ARJUNA JEE AIR D1 (2026)
Rotation
Till so far we have learnt kinematics and kinetics of translation motion in which all the particles of a body
undergo identical motions i.e. at any instant of time all of them have equal velocities and equal
accelerations and in any interval of time they all follow identical trajectories. Therefore kinematics of any
particle of a body or of its mass center in translation motion is representative of kinematics of the whole
body. But when a body is in rotation motion, all of its particles and the mass center do not undergo
identical motions. Newton’s laws of motion, which are the main guiding laws of mechanics, are
applicable to a point particle and if applied to a rigid body or system of particles, they predict motion of
the mass center. Therefore, it becomes necessary to investigate how mass center and different particles of
a rigid body move when the body rotates. In kinematics of rotation motion we investigate relations
existing between time, positions, velocities and accelerations of different particles and mass center of a
rigid body in rotation motion.
Rigid Body
A rigid body is an assemblage of a large number of material particles, which do not change their mutual
distances under any circumstance or in other words, they are not deformed under any circumstance.
Actual material bodies are never perfectly rigid and are deformed under action of external forces. When
these deformations are small enough to be considered during their course of motion, the body is assumed
a rigid body. Hence, all solid objects such as stone, ball, vehicles etc are considered as rigid bodies while
analyzing their translation as well as rotation motion.
To analyze rotation of a body relative motion between its particles cannot be neglected and size of the
body becomes a considerable factor. This is why study of rotation motion is also known as mechanics of
rigid bodies.
Rotation Motion of a Rigid Body
Any kind of motion of a body is identified by change in position or change in orientation or change in
both. If a body changes its orientation during its motion it said to be in rotation motion.
In the following figures, a rectangular plate is shown moving in the x-y plane. The point C is its mass
center. In the first case it does not changes orientation, therefore is in pure translation motion. In the
second case it changes its orientation by during its motion. It is a combination of translation and rotation
motion.
Rotation i.e. change in orientation is identified by the angle through which a linear dimension or a
straight line drawn on the body turns. In the figure this angle is shown by θ.
PHYSICS WALLAH 1
, ROTATIONAL MOTION
Example
Identify Translation and rotation motion
A rectangular plate is suspended from the ceiling by two parallel rods each pivoted at one end on the
plate and at the other end on the ceiling. The plate is given a side-push to oscillate in the vertical plane
containing the plate. Identify motion of the plate and the rods.
Solution
Neither of the linear dimensions of the plate turns during the motion. Therefore, the plate does not change
its orientation. Here edges of the body easily fulfill our purpose to measure orientation; therefore, no line
is drawn on it.
The plate is in curvilinear translation motion and the rods are in rotation motion.
Types of Motions involving Rotation
Motion of body involving rotation can be classified into following three categories.
I Rotation about a fixed axis.
II Rotation about an axis in translation.
III Rotation about an axis in rotation
Rotation about a fixed axis
Rotation of ceiling fan, potter’s wheel, opening and closing of doors and needles of a wall clock etc.
come into this category.
When a ceiling fan it rotates, the vertical rod supporting it remains stationary and all the particles on the
fan move on circular paths. Circular path of a particle P on one of its blades is shown by dotted circle.
Centers of circular paths followed by every particle are on the central line through the rod. This central
line is known as axis of rotation and is shown by a dashed line. All the particles on the axis of rotation
are at rest, therefore the axis is stationary and the fan is in rotation about this fixed axis.
A door rotates about a vertical line that passes through its hinges. This vertical line is the axis of rotation.
In the figure, the axis of rotation is shown by dashed line.
Axis of rotation
An imaginary line perpendicular to plane of circular paths of particles of a rigid body in rotation and
containing the centers of all these circular paths is known as axis of rotation.
It is not necessary that the axis of rotation pass through the body. Consider system shown in the figure,
where a block is fixed on a rotating disk. The axis of rotation passes through the center of the disk but not
through the block.
PHYSICS WALLAH 2
, ROTATIONAL MOTION
Important observations
Let us consider a rigid body of arbitrary shape rotating about a fixed axis PQ passing through the body.
