ROTATIONAL MOTION_BLC_SWE_FALL23.pptx

CLASSIFICATION OF MOTION
1. Based on duration: Periodic, Non periodic
2. Based on Speed: Uniform, non-uniform
3. Based on Path: Translatory, Circular, Rotatory, Oscillatory and
Vibratory

Translatory Motion What is Translatory Motion?
• If a body moves as a whole such that all particles of the
body moves with same velocity in straight parallel path,
then the body is said to be in translatory motion.

CLASSIFICATION OF TRANSLATORY
MOTION
• Translatory motion
can be of two types:
rectilinear and
curvilinear.
• Rectilinear Motion
A car moving along a
straight path and the
train moving in a straight
track are examples of
rectilinear motion
• Curvilinear Motion
A stone thrown up in the
air at a certain angle and
a car taking a turn are
examples of curvilinear
motion

Circular Motion
• In physics, circular motion is a movement of an
object along the circumference of a circle or rotation
along a circular path. Circular motion is one type of
rotational motion and the axis through which the
object rotates is called axis of rotation.
• Examples of circular motion include:
• an artificial satellite orbiting the Earth at constant
height, a stone which is tied to a rope and is being
swung in circles, a car turning through a curve in a
race track, an electron moving perpendicular to a
uniform magnetic field

Angular displacement θ
Consider a particle revolving around a point O in a circle of
radius r. Let the position of the particle at time t = 0 be A and
after time t, its position is B.
It can be measured by using a simple formula. The formula is:
where,
θ is the angular displacement,
s is the distance travelled by the body, and
r is the radius of the circle along which it is moving.
In simpler words, the displacement of an object is the
distance travelled by it around the circumference of a circle
divided by its radius.

Angular velocity(ω)
• The rate of change of angular displacement is called
angular velocity.

Acceleration In Circular Motion
Acceleration Type Definition Direction Cause
Angular acceleration
(α = d/dt)
Measure of change in
angular velocity
Around the axis
of rotation
Caused by torque or force
applied to rotating object
Tangential
acceleration
at = rα=d|v|/dt Acceleration along the
tangent to the circular
path
Tangent to the
circular path
Caused by change in linear
speed (magnitude of
velocity)
Centripetal
acceleration
ac=v2/r=r2
Acceleration directed
towards the center of
the circular path
Towards the
center of the
circle
Required to keep an object
moving in a circular path

Acceleration In Circular Motion
Angular
acceleration
α
Tangential
acceleration
at
Centripetal
Acceleration
ac
Net
acceleration
anet
α = d/dt
angular
acceleration α is
rate of change in
angular velocity 
at = rα
=d|v|/dt
Due to magnitude
change of velocity
tangential
acceleration is rate of
change in linear
velocity by time for
both term
ac=v2/r=r2
Due to direction
change of velocity
anet
2 = at
2 + ac
2

COMMON MISTAKES AND MISCONCEPTIONS
• People sometimes mix up angular and tangential (or linear)
acceleration. Angular acceleration is the change in angular velocity divided
by time, while tangential acceleration is the change in linear velocity
divided by time.
• People sometimes forget that angular acceleration does not change with
radius, but tangential acceleration does. For example, for a rotating wheel
that is speeding up, a point on the outside covers more distance in the
same amount of time as a point closer to the center. It has a much larger
tangential acceleration than the portion closer to the axis of rotation.
However, the angular acceleration of every part of the wheel is the same
because the entire object moves as a rigid body through the same angle in
the same amount of time.

Example
A particle is in circular motion with an acceleration α = 0.2 rad s−2.
a) What is the angular displacement made by the particle after 5 s?
b) What is the angular velocity at t = 5 s?. Assume the initial angular velocity is zero.
Solution
Since the initial angular velocity is zero
(ω0 = 0). The angular displacement made
by the particle is given by
Now, angular velocity will be :

Example
• A particle moves in a
circle of radius 10 m.
Its linear speed is
given by v = 3t where
t is in second and v is
in m s-1.
• a) Find the
centripetal and
tangential
acceleration at t = 2
s.
• b) Calculate the
angle between the
resultant
acceleration and the

RIGID BODY
Rigid body is a solid body with a fixed geometrical shape and size, both of which
have negligible change during the motion or under the action of the applied
forces.
• The distance between any two given points on a rigid body remains constant in
time regardless of external forces exerted on it.

