1. The document discusses rotational dynamics and circular motion. It defines concepts like angular velocity, moment of inertia, centripetal force, and radius of gyration.
2. Examples of circular motion discussed include vehicles moving in circular tracks, wells of death, and vehicles on banked roads. The forces and equations of motion are analyzed.
3. Vertical circular motion under gravity is also examined, like a point mass attached to a string or rod moving in a vertical circle. Dynamics of a vehicle on a convex overbridge are also covered.
4. Moment of inertia is introduced as an analogous concept to mass for rotational motion. Formulas are given for moment of inertia of objects like rings, discs, and
2. 2
1.1 INTRODUCTION
Every day we come across several revolving or rotating (rigid) objects. During revolution,
the object undergoes circular motion about some point outside the object or about some other
object, while during rotation the motion is about an axis of rotation passing through the object.
1.2 Characteristics of Circular Motion:
1) It is an accelerated motion
2) It is a periodic motion
1.2.1 Kinematics of Circular Motion:
The tangential velocity is, ๐
โ
โ = ๐
โโโ x ๐
โ , where ๐
โโโ is the angular velocity.
The magnitude of ๐
โ
โ is v = ฯ r.
Direction of ๐
โโโ is always along the axis of rotation and is given by the right-hand thumb rule.
Fig. Directions of angular velocity
If T is period of circular motion and n is the frequency, w = 2๐ n =
๐๐
๐ป
Uniform circular motion: During circular motion if the speed of the particle remains constant, it is
called Uniform Circular Motion (UCM). The acceleration responsible for this is the centripetal or
radial acceleration ๐๐
โโโโ = - ๐๐
๐
โ
For UCM, its magnitude is constant and it is a = ๐๐
r =
๐๐
๐
= vw. It is always directed towards the
centre of the circular motion, hence called centripetal.
Fig. Directions of linear velocity and acceleration.
Non-uniform circular motion: For non-uniform circular motion, the magnitude of ๐๐
โโโโ is not
constant. This is due to the angular acceleration ๐ถ
โโ =
๐ ๐
โโ
โ
๐ ๐
For increasing speed, it is along the direction of ๐
โโโ while during decreasing speed, it is opposite to
that of ๐
โโโ .
Fig. Direction of angular acceleration.
If the angular acceleration ๐ถ
โโ is constant and along the axis of rotation, all ๐ฝ
โ
โ , ๐
โโโ and ๐ถ
โโ will be
directed along the axis.
1.2.2 Dynamics of Circular Motion (Centripetal Force and Centrifugal Force):
3. 3
i) Centripetal force (CPF): The acceleration responsible for circular motion is the centripetal or
radial acceleration ๐๐
โโโโ = - ๐๐
๐
โ . The force providing this acceleration is the centripetal or radial
force, CPF = โ๐๐๐
๐
โ
ii) Centrifugal force (c.f.f.): This force, (+๐๐๐
๐
โ ) the centrifugal force. It is a pseudo force arising
due to the centripetal acceleration of the frame of reference.
It must be understood that centrifugal force is a non-real force, but NOT an imaginary force.
Resultant force = โ๐๐๐
๐
โ Or ๐๐๐
๐
โ + โ(๐๐๐๐ ๐๐๐๐๐๐) = 0
1.3 Applications of Uniform Circular Motion:
1.3.1 Vehicle Along a Horizontal Circular Track:
Forces acting on the car (considered to be a particle) are
(i) weight mg, vertically downwards,
(ii) normal reaction N, vertically upwards that balances the weight mg and
(iii) force of static friction ๐๐ between road and the tyres. This is static friction because it prevents
the vehicle from outward slipping or skidding. This is the resultant force which is centripetal.
Fig. Vehicle on a horizontal road
โด ๐๐ = ๐ต ๐๐๐ ๐๐ = ๐๐๐๐
=
๐๐๐
๐
โด
๐๐
๐ต
=
๐๐๐
๐
=
๐๐
๐๐
For a given track, radius r is constant.
For given vehicle, mg = N is constant.