Two of its particles A and B shown are moving on their circular paths.
⇒ All of its particles, not on the axis of rotation, move on circular paths with centers on the axis or rotation.
All these circular paths are in parallel planes that are perpendicular to the axis of rotation.
⇒ All the particles of the body cover same angular displacement in the same time interval, therefore all of
them move with the same angular velocity and angular acceleration.
⇒ Particles moving on circular paths of different radii move with different speeds and different magnitudes
of linear acceleration. Furthermore, no two particles in the same plane perpendicular to the axis of
rotation have same velocity and acceleration vectors.
⇒ All the particles on a line parallel to the axis of rotation move circular paths of the same radius therefore
have same velocity and acceleration vectors.
⇒ Consider two particles in a plane perpendicular to the rotational axis. Every such particle on a rigid body
in rotation motion moves on circular path relative to another one. Radius of the circular path equals to the
distance between the particles. In addition, angular velocity and angular acceleration equals to that of
rotation motion of the body.
Rotation about an axis in translation
Rotation about an axis in translation includes a broad category of motions. Rolling is an example of this
kind of motion. A rod lying on table when pushed from its one end in its perpendicular direction also
executes this kind of motion. To understand more let us discuss few examples.
Consider rolling of wheels of a vehicle, moving on straight level road. Relative to a reference frame,
moving with the vehicle wheel appears rotating about its stationary axel. The rotation of the wheel from
this frame is rotation about fixed axis. Relative to a reference frame fixed with the ground, the wheel
appears rotating about the moving axel, therefore, rolling of a wheel is superposition of two simultaneous
but distinct motions – rotation about the axel fixed with the vehicle and translation of the axel together
with the vehicle.
Important observations
⇒ Every particle of the body always remains in a plane perpendicular to the rotational axis. Therefore, this
kind of motion is also known as general plane motion.
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, ROTATIONAL MOTION
⇒ Relative to every particle another particle in a plane perpendicular to axis of rotation moves on circular
path. Radius of the circular path equals to the distance between the particles and angular velocity and
angular acceleration equals to that of rotation motion of the body.
⇒ Rotation about axis in translation is superposition of pure rotation about the axis and simultaneous
translation motion of the axis.
Rotation about an axis in rotation.
In this kind of motion, the body rotates about an axis that also rotates about some other axis. Analysis of
rotation about rotating axes is not in the scope of JEE, therefore we will discus it to have an elementary
idea only.
As an example consider a rotating top. The top rotates about its central axis of symmetry and this axis
sweeps a cone about a vertical axis. The central axis continuously changes its orientation, therefore is in
rotation motion. This type of rotation in which the axis of rotation also rotates and sweeps out a cone is
known as precession.
Another example of rotation about axis in rotation is a table-fan swinging while rotating. Table-fan
rotates about its horizontal shaft along which axis of rotation passes. When the rotating table-fan swings,
its shaft rotates about a vertical axis.
Angular displacement, angular velocity and angular acceleration
Rotation motion is the change in orientation of a rigid body with time. It is measured by turning of a
linear dimension or a straight line drawn on the body. In the figure is shown at two different instants t = 0
and t a rectangular plate moving in its own plane. Change in orientation during time t equals to the angle
θ through which all the linear dimensions of the plate or a line AB turns.
If the angle θ continuously changes with time t, instantaneous angular velocity ω and angular
acceleration α for rotation of the body are defined by the following equations.
d
= ...[1]
dt
d 2 d d
= = = ...[2]
dt 2
dt d
Direction of angular motion quantities
Angular displacement, angular velocity and angular acceleration are known as angular motion quantities.
Infinitesimally small angular displacement, instantaneous angular velocity and angular acceleration are
vector quantities. Direction of infinitesimally small angular displacement and instantaneous angular
velocity is given by the right hand rule. For a disk rotating as shown in the figure, the angular velocity
points upwards along the axis of rotation.
