Rotational Motion
- refers to motion of a
body about a fixed
axis of rotation
wherein, the particles
have the same
instantaneous angular
velocity.
INTRODUCTION
Rotational inertia, which is also termed as
moment of inertia, is a similar concept
applied to objects whose motion is
rotational instead of linear. The rotational
inertia of an object is that object’s
resistance to a change in its angular
velocity. That is, if an object is spinning, it
has a tendency to keep spinning until a net
torque acts upon it.
FORMULA
For the very special case of the
moment of inertia of a single mass
m, rotating about an axis, a
distance r from m, we have
I=mr2
Figure
Axis
Formula
Thin ring
Axis through center
I=mr2
Disk
Axis through center
I=½ mr2
Thin rod
Axis through one
end perpendicular
to length
I=1/3 ml2
Sample Problem
Find the moment of inertia of a solid
cylinder of mass 3.0 kg and radius 0.50 m,
which is free to rotate about an axis
through its center.
Given: m =
r
Find:I
=
3.0 kg
0.50 m
Solution:
I = ½ mr2
= ½ (3.0 kg) (0.50 m)2
= ½ (3.0 kg) (0.25m2)
= 0.38 kg m2
Laws for Rotational Motion
1st law for rotational motion:
A body in motion at a constant angular
velocity will continue in motion at that
same angular velocity, unless acted upon
by some unbalanced external torque.
2nd law for rotational motion:
When an unbalanced external torque
acts on a body with moment of inertia I, it
gives that body an angular acceleration a,
which is directly proportional to the
torque T and inversely proportional to the
moment of inertia.
3rd law for rotational motion:
If body A and body B have the same
axis of rotation, and if body A exerts a
torque on body B, then body B exerts an
equal but opposite torque on body A.
Torque is the
quantity that
measures how
effectively a force
(F) causes
acceleration.
The magnitude of
the torque can be
calculated by:
T= F x l
a train wheel
Sample Problem
The width of a door is 40 cm. If it is opened by applying a
force of 2 N at its edge (away from the hinges), calculate
the torque produced which causes the door to open.
Solution :
Length of lever arm = 40 cm = 0.40 m (since distance
between axis of rotation and line of action of force is 40
cm)
Force applied = 2 N
Torque = Lever arm x force applied
= 0.40 x 20
= 8 Nm (8 Newton meter)
The center of
gravity is the
point from
where all the
weight of an
object is said
to be
concentrated.
Locating the Center of Gravity
A system is provided for determining the
center of gravity of a body by providing
a balance arm supported from a single
pivot point, then positioning the body on
the balance arm and measuring the
rotation of the balance arm in two
orthogonal planes.
We have just seen that a body in uniform circular
motion experiences a centripetal acceleration. From
Newton’s second law of motion, there must be a
force that gives an object this acceleration.
Applying Newton’s Second Law to an object in uniform
circular motion,
F = ma
Fc = mac
= mv2
r
Fc = mv2
r
Sample Problems
A 400-g rock attached to a 1.0 m string is whirled in a
horizontal circle at the constant speed of 10.0 m/s.
Neglecting the effects of gravity, what is the centripetal
force acting on the rock?
Given: v = 10.0 m/s
r = 1.0 m
m = 400 g x 1kg = 0.4 kg
1000 g
Find: Fc
Solution:
Fc = mv2
r
= (0.4 kg) (100 m2/s2)
m
= 40 kg ∙ m/s2
= 40 N
An object traveling
in a circle behaves
as if it is
experiencing an
outward force.
This force known
as the centrifugal
force.
Fc = mv2/r,
where Fc = centrifugal
force,
m = mass,
v = speed, and
r = radius.
Simulated gravity is the
varying (increase or
decrease) of
apparent gravity (gforce) via artificial
means, particularly in
space, but also on the
Earth.
An angular momentum is the property
characterizing the rotary inertia of an object or
system of objects in motion about an axis that
may or may not pass through the object or
system. Angular momentum is a vector quantity,
requiring the specification of both a magnitude
and a direction for its complete description.
H=Iω
H = angular momentum
I = inertia
w = inertial space
Conservation of Angular Momentum
Momentum is the product of inertia and velocity. Inertia
means the tendency of something not to change, and
velocity means how fast it moves. So momentum means
the tendency of an object in motion not to slow
down. Momentum is of two kinds, angular and
linear. Both kinds of momentum are conserved in any
collision. Conservation means that none is lost, so the
total momentum of the system before the collision, plus
any additional impulse from outside the system, will
equal the total momentum after the collision. This
conservation principle is key to deriving formulas.