(๐๐)๐๐๐ = ๐๐. ๐ต, where, ๐๐ is the coefficient of static friction
At the maximum possible speed ๐๐,
(๐๐)๐๐๐
๐ต
= ๐๐ =
๐๐๐๐
๐
๐๐
โด ๐๐๐๐ = โ๐๐๐๐
1.3.2 Well (or Wall) of Death: (เคฎเฅเคค เคเคพ เค
เฅ เคเค):
The forces acting on the vehicle are
(i) Normal reaction N acting horizontally and towards the centre,
(ii) Weight mg acting vertically downwards, and
(iii) Force of static friction ๐๐ acting vertically upwards between vertical wall and the tyres.
Fig. Well of death
N = mr๐๐
=
๐๐๐
๐
and mg = ๐๐
Force of static friction ๐๐ is always less than or equal to ๐๐N.
โด ๐๐ โค ๐๐ (
๐๐๐
๐
)
4. 4
โด ๐ โค
๐๐๐๐
๐
โด ๐๐
โฅ
๐๐
๐๐
โด ๐ฝ๐๐๐ = โ
๐๐
๐๐
1.3.3 Vehicle on a Banked Road:
โDefined as the phenomenon in which the edge is raised for the curved roads above the inner
edge to provide the necessary centripetal force to the vehicle to that they take a safe turnโ
There are two forces acting on the vehicle,
(i) weight mg, vertically downwards and
(ii) normal reaction N, perpendicular to the surface of the road.
Its vertical component Ncosฮธ balances weight mg. Horizontal component Nsinฮธ being the
resultant force, must be the necessary centripetal force.
Fig: Vehicle on a banked road.
N cos ๐ฝ = mg and
N sin ๐ฝ = mr๐๐
=
๐๐๐
๐
โด ๐๐๐ ๐ฝ =
๐๐
๐๐
(a) Most safe speed: For a particular road, r and ฮธ are fixed.
๐๐ = โ๐๐ ๐๐๐ ๐ฝ
(b) Banking angle: While designing a road, this expression helps us in knowing the angle of banking
as ๐ฝ = ๐๐๐โ๐
(
๐๐
๐๐
)
(c) Speed limits: If the vehicle is running exactly at the speed ๐๐ = โ๐๐ ๐๐๐ ๐ฝ, the forces acting
on the vehicle are
(i) weight mg acting vertically downwards and
(ii) normal reaction N acting perpendicular to the road.
Fig: Banked road: lower speed limit
For speeds ๐๐ < โ๐๐ ๐๐๐ ๐ฝ,
๐๐๐
๐
๐
< ๐ต ๐๐๐ ๐ฝ.
5. 5
Fig: Banked road: upper speed limit.
These two forces take care of the necessary centripetal force.
โด ๐๐ = ๐๐ ๐๐๐ ๐ฝ + ๐ต ๐๐๐ ๐ฝ and
For minimum possible speed, ๐๐ is maximum and equal to ๐๐N.
For ๐๐ โฅ tan ๐ฝ, ๐๐๐๐ = 0. This is true for most of the rough roads, banked at smaller angles.
(d) For speeds ๐๐ = โ๐๐ ๐๐๐ ๐ฝ,
๐๐๐
๐
๐
> ๐ต ๐๐๐ ๐ฝ.
For maximum possible speed, ๐๐ is maximum and equal to ๐๐N.
If ๐๐ = cot ฮธ, ๐๐๐๐ = โ. But (๐๐)๐๐๐ = 1. Thus, for ฮธ โฅ 45ยฐ, ๐๐๐๐ = โ.
(e) For ๐๐ = 0, both the equations give us v = โ๐๐ ๐๐๐ ๐ฝ which is the safest speed
1.3.4 Conical Pendulum:
A tiny mass connected to a long, flexible, massless, inextensible string, and suspended to a rigid
support is called a pendulum.
The string moves along the surface of a right circular cone of vertical axis and the point object
performs a (practically) uniform horizontal circular motion is called a conical pendulum.
Fig: In an inertial frame
The forces acting on the bob are
(i) its weight mg directed vertically downwards and
(ii) the force ๐ป๐ due to the tension in the string, directed along the string, towards the support A.
6. 6
Fig: In a non- inertial frame
Radius r of the circular motion is r = L sin ๐ฝ.
1.4 Vertical Circular Motion:
Two types of vertical circular motions are:
(a) A controlled vertical circular motion such as a giant wheel or similar games. In this case the
speed is either kept constant or NOT totally controlled by gravity.
(b) Vertical circular motion controlled only by gravity. In this case, we initially supply the necessary
energy (mostly) at the lowest point.