PHYSICS WALLAH 4
Rotation
Till so far we have learnt kinematics and kinetics of translation motion in which all the particles of a body
undergo identical motions i.e. at any instant of time all of them have equal velocities and equal
accelerations and in any interval of time they all follow identical trajectories. Therefore kinematics of any
particle of a body or of its mass center in translation motion is representative of kinematics of the whole
body. But when a body is in rotation motion, all of its particles and the mass center do not undergo
identical motions. Newton’s laws of motion, which are the main guiding laws of mechanics, are
applicable to a point particle and if applied to a rigid body or system of particles, they predict motion of
the mass center. Therefore, it becomes necessary to investigate how mass center and different particles of
a rigid body move when the body rotates. In kinematics of rotation motion we investigate relations
existing between time, positions, velocities and accelerations of different particles and mass center of a
rigid body in rotation motion.
Rigid Body
A rigid body is an assemblage of a large number of material particles, which do not change their mutual
distances under any circumstance or in other words, they are not deformed under any circumstance.
Actual material bodies are never perfectly rigid and are deformed under action of external forces. When
these deformations are small enough to be considered during their course of motion, the body is assumed
a rigid body. Hence, all solid objects such as stone, ball, vehicles etc are considered as rigid bodies while
analyzing their translation as well as rotation motion.
To analyze rotation of a body relative motion between its particles cannot be neglected and size of the
body becomes a considerable factor. This is why study of rotation motion is also known as mechanics of
rigid bodies.
Rotation Motion of a Rigid Body
Any kind of motion of a body is identified by change in position or change in orientation or change in
both. If a body changes its orientation during its motion it said to be in rotation motion.
In the following figures, a rectangular plate is shown moving in the x-y plane. The point C is its mass
center. In the first case it does not changes orientation, therefore is in pure translation motion. In the
second case it changes its orientation by during its motion. It is a combination of translation and rotation
motion.
Rotation i.e. change in orientation is identified by the angle through which a linear dimension or a
straight line drawn on the body turns. In the figure this angle is shown by θ.
PHYSICS WALLAH 1
, ROTATIONAL MOTION
Example
Identify Translation and rotation motion
A rectangular plate is suspended from the ceiling by two parallel rods each pivoted at one end on the
plate and at the other end on the ceiling. The plate is given a side-push to oscillate in the vertical plane
containing the plate. Identify motion of the plate and the rods.
Solution
Neither of the linear dimensions of the plate turns during the motion. Therefore, the plate does not change
its orientation. Here edges of the body easily fulfill our purpose to measure orientation; therefore, no line
is drawn on it.
The plate is in curvilinear translation motion and the rods are in rotation motion.
Types of Motions involving Rotation
Motion of body involving rotation can be classified into following three categories.
I Rotation about a fixed axis.
II Rotation about an axis in translation.
III Rotation about an axis in rotation
Rotation about a fixed axis
Rotation of ceiling fan, potter’s wheel, opening and closing of doors and needles of a wall clock etc.
come into this category.
When a ceiling fan it rotates, the vertical rod supporting it remains stationary and all the particles on the
fan move on circular paths. Circular path of a particle P on one of its blades is shown by dotted circle.
Centers of circular paths followed by every particle are on the central line through the rod. This central
line is known as axis of rotation and is shown by a dashed line. All the particles on the axis of rotation
are at rest, therefore the axis is stationary and the fan is in rotation about this fixed axis.
A door rotates about a vertical line that passes through its hinges. This vertical line is the axis of rotation.
In the figure, the axis of rotation is shown by dashed line.
Axis of rotation
An imaginary line perpendicular to plane of circular paths of particles of a rigid body in rotation and
containing the centers of all these circular paths is known as axis of rotation.
It is not necessary that the axis of rotation pass through the body. Consider system shown in the figure,
where a block is fixed on a rotating disk. The axis of rotation passes through the center of the disk but not
through the block.
PHYSICS WALLAH 2
, ROTATIONAL MOTION
Important observations
Let us consider a rigid body of arbitrary shape rotating about a fixed axis PQ passing through the body.