RIGID BODY
The motion of a rigid body which is not pivoted or fixed in some way is
either
• PURE TRANSLATION
• or a COMBINATION OF TRANSLATION AND ROTATION.
• The motion of a rigid body which is pivoted or fixed in some way is
rotation.
• The rotation may be about an axis that is fixed (e.g. a ceiling fan) or
moving (e.g. an oscillating table fan.
• We shall consider rotational motion about a fixed axis only

Rotational Motion
• In physics, Rotational motion refers to
the movement of an object around an
axis or center point. It involves spinning
or turning about an imaginary line known
as the axis of rotation.
• Each particles of the object move in a
circular path and the center of all circles
lie on the axis of rotation
• Examples of rotational motion include a
spinning top, the Earth rotating on its
axis, the wheels of a car in motion, or a
rotating fan.
axis of rotation

CIRCULAR AND ROTATIONAL MOTION
Circular motion Rotational motion
• The object in a circular motion just
moves in a circle.
• The object rotates around an axis
in rotational motion.
• The distance between the center
of mass and the axis of rotation
remains fixed.
• Rotational movement is based on
the rotation of the body around
the center of mass.
• The axis of rotation remains fixed. • The axis of rotation can change.
• Artificial satellites, for example,
orbit the Earth at a fixed altitude.
• The Earth, for example, rotates on
its own axis.

Rigid body: Rigid body is a body with a fixed
geometrical shape and size, both of which do
not change during the motion or under the
action of the applied forces.
Difference between circular motion and
rotational motion:
Difference Between Circular Motion And Rotational Motion:

COMPARISON-
PARAMETERS
OF MOTION
(LINEAR AND
ROTATIONAL)

CENTER OF MASS (COM)
Center of mass of a system is the point
that behaves as whole mass of the
system is concentrated at it and all
external forces are acting on it.
• For rigid bodies, center of mass is
independent of the state of the body
i.e., Whether it is in rest or in
accelerated motion center of mass will
remain same

WHAT IS USEFUL ABOUT COM?
• The interesting thing
about the COM of an
object or system is that it
is the point where any
uniform force on the
object acts.
This is useful because
it makes it easy to
solve mechanics
problems where we
have to describe the
motion of oddly-
shaped objects and
complicated systems.

Moment of Inertia
• Moment of inertia: The inertia of rotational motion is called moment
of inertia. It is denoted by L.
• Moment of inertia is the property of an object by virtue of which it
opposes any change in its state of rotation about an axis.
• The moment of inertia of a body about a given axis is equal to the
sum of the products of the masses of its constituent particles and the
square of their respective distances from the axis of rotation.

Moment of Inertia of a Rigid Body

Physical Significance of
Moment of Inertia (I)
• We know that,
• In translational motion, F = m.a - - - - -- - - - - (i)
• In rotational motion, τ=Iα- - - - -- - - - - (ii)
• Where τ is the torque acting on the rigid body and α
is the angular acceleration.
• By comparison of these two expressions, it is clear
that force and acceleration in translational motion
are analogous to torque and angular acceleration in
rotational motion respectively.
• Therefore, we can say that moment of inertia in
rotational motion must be analogous to mass in
translational motion. In translational motion, the
mass of a body represents its oppose to change in
its state of translational motion. Similarly, moment of
inertia in rotational motion represents its oppose to
change in its state of rotation. Thus, moment of
inertia is inertia in rotational motion.
• The physical significance of the moment of inertia is
same in rotational motion as the mass in linear
motion.
The moment of inertia is only one of the numerous mass
characteristics that may be used to quantify the stability of
a structure as well as the amount of force required to
change its motion. When it comes to building construction,
steadiness is an essential component that must be
considered throughout the design and production of
various buildings. Understanding the moment of inertia
along different axes is crucial for evaluating
a structure’s robustness against both external forces and
internal forces. In this article, we will go over all of the
different features of the moment of inertia, as well as
determine the moment of inertia of the disc.

• What is the moment of inertia?
• Why is the moment of inertia important?
• Aspects influencing moment of inertia
• Physical significance of the moment of
inertia
• Moment of inertia of a disc
• Solid disk
• Axis at the periphery
• A disc having a hole
• FAQs
• Why is the moment of inertia calculated?
• When compared to a circular disc, why does
a ring have a larger moment of inertia?
• Related Posts

• The moment of inertia is only one
of the numerous mass
characteristics that may be used to
quantify the stability of a structure
as well as the amount of force
required to change its motion.
• When it comes to building
construction, steadiness is an
essential component that must be
considered throughout the design
and production of various
buildings.
• Understanding the moment of inertia along
different axes is crucial for evaluating
a structure’s robustness against both
external forces and internal forces.
• In this article, we will go over all of the
different features of the moment of inertia,
as well as determine the moment of inertia
of the disc.

