EP015 Lecture 8

Chapter 8

8.1 Rotational Kinematics.
8.2 Equilibrium of Rigid Body.

8.3 Rotational Dynamics.
8.4 Conservation of Angular Momentum.

CONCEPTUAL MAP : Rotational of Rigid Body

CONCEPTUAL MAP : Rotational of Rigid Body

8.0 INTRODUCTION

8.0 INTRODUCTION

A rigid body refers to a solid,
which consists of millions of
particles and never changes
its shape when exerted with
external forces.

Rotational motion of a rigid
body refers to a type of motion
where all particles of the body
undergo circular motion about

an axis of rotation.

LEARNING OUTCOMES : Rotational of Rigid Body

8.1 ROTATIONAL KINEMATICS

In kinematics, there are three main parameters
concerning to rotational motion:

8.1 ROTATIONAL KINEMATICS
ANGULAR DISPLACEMENT

The angular displacement,  of a body is the angle

(measured in radians) through which a point has been
rotated about a specified axis.

Note that 1 rotation equivalents 360o or 2 rad.

8.1 ROTATIONAL KINEMATICS
ANGULAR VELOCITY

The angular velocity,  is defined as the rate of
change of angular displacement.

Its SI unit is rad s-1.
The direction depends on the direction of rotation

(use right hand rule).

8.1 ROTATIONAL KINEMATICS
ANGULAR VELOCITY

The average angular velocity:

The instantaneous angular velocity:

8.1 ROTATIONAL KINEMATICS
ANGULAR ACCELERATION

The angular acceleration,  is defined as the rate of

change of angular velocity.

Its SI unit is rad s-2. The direction depends on either
the body is speeding up or slowing down.

8.1 ROTATIONAL KINEMATICS
ANGULAR ACCELERATION
The average angular acceleration:

The instantaneous angular acceleration:

8.1 ROTATIONAL KINEMATICS

Relationship between parameters of rotational motion
and linear motion are as follow:

8.1 ROTATIONAL KINEMATICS

As linear motion, rotational motion is also concerning
with kinematics (of uniform acceleration) and
formulated as follow:

8.2 EQUILIBRIUM OF RIGID BODY

A motion of a rigid body is dependence on the
presence of resultant force or/ and the resultant torque.

Based on the 2nd Newton’s Law, the present of the
resultant force will make a rigid body accelerates in a
straight line, while the present of the resultant torque

will make the body rotate.

Furthermore, the present of both resultant force and
torque will cause the body to roll.

8.2 EQUILIBRIUM OF RIGID BODY

As a conclusion, a rigid body is in equilibrium (at rest
or moving / rotate with constant velocity) if:

Resultant Force Resultant Torque

8.2 EQUILIBRIUM OF RIGID BODY
RESULTANT FORCE

Based on the 1st Newton’s Law, in order to prevent the body to
accelerate, then :

in a 1-D system, or :

in a 2-D system.

8.2 EQUILIBRIUM OF RIGID BODY
RESULTANT FORCE

Based on the 1st Newton’s Law, in order to prevent the
body to undergo accelerating / decelerating rotational

motion, then :

which means that :

8.2 EQUILIBRIUM OF RIGID BODY

TORQUE

The torque (or also known as
‘moment’) is produced by force,

which exerts not at the axis of
rotation with allowed angle.

where  = angle between r and F

8.2 EQUILIBRIUM OF RIGID BODY
TORQUE

The basic unit of torque is kg m2 s-2 but normally
known as Newton meter (N m).

Torque is a vector quantity. The direction of  can be
determined by using the “right-hand rule”.

8.2 EQUILIBRIUM OF RIGID BODY
TORQUE

8.2 EQUILIBRIUM OF RIGID BODY

TORQUE

The relationship between the direction of torque and
the direction of rotation can be determined by using

‘Right Hand Grip’.

Direction of Torque

Direction of Rotation

8.2 EQUILIBRIUM OF RIGID BODY
TORQUE

• Rotation : Following clockwise
Torque : Inwards (-ve)
Rotation : Anti-clockwise
Torque : Outwards (+ve)

8.2 EQUILIBRIUM OF RIGID BODY
TORQUE

8.2 EQUILIBRIUM OF RIGID BODY

8.3 ROTATIONAL DYNAMICS

Based on the Newton’s 2nd Law, the relationship
between Torque and moment of inertia is respectively

as follow:

where I is the moment of inertia of the body.

Torque is a vector quantity. The direction of torque
depends on either the body is speeding up or slowing
down (the same direction as the angular acceleration).

8.3 ROTATIONAL DYNAMICS

,
,

8.3 ROTATIONAL DYNAMICS
MOMENT OF INERTIA

Moment of inertia, I refers to the physical property of a
body to resist the any change in its rotational motion.

The SI unit is kg m2.

In this topic, determination of the moment of inertia is
made for the following system:
• System of a particle.
• System of a rigid body.

• System of discrete particle / rigid body.

8.3 ROTATIONAL DYNAMICS
MOMENT OF INERTIA OF A PARTICLE
Moment of inertia of a particle is formulated as follow:

with r = distance between the particle
and the axis of rotation.

8.3 ROTATIONAL DYNAMICS
MOMENT OF INERTIA OF A PARTICLE

8.3 ROTATIONAL DYNAMICS

MOMENT OF INERTIA OF A RIGID BODY

Moment of inertia of a rigid body (uniform shape)
depends on it shape and formulated as follow:

Note:
IC means the moment of
inertia of a rigid body
where the axis of rotation
is located at the centre of
mass.

Moment inertia of a rigid
body is depending on :
• Its mass, m.
• Radius of rotation, r.
• The shape of the body.

8.3 ROTATIONAL DYNAMICS
MOMENT OF INERTIA OF A DISCRETE SYSTEM

Moment of inertia of a discrete system
(particles or rigid bodies) is formulated as follow:

with m = mass of each particle or rigid body.

8.3 ROTATIONAL DYNAMICS
MOMENT OF INERTIA OF A DISCRETE SYSTEM

Moment of inertia of a discrete particles
is formulated as follow :

8.4 CONSERVATION OF ANGULAR MOMENTUM
ANGULAR MOMENTUM

Angular momentum, L is defined as a vector product of
the position vector and its linear momentum.

Mathematically, the magnitude of angular momentum
is represented as follow :

The direction is the same as the angular velocity.
The basic unit of angular momentum is kg m2 s-1

but normally written as kg m2 rad s-1.

8.4 CONSERVATION OF ANGULAR MOMENTUM
ANGULAR MOMENTUM

8.4 CONSERVATION OF ANGULAR MOMENTUM

The principle states that:
“The total angular momentum about any axis for a
system that consists of many rigid particles remains
constant, unless there is an external torque acts on it”.

8.4 CONSERVATION OF ANGULAR MOMENTUM

ANALOGOUS DESCRIPTION FOR
TRANSLATIONAL AND ROTATIONAL MOTION :

The following table shows the comparison between quantities of
translational and rotational motion :