Two of its particles A and B shown are moving on their circular paths.
⇒ All of its particles, not on the axis of rotation, move on circular paths with centers on the axis or rotation.
All these circular paths are in parallel planes that are perpendicular to the axis of rotation.
⇒ All the particles of the body cover same angular displacement in the same time interval, therefore all of
them move with the same angular velocity and angular acceleration.
⇒ Particles moving on circular paths of different radii move with different speeds and different magnitudes
of linear acceleration. Furthermore, no two particles in the same plane perpendicular to the axis of
rotation have same velocity and acceleration vectors.
⇒ All the particles on a line parallel to the axis of rotation move circular paths of the same radius therefore
have same velocity and acceleration vectors.
⇒ Consider two particles in a plane perpendicular to the rotational axis. Every such particle on a rigid body
in rotation motion moves on circular path relative to another one. Radius of the circular path equals to the
distance between the particles. In addition, angular velocity and angular acceleration equals to that of
rotation motion of the body.
Rotation about an axis in translation
Rotation about an axis in translation includes a broad category of motions. Rolling is an example of this
kind of motion. A rod lying on table when pushed from its one end in its perpendicular direction also
executes this kind of motion. To understand more let us discuss few examples.
Consider rolling of wheels of a vehicle, moving on straight level road. Relative to a reference frame,
moving with the vehicle wheel appears rotating about its stationary axel. The rotation of the wheel from
this frame is rotation about fixed axis. Relative to a reference frame fixed with the ground, the wheel
appears rotating about the moving axel, therefore, rolling of a wheel is superposition of two simultaneous
but distinct motions – rotation about the axel fixed with the vehicle and translation of the axel together
with the vehicle.
Important observations
⇒ Every particle of the body always remains in a plane perpendicular to the rotational axis. Therefore, this
kind of motion is also known as general plane motion.
PHYSICS WALLAH 3
, ROTATIONAL MOTION
⇒ Relative to every particle another particle in a plane perpendicular to axis of rotation moves on circular
path. Radius of the circular path equals to the distance between the particles and angular velocity and
angular acceleration equals to that of rotation motion of the body.
⇒ Rotation about axis in translation is superposition of pure rotation about the axis and simultaneous
translation motion of the axis.
Rotation about an axis in rotation.
In this kind of motion, the body rotates about an axis that also rotates about some other axis. Analysis of
rotation about rotating axes is not in the scope of JEE, therefore we will discus it to have an elementary
idea only.
As an example consider a rotating top. The top rotates about its central axis of symmetry and this axis
sweeps a cone about a vertical axis. The central axis continuously changes its orientation, therefore is in
rotation motion. This type of rotation in which the axis of rotation also rotates and sweeps out a cone is
known as precession.
Another example of rotation about axis in rotation is a table-fan swinging while rotating. Table-fan
rotates about its horizontal shaft along which axis of rotation passes. When the rotating table-fan swings,
its shaft rotates about a vertical axis.
Angular displacement, angular velocity and angular acceleration
Rotation motion is the change in orientation of a rigid body with time. It is measured by turning of a
linear dimension or a straight line drawn on the body. In the figure is shown at two different instants t = 0
and t a rectangular plate moving in its own plane. Change in orientation during time t equals to the angle
θ through which all the linear dimensions of the plate or a line AB turns.
If the angle θ continuously changes with time t, instantaneous angular velocity ω and angular
acceleration α for rotation of the body are defined by the following equations.
d
= ...[1]
dt
d 2 d d
= = = ...[2]
dt 2
dt d
Direction of angular motion quantities
Angular displacement, angular velocity and angular acceleration are known as angular motion quantities.
Infinitesimally small angular displacement, instantaneous angular velocity and angular acceleration are
vector quantities. Direction of infinitesimally small angular displacement and instantaneous angular
velocity is given by the right hand rule. For a disk rotating as shown in the figure, the angular velocity
points upwards along the axis of rotation.
PHYSICS WALLAH 4