What Is The Moment Of Inertia?
• A body’s moment of inertia is equal
to the product of the masses of all
its particles multiplied by the square
of their proximity from the rotation’s
axis. Or, to put it another way,
• it is the “quantity” that determines
how much torque is required to
achieve a certain angular
acceleration around a rotating axis.
• An object’s rotational inertia, or
moment of inertia, is sometimes
referred to as its angular mass.
• When calculating moments of
inertia, it is common practice
to do so with reference to a
particular axis of rotation.
• The concentration of mass
around a rotational axis is a
primary factor in determining
the outcome.
• The moment of Inertia might
be different depending on
which axis is selected.

Why is the moment of inertia important?
• The moment of inertia establishes how much torque is required for a
given angular acceleration.
• Torque (or rotating force) is determined by the mass moment of inertia.
• The magnitude of torque needed to get a certain angular acceleration
may be calculated by multiplying the moment of inertia by the angular
acceleration. For a given acceleration, a higher moment of inertia
number means more torque is needed.
• The designer’s ability to accurately identify these values is crucial for
meeting the stringent performance requirements of the construction
industry.
• The designer’s ability to strike the right balance between compactness,
lightness, and efficiency is crucial to the success of any endeavour.
• Measuring MOI may also be used to ensure that the tolerances and
targets of the production and assembly processes are acceptable.

Aspects Influencing Moment Of Inertia
• The following is a list of the fundamental elements that
influence the moment of inertia:
• The density of the material
• The dimensions of the material
• The form that the substance takes
• Axis of rotation

Physical significance of the
moment of inertia
• The moment of inertia carries the same weight in terms of its
physical implications as a mass that is moving in a linear
direction.
• When determining a body’s inertia during translational motion,
mass is the most important factor to consider.
• The magnitude of an object’s moment of inertia grows as its
mass does.
• The force that is necessary to produce linear acceleration will,
as a result, increase.
• When anything is moving in a rotating motion, the angular
acceleration will be higher if the moment of inertia is larger.

Radius of Gyration (K)
• Radius of gyration or gyradius of a body
about an axis of rotation is defined as
the radial distance to a point which
would have a moment of inertia the
same as the body's actual distribution
of mass, if the total mass of the body
were concentrated.
• It is the imaginary radius from the
reference axis where the whole mass is
assumed to be concentrated.

Torque τ is defined as a quantity in rotational motion,
which when multiplied by a small angular
displacement gives us work done in rotational
motion. This quantity corresponds to force in linear
motion, which when multiplied by a small linear
displacement gives us work done in linear motion.
PHYSICAL SIGNIFICANCE OF TORQUE:
Torque in rotational motion is same as force
in linear motion! All it does is include the
angular rotation. Otherwise torque is the
force that would cause displacement.
Torque is the turning effect of a force about
the axis of rotation.
τ = r x F = rF sinθ n
It is a vector quantity. If the nature of the
force is to rotate the object clockwise, then
torque is called negative and if rotate the
object anticlockwise, then it is called
positive.
Its SI unit is ‘newton-metre’ and its
dimension is [ML2T-2].
In rotational motion, torque, τ = Iα where a
is angular acceleration and 1is moment of
inertia.
• Torque is the twisting effect of the
force applied to a rotating object
which is at a position r from its
axis of rotation. Mathematically,
this relationship is represented as
follows:
TORQUE τ

TORQUE τ
• Physicists often discuss
torque within the context of
equilibrium, even though an
object experiencing net
torque is definitely not in
equilibrium.
• In fact, torque provides a convenient
means for testing and measuring the
degree of rotational or circular
acceleration experienced by an object,
just as other means can be used to
calculate the amount of linear
acceleration.
• In equilibrium, the net sum of all
forces acting on an object should be
zero; thus in order to meet the
standards of equilibrium, the sum of
all torques on the object should also
be zero.

TORQUE τ
• Balance and Equilibrium: Torque is
essential in determining the rotational
equilibrium of an object.
• For an object to be in equilibrium, the
sum of the torques acting on it must
be zero. This is known as the
"principle of moments."
• In practical terms, this means that an
object will remain at rest or rotate at a
constant angular velocity if the total
torque acting on it is balanced.
• This is why a balanced seesaw or a
balanced bicycle wheel stays
stationary or rotates steadily.
• Applications: Torque has numerous
applications in everyday life and
various fields, including
engineering, mechanics, and
physics. It is critical in designing
and analyzing machines, engines,
vehicles, and other mechanical
systems that involve rotational
motion.