1.4.1 Point Mass Undergoing Vertical Circular Motion Under Gravity:
Case I: Mass tied to a string:
Both, weight (mg) and force due to tension are downward i.e., towards the center. There are only
resultant centripetal force acts towards centre.
Fig: Vertical circular motion
Uppermost position (A): Thus, if ๐ฝ๐จ is the speed at the uppermost point,
Radius r of the circular motion is the length of the string. For minimum possible speed,
Lowermost position (B): Force due to the tension, ๐ป๐ฉ is vertically upwards, and opposite to mg. If
๐ฝ๐ฉ is the speed at the lowermost point,
The vertical displacement is 2r and the motion is governed only by gravity.
7. 7
Positions when the string is horizontal (C and D): Force due to the tension is the only force towards
the centre as weight mg is perpendicular to the tension.
Arbitrary positions: Force due to the tension and weight are neither along the same line, nor
perpendicular. It decreases the speed while going up and increases it while coming down.
Case II: Mass tied to a rod: Consider a bob tied to a rod and whirled along a vertical circle. Zero
speed is possible at the uppermost point.
1.4.2 Sphere of Death (เคฎเฅเคคเฅเคฏเฅ เคเฅเคฒ):
Two-wheeler rider undergo rounds inside a hollow sphere. Starting with small horizontal circles,
they eventually perform revolutions along vertical circles. The dynamics of this vertical circular
motion is the same as that of the point mass tied to the string, except that the force due to
tension T is replaced by the normal reaction force N.
The linear speed is more for larger circles but angular speed is more for smaller circles. This is as
per the theory of conical pendulum.
1.4.3 Vehicle at the Top of a Convex Over-Bridge:
Fig: Vehicle on a convex over-bridge
Forces acting on the vehicle are (a) Weight mg and (b) Normal reaction force N, both along the
vertical line.
Thus, if v is the speed at the uppermost point,
As the speed is increased, N goes on decreasing. Thus, for just maintaining contact, N = 0.
This imposes an upper limit on the speed as ๐๐๐๐ = โ๐๐
1.5 Moment of Inertia as an Analogous Quantity for Mass:
Fig: A body of N particles
Let us consider the object to be consisting of N particles of masses ๐๐, ๐๐ .โฆโฆโฆ๐๐ at respective
perpendicular distances ๐๐,๐๐,โฆ โฆ โฆ ๐๐ from the axis of rotation.
Translational K.E. of the first particle is
8. 8
Thus, rotational K.E.
If I = โ ๐๐๐๐
๐
replaces mass m and angular speed ฯ replaces linear speed v, rotational K.E. =
๐
๐
๐ฐ๐๐
is analogous to translational K.E. =
๐
๐
๐๐๐
Thus, I is defined to be the rotational inertia or moment of inertia (M.I.)
The moment of inertia of an object depends upon (i) individual masses and (ii) the distribution of
these masses about the given axis of rotation.
The moment of inertia is to be obtained by integration as I = โซ ๐๐
๐ ๐ .
Fig: Moment of Inertia of a ring
1.5.1 Moment of Inertia of a Uniform Ring:
An object is called a uniform ring if its mass is situated uniformly on the circumference of a circle.
If it is rotating about its own axis, its entire mass M is practically at a distance equal to its radius R
form the axis. The moment of inertia of a uniform ring of mass M and radius R is I = ๐ด๐น๐
.
1.5.2 Moment of Inertia of a Uniform Disc:
The ratio ๐ =
๐
๐จ
=
๐๐๐๐
๐๐๐๐
is called the surface density.
Consider a uniform disc of mass M and radius R rotating about its own axis, which is the line
perpendicular to its plane and passing through its centre ๐ =
๐ด
๐ ๐น๐.
Fig: Moment of Inertia of a disk
Width of this ring is dr, which is so small that the entire ring can be considered to be of average
radius r.
Area of this ring is A = 2ฯr.dr
โด ๐ =
๐ ๐
๐๐ ๐.๐ ๐
โด dm = 2ฯฯr.dr.
The moment of inertia of this ring is ๐ฐ๐ = ๐ ๐ (๐๐
)
Moment of inertia (I) of the disc can now be obtained by integrating ๐ฐ๐ from r = 0 to r = R.
9. 9
1.6 Radius of Gyration:
It depends upon mass of that object and how that mass is distributed around the axis of rotation.
