservative forces every point is associated with certain energy which is
called the energy of position or potential energy. Potential energy generally
are of three types : Elastic potential energy and Gravitational potential energy
Page 4
89
4.1 Introduction
Work is said to be done when a force applied on the body displaces the body
through a certain distance in the direction of force.
4.2 Work Done by a Constant Force
Let a constant force F be applied on the body such that it makes an angle
? with the horizontal and body is displaced through a distance s.
Then work done by the force in displacing the body through a distance s is
given by
W = (F cos ?) s = Fs cos ? ? W = (F cos ?) s = Fs cos ?
W =
4.3 Nature of Work Done
Positive work Negative work
Positive work means that force (or its Negative work means that force (or its
component) is parallel to displacement component) is opposite to displacement
0º = ? < 90º i.e., 90º < ? = 180º
The positive work sigmnes that the external The negative work sigmnes that the
external force favours the motion force opposes the motion of the body.
of the body.
4.4 Work Done by aVariable Force
When the magnitude and direction of a force varies with position, the work
done by such a force for an infinite simal displacement is given by
dW = .
The total work done in going from A to B is W = .
Area under force displacement curve with proper algebraic sign represents
work done by the force.
4.5 Work Depends on Frame of Reference
With change of frame of reference (inertial) force does not change while
displacement may change. So the work done by a force will be different in
different frames.
Examples : If a person is pushing a box inside a moving train, the work done
in the frame of train will while in the frame of earth will be
where is the displacement of the train relative to the ground.
4.6 Energy
The energy of a body is defined as its capacity for doing work.
(1) It is a scalar quantity.
(2) Dimension : [ML
2
T
2
] it is same as that of work or torque.
(3) Units : Joule [S.I.], erg [C.G.S.]
Practical units : electron volt (eV), Kilowatt hour (KWh), Calories (Cal)
Relation between different units :
1 Joule = 10
7
erg
1 eV = 1.6 × 10
–19
Joule
1 KWh = 3.6 × 10
6
Joule
1 Calorie = 4.18 Joule
(4) Mass energy equivalence : The relation between the mass of a particle
m and its equivalent energy is given as E = mc
2
where c = velocity of
light in vacuum.
4.7 Kinetic Energy
The energy possessed by a body by virtue of its motion is called kinetic
energy.
Let m = mass of the body, v = velocity of the body then K.E. = .
(1) Kinetic energy depends on frame of reference : The kinetic energy
of a person of mass m, sitting in a train moving with speed v, is zero in
the frame of train but in the frame of the earth.
(2) Work-energy theorem : It states that work done by a force acting on a
body is equal to the change produced in the kinetic energy of the body.
This theorem is valid for a system in presence of all types of forces
(external or internal, conservative or non-conservative).
(3) Relation of kinetic energy with linear momentum : As we know
E= p= 2mE
p2
2m
(4) Various graphs of kinetic energy
4.8 Potential Energy
4.9 Potential Energy
Potential energy is defined only for conservative forces. In the space occupied
by conservative forces every point is associated with certain energy which is
called the energy of position or potential energy. Potential energy generally
are of three types : Elastic potential energy and Gravitational potential energy
etc.
(1) Change in potential energy : Change in potential energy between any
two points is defined in terms of the work done by the fo rce in displacing
the particle between these two points without any change in kinetic
energy.
U
2
– U
1
= ...(1)
(2) Potential energy curve : A graph plotted between the potential energy of
a particle and its displacement from the centre of force is called potential
energy curve. Negative gradient of the potential energy gives force.
= F
(5) Types of equilibrium : If net force acting on a particle is zero, it is said
to be in equilibrium.
For equilibrium, = 0, but the equilibrium of particle can be of three
types :
Stable Unstable Neutral
When a particle is displaced
slightly from a position, then
a force acting on it brings it
back to the initial position,
it is said to be in stable
equilibrium position.
Potential energy is minimum.
i.e., rate of change of is
positive.
Example : A marble placed
at the bottom of a hemi-
spherical bowl.
4.10 Elastic Potential Energy
When a particle is displaced
slightly from a position, then
a force acting on it tries to
displace the particle further
away from the equilibrium
position, it is said to be in
unstable equilibrium.
Potential energy is maximum.
i.e., rate of change of
is negative.
Example : A marble
balanced on top of a hemi-
spherical bowl.
When a particle is slightly
displaced from a position
then it does not experience
any force acting on it and
continues to be in equili-
brium in the displaced
position, it is said to be in
neutral equilibrium.
Potential energy is constant.
i.e., rate of change of
is zero.
Example : A marble placed
on horizontal table.
Page 5
89
4.1 Introduction
Work is said to be done when a force applied on the body displaces the body
through a certain distance in the direction of force.
4.2 Work Done by a Constant Force
Let a constant force F be applied on the body such that it makes an angle
? with the horizontal and body is displaced through a distance s.
Then work done by the force in displacing the body through a distance s is
given by
W = (F cos ?) s = Fs cos ? ? W = (F cos ?) s = Fs cos ?