Consider a rigid
body rotating
with a uniform
angular
acceleration 
about a fixed axis
passing through
point O.
Suppose that the
body consists of
n particles of
masses m1, m2,
m3 ……., mn
situated at
distance r1, r2, r3
………rn from the
axis of rotation
respectively. As
the body rotates,
all the particles
perform circular
motion with the
same angular
acceleration .

Expression for K.E. of Rotation

• Angular momentum, property characterizing the rotary inertia of an object or
system of objects in motion about an axis that may or may not pass through
the object or system.
• The Earth has orbital angular momentum by reason of its annual revolution
about the Sun and spin angular momentum because of its daily rotation
about its axis.
Angular Momentum

Angular momentum L = I
The
moment of
linear
momentum
is called
angular
momentum.
It is
denoted by
L.

• The angular momentum of a rigid object is defined as the product of
the moment of inertia and the angular velocity.
• It is analogous to linear momentum and is subject to the
fundamental constraints of the conservation of angular momentum
principle if there is no external torque on the object.
• Angular momentum is a vector quantity. It is derivable from the
expression for the angular momentum of a particle
Angular momentum L = I

• Linear momentum is a product of the
mass (m) of an object and the velocity (v)
of the object. If an object has higher
momentum, then it harder to stop it.
• The formula for linear momentum is p =
mv.
• The total amount of momentum never
changes, and this property is called
conservation of momentum. Let us study
more about Linear momentum and
conservation of momentum.
Linear Momentum of a System of Particles
Linear Momentum of System of
Particles

Angular momentum and linear momentum are examples of the parallels
between linear and rotational motion.
They have the same form and are subject to the fundamental constraints
of conservation laws, the conservation of momentum and the
conservation of angular momentum.
Angular and Linear Momentum

Principle of Conservation of Angular Momentum
If no external torque acts on a rotating body then angular momentum
of the body remains constant.
We know that,
Example of principle of
conservation of angular
momentum:
• Ballet dancers rotate
themselves about their feet as
axis with constant angular
momentum. When they fold
their hands near the body, the
M.I. decreases, and the
angular velocity increases and
they rotates fast. But, if they
stretch their arms away from
the body, the M.I. increases
and angular velocity
decreases and they rotate
slow.

MOMENT OF INERTIA OF A CIRCULAR DISC
• Consider a uniform circular disc of mass M and radius R, rotating
about an axis passing through its centre and perpendicular to its
plane.
• Now, the area of the disc, A = π r 2 Mass of the disc = M Mass per
unit area of the disc, 𝜌 = M π R
Let us consider, a small
circular strip of width dx at
a distance x from the
centre of the disc. The
area of the strip dA =
Circumference of the strip
× width of the strip
dA = 2 𝜋 x × dx If dm is the
mass of the strip, then
dm = 𝜌 × 2 𝜋 x dx dm
= M π R 2 × 2 𝜋 x dx
dm = 2M R 2 x dx

Why is the moment of inertia calculated?
• In terms of how it affects motion in a straight line, the
moment of inertia plays the same part as mass.
• A body's rotational inertia is the amount of force it takes to
alter its direction of rotation. It remains the same for each
given rigid frame and any given rotational axis.

When compared to a circular disc, why
does a ring have a larger moment of
inertia?
• When compared to a circular disc of the same radius and
mass, a ring's moment of inertia is larger along an axis that
passes through its centre of mass and is transverse to its
plane. As a result of its mass being concentrated at its
outer edge, furthest from its central axis, a ring has a
greater moment of inertia.

Summary of moment of inertia
• In physics, moment of inertia is a crucial concept that
describes how an object's mass is distributed relative to its
rotational axis. It plays a significant role in understanding
rotational motion and is particularly important in the study
of classical mechanics and rigid body dynamics.
• In summary, the moment of inertia is a fundamental
property that governs rotational motion and helps us
understand how objects behave when subjected to
rotational forces. Its physical significance lies in describing
rotational inertia, angular acceleration, conservation of
angular momentum, rigid body rotation, and gyroscopic
effects.

ROTATIONAL MOTION_BLC_SWE_FALL23.pptx

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