The moment of inertia of any object as I = ๐ด๐ฒ๐
, where M is mass of that object. K is defined as
the radius of gyration of the object about the given axis of rotation, I = ๐ด๐ฒ๐
. Larger the value of K,
farther is the mass from the axis.
1.7 Theorem of Parallel Axes and Theorem of Perpendicular Axes:
Expressions of moment of inertias of regular geometrical shapes are about their axes of symmetry.
1.7.1 Theorem of Parallel Axes:
Fig: Theorem of parallel axes
Consider, MOP is any axis passing through point O. Axis ACB is passing through the centre of mass
C of the object, parallel to the axis MOP, and at a distance h from it (โด h = CO).
Consider a mass element dm located at point D. Perpendicular on OC from point D is DN. Moment
of inertia of the object about the axis ACB is ๐ฐ๐ช = โซ(๐ซ๐ช)๐
๐ ๐, and about the axis MOP it is ๐ฐ๐ =
โซ(๐ซ๐)๐
๐ ๐ .
From the definition of the centre of mass, โซ ๐ต๐ช. ๐ ๐ = ๐ .
It states that, โThe moment of inertia (๐ฐ๐) of an object about any axis is the sum of its moment of
inertia (๐ฐ๐ช) about an axis parallel to the given axis, and passing through the centre of mass and the
product of the mass of the object and the square of the distance between the two axes (M๐๐
).โ
1.7.2 Theorem of Perpendicular Axes:
Consider, ๐ฐ๐ and ๐ฐ๐ be the moment of Inertia of axis passing through point โMโ and โNโ.
Fig: Theorem of perpendicular axes
Consider a mass element dm located at any point P. PM = y and PN = x are the perpendiculars
drown from P respectively on the x and y axes.
10. 10
z = โ๐๐ + ๐๐. If ๐ฐ๐, ๐ฐ๐ and ๐ฐ๐ are the respective moment of inertias
It states that, โThe moment of inertia (๐ฐ๐) of a laminar object about an axis (z) perpendicular to its
plane is the sum of its moment of inertias about two mutually perpendicular axes (x and y) in its
plane, all the three axes being concurrentโ.
1.8 Angular Momentum or Moment of Linear Momentum:
๐ณ
โ
โ = ๐
โ ร ๐
โ
โ , were ๐
โ is the position vector from the axis of rotation.
In magnitude, L = P ร r sin ฮธ, where ฮธ is the smaller angle between the directions of ๐ท
โโ and ๐
โ .
1.8.1 Expression for Angular Momentum in Terms of Moment of Inertia:
A rigid object rotating with a constant angular speed ฯ about an axis perpendicular to the plane of
paper.
Let us consider the object to be consisting of N number of particles of masses ๐๐, ๐๐,โฆ โฆ โฆ . ๐๐ต
at respective perpendicular distances ๐๐, ๐๐,โฆ โฆ โฆ . ๐๐ต from the axis of rotation.
As the object rotates, all these particles perform UCM with same angular speed ฯ, but with
different linear speeds ๐๐ = ๐๐๐, ๐๐ = ๐๐๐,โฆ โฆ โฆ . ๐๐ต = ๐๐ต๐.
Directions of individual velocities ๐๐
โโโโ , ๐๐
โโโโ , etc., are along the tangents to their respective tracks.
Linear momentum of the first particle is of magnitude ๐๐ = ๐๐๐๐ = ๐๐๐๐๐. Its direction is
along that of ๐๐
โโโโ .
Its angular momentum is thus of magnitude ๐ณ๐ = ๐๐๐๐ = ๐๐๐๐
๐
๐
Similarly, ๐ณ๐ = ๐๐๐๐
๐
๐, ๐ณ๐ = ๐๐๐๐
๐
๐, โฆโฆโฆ., ๐ณ๐ต = ๐๐ต๐๐ต
๐
๐
Thus, magnitude of angular momentum of the body is,
Where, I = ๐๐๐๐
๐
+ ๐๐๐๐
๐
+ โฆ. + ๐๐ต๐๐ต
๐
is the moment of inertia.
1.9 Expression for Torque in Terms of Moment of Inertia:
Fig: Expression for torque
A rigid object rotating with a constant angular acceleration ฮฑ about an axis perpendicular to the
plane of paper.