W =
4.3 Nature of Work Done
Positive work Negative work
Positive work means that force (or its Negative work means that force (or its
component) is parallel to displacement component) is opposite to displacement
0º = ? < 90º i.e., 90º < ? = 180º
The positive work sigmnes that the external The negative work sigmnes that the
external force favours the motion force opposes the motion of the body.
of the body.
4.4 Work Done by aVariable Force
When the magnitude and direction of a force varies with position, the work
done by such a force for an infinite simal displacement is given by
dW = .
The total work done in going from A to B is W = .
Area under force displacement curve with proper algebraic sign represents
work done by the force.
4.5 Work Depends on Frame of Reference
With change of frame of reference (inertial) force does not change while
displacement may change. So the work done by a force will be different in
different frames.
Examples : If a person is pushing a box inside a moving train, the work done
in the frame of train will while in the frame of earth will be
where is the displacement of the train relative to the ground.
4.6 Energy
The energy of a body is defined as its capacity for doing work.
(1) It is a scalar quantity.
(2) Dimension : [ML
2
T
2
] it is same as that of work or torque.
(3) Units : Joule [S.I.], erg [C.G.S.]
Practical units : electron volt (eV), Kilowatt hour (KWh), Calories (Cal)
Relation between different units :
1 Joule = 10
7
erg
1 eV = 1.6 × 10
–19
Joule
1 KWh = 3.6 × 10
6
Joule
1 Calorie = 4.18 Joule
(4) Mass energy equivalence : The relation between the mass of a particle
m and its equivalent energy is given as E = mc
2
where c = velocity of
light in vacuum.
4.7 Kinetic Energy
The energy possessed by a body by virtue of its motion is called kinetic
energy.
Let m = mass of the body, v = velocity of the body then K.E. = .
(1) Kinetic energy depends on frame of reference : The kinetic energy
of a person of mass m, sitting in a train moving with speed v, is zero in
the frame of train but in the frame of the earth.
(2) Work-energy theorem : It states that work done by a force acting on a
body is equal to the change produced in the kinetic energy of the body.
This theorem is valid for a system in presence of all types of forces
(external or internal, conservative or non-conservative).
(3) Relation of kinetic energy with linear momentum : As we know
E= p= 2mE
p2
2m
(4) Various graphs of kinetic energy
4.8 Potential Energy
4.9 Potential Energy
Potential energy is defined only for conservative forces. In the space occupied
by conservative forces every point is associated with certain energy which is
called the energy of position or potential energy. Potential energy generally
are of three types : Elastic potential energy and Gravitational potential energy
etc.
(1) Change in potential energy : Change in potential energy between any
two points is defined in terms of the work done by the fo rce in displacing
the particle between these two points without any change in kinetic
energy.
U
2
– U
1
= ...(1)
(2) Potential energy curve : A graph plotted between the potential energy of
a particle and its displacement from the centre of force is called potential
energy curve. Negative gradient of the potential energy gives force.
= F
(5) Types of equilibrium : If net force acting on a particle is zero, it is said
to be in equilibrium.
For equilibrium, = 0, but the equilibrium of particle can be of three
types :
Stable Unstable Neutral
When a particle is displaced
slightly from a position, then
a force acting on it brings it
back to the initial position,
it is said to be in stable
equilibrium position.
Potential energy is minimum.
i.e., rate of change of is
positive.
Example : A marble placed
at the bottom of a hemi-
spherical bowl.
4.10 Elastic Potential Energy
When a particle is displaced
slightly from a position, then
a force acting on it tries to
displace the particle further
away from the equilibrium
position, it is said to be in
unstable equilibrium.
Potential energy is maximum.
i.e., rate of change of
is negative.
Example : A marble
balanced on top of a hemi-
spherical bowl.
When a particle is slightly
displaced from a position
then it does not experience
any force acting on it and
continues to be in equili-
brium in the displaced
position, it is said to be in
neutral equilibrium.
Potential energy is constant.
i.e., rate of change of
is zero.
Example : A marble placed
on horizontal table.
93
(1) Restoring force and spring constant : When a spring is stretched or
compressed from its normal position (x = 0) by a small distance x, a
restoring froce is produced in the spring to bring it to the normal position.
According to Hooke’s law this restoring force is proportional to the
displacement x and its direction is always opposite to the displacement.
i.e., a
or = ...(i)
where k is called spring constant.
(2) Expression for elastic potential energy :
Elastic potential energy U =
Note :
• If spring is stretched from initial position x
1
to final position x
2
then
work done = Increment in elastic potential energy
=
(3) Energy graph for a spring : It mean kinetic energy changes parabolically
w.r.t. position but total energy remain always constant irrespective to
position of the mass.
4.11 Law of Conservation of Energy
(1) Law of conservation of energy : For an isolated system or body in
presence of conservative forces the sum of kinetic and potential energies
at any point remains constant throughout the motion. It does not depends
upon time. This is known as the law of conservation of mechanical energy.
(2) Law of conservation of total energy : If the forces are conservative
and non-conservative both, it is not the mechanical energy alone which
is conserved, but it is the total energy, may be heat, light, sound or
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