rotational motion physics class 11Th crash course

 

 

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1. 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.

2. 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. 

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 q.

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.

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

Important observations
Let us consider a rigid body of arbitrary shape rotating about a fixed A
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.

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

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.
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.