Let us consider the object to be consisting of N number of particles of masses
๐๐,๐๐, โฆ โฆ . . , ๐๐ต at respective perpendicular distances ๐๐, ๐๐,โฆ โฆ . . , ๐๐ต from the axis of
rotation. Linear accelerations ๐๐ = ๐๐๐ถ, ๐๐ = ๐๐๐ถ,โฆโฆโฆ., ๐๐ต = ๐๐ต๐ถ etc.
Force experienced by the first particle is ๐๐ = ๐๐๐๐ = ๐๐๐๐๐ถ
As these forces are tangential, their respective perpendicular distances from the axis are ๐๐,
๐๐,โฆ . . ๐๐ต.
Thus, the torque experienced by the first particle is of magnitude ๐๐ = ๐๐๐๐ = ๐๐๐๐
๐
๐ถ
Similarly, ๐๐ = ๐๐๐๐
๐
๐ถ, ๐๐ = ๐๐๐๐
๐
๐ถ,โฆโฆโฆ, ๐๐ต = ๐๐ต๐๐ต
๐
๐ถ
Magnitude of the resultant torque is,
11. 11
where, I = ๐๐๐๐
๐
+ ๐๐๐๐
๐
โฆ โฆ . . +๐๐ต๐๐ต
๐
is the moment of inertia of the object about the given
axis of rotation.
The relation ๐ = ๐ฐ๐ถ is analogous to f = ma for the translational motion if the moment of inertia I
replaces mass, which is its physical significance.
1.10 Conservation of Angular Momentum:
Angular momentum of a system is, ๐ณ
โ
โ = ๐
โ ร ๐
โ
โ
Where, ๐
โ is the position vector from the axis of rotation and ๐
โ
โ is the linear momentum.
Hence, angular momentum ๐ณ
โ
โ is conserved in the absence of external unbalanced torque ๐
โ . This is
the principle of conservation of angular momentum.
Examples of conservation of angular momentum: During some shows of ballet dance, acrobat in a
circus, sports like ice skating, diving in a swimming pool, etc., the principle of conservation of
angular momentum is realized. In all these applications the product L = Iw = I (2๐ ๐) is constant.
(i) Ballet dancers: During ice ballet, the dancers have to undertake rounds of smaller and larger
radii. The dancers come together while taking the rounds of smaller radius. The moment of inertia
of their system becomes minimum and the frequency increases, to make it thrilling. While outer
rounds, the dancers outstretch their legs and arms. This increases their moment of inertia that
reduces the angular speed and hence the linear speed. This is essential to prevent slipping.
(ii) Diving in a swimming pool (during competition): While on the diving board, the divers stretch
their body so as to increase the moment of inertia. Immediately after leaving the board, they fold
their body. This reduces the moment inertia considerably. As a result, the frequency increases and
they can complete more rounds in air to make the show attractive. Again, while entering into
water they stretch their body into a streamline shape. This allows them a smooth entry into the
water.
1.11 Rolling Motion:
The objects like a cylinder, sphere, wheels, etc. are quite often seen to perform rolling motion.
(i) circular motion of the body as a whole, about its own symmetric axis and
(ii) linear motion of the body assuming it to be concentrated at its centre of mass.
Consider an object of moment of inertia I, rolling uniformly.
v = Linear speed of the centre of mass, R = Radius of the body, w = Angular speed of rotation
โด ๐ =
๐ฝ
๐น
for any particle
M = Mass of the body, K = Radius of gyration of the body
โด ๐ฐ = ๐ด๐ฒ๐
Total kinetic energy of rolling = Translational K.E. + Rotational K.E.
12. 12
Static friction is essential for a purely rolling motion. It prevents the sliding motion.
1.11.1 Linear Acceleration and Speed While Pure Rolling Down an Inclined Plane:
Inclination of the plane with the horizontal is ฮธ.
Fig: Rolling along an incline
As the objects starts rolling down, its gravitational P.E. is converted into K.E. of rolling.
Linear distance travelled along the plane is s =
๐
๐๐๐ ๐ฝ
During this distance, the linear velocity has increased from zero to v.
For pure sliding, without friction, the acceleration is g sin ฮธ and final velocity is โ๐๐๐ . Thus,
during pure rolling, the factor (1 +
๐ฒ๐
๐น๐) is effective for both the expressions.