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Good quality notes on Physics - Kinematics of Rotational Motion. Very helpful for students preparing for Engineering and medical entrance examination and also who are studying in Class XI and XII€2,94
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Good quality notes on Physics - Kinematics of Rotational Motion. Very helpful for students preparing for Engineering and medical entrance examination and also who are studying in Class XI and XII
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Good quality notes on Physics - Kinematics of rotational Motion. Very helpful for students preparing for Engineering and medical entrance examination and also who are studying in Class XI and XII
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, the body is not deformed under any circumstance.
Actual material bodies are never perfectly rigid and are deformed under the action of external forces. When
these deformations are small enough not to be considered during the course of motion, the body is assumed
to be a rigid body. Hence, all solid objects such as stone, ball, vehicles etc are considered as rigid bodies while
analyzing their translational as well as rotational motion.
Rotational motion of a rigid body
Any kind of motion 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 rotational motion.
In the following figures, a rectangular plate is shown moving in the x-y plane. The point C is its centre of mass.
In the first case it does not change its orientation, therefore is in pure translation motion. In the second case it
changes its orientation during its motion. It is a combination of translational and rotational motion.
y New
y orientation
A A
t+t •C
t •C B
•C
•C B t+t Original
t orientation
x x
O Pure Translation O
Combination of translation and rotation
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 .
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, opening and closing of doors and rotation of needles of a wall clock etc. come into this
category.
When a ceiling fan 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. Centres of circular
paths followed by every particle on the central line through the rod. This central line is known as the 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.
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Rotational Motion Part-01
Door
Ceiling Fan
P
Axis of rotation Axis of rotation
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 the plane of circular paths of particles of a rigid body in rotation and
containing the centres of all these circular paths is known as axis of rotation.
It is not necessary that the axis of rotation should pass through the body. Consider a system shown in the figure,
where a block is fixed on a rotating disc. The axis of rotation passes through the center of the disc but not
through the block.
Axis of rotation
r
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 are shown moving on their circular paths.
All its particles, not lying on the axis of rotation, move along circular paths with centres 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 undergo 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.
P
A
B
Axis of rotation
Q
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Rotational Motion Part-01
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.
Consider the rolling of wheels of a vehicle, moving on straight levelled road. The wheel appears rotating about
its stationary axle relative to a reference frame, attached with the vehicle. The rotation of the wheel as observed
from this frame is rotation about a fixed axis. Relative to a reference frame fixed with the ground, the wheel
appears rotating about the moving axle, therefore, rolling of a wheel is superposition of two simultaneous but
distinct motions – rotation about the axle fixed with the vehicle and translation of the axle together with the
vehicle.
C
Rotation about an axis in rotation.
In this kind of motion, the body rotates about an axis which in turn rotates about some other axis. Analysis of
rotation about rotating axes is beyond our scope, therefore we shall keep our discussion elementary level 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 it is in rotational
motion. This type of rotation in which the axis of rotation also rotates and sweeps out a cone is known as
precession.
Precession of the
Rotation about central axis
central axis
Another example of rotation about an axis in rotation is a swinging table-fan while running. Table-fan rotates
about its shaft along which its axis of rotation passes. When running swings, its shaft rotates about a certain
axis.
Note
In a rigid body angular velocity of any point w.r.t. any other point is constant and is equal to the angular velocity
of the rigid body.
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Rotational Motion Part-01
Illustration 1.
In Rotational motion of a rigid body, all particles move with
(1) same linear and angular velocity
(2) same linear and different angular velocity.
(3) with different linear velocities & same angular velocities
(4) with different linear velocity & different angular velocities.
Solution. (3)
In rotational motion of the rigid body all particles cover the same angular displacements in a particular
interval. So angular velocity of all the particle will be same. But linear velocity is also dependent on the
distance of particles from the axis of rotation so linear velocity will be different for all particles as the distances
are different for all the particles.
Illustration 2.
On account of the rotation of earth about its axis –
(1) the linear velocity of objects at equator is greater than that at other places
(2) the angular velocity of objects at equator is more than that of objects at poles
(3) the linear velocity of objects at all places on the earth is equal, but angular velocity is different
(4) the angular velocity and linear velocity are uniform at all places
Solution. (1)
v = r
Where r is the distance of the particle from the axis and is the angular velocity of the each which will be same
for all the particles
VE greater than that at any other places as r will be highest at equator
Kinematics of Rotational Motion
Time period (T) : Time taken by the particle to complete one rotation.
frequency (f) : No. of cycles completed by a particle per second is know as frequency
rpm = rotations per minute(N)
N
f=
60
Angular Displacement ()
• When a particle moves in a curved path, the change in the angle traced by its position vector about a fixed point
is known as angular displacement.
• Unit : radian
• Dimension : M0L0T0 i.e. dimensionless.
• Elementary (small) angular displacement is a vector whereas other (large) angular displacements is a scalar.
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