ROTATIONAL DYNAMICS
Lesson
Objective:
To verify the Law of Conservation of Angular Momentum in a
rotational dynamics system with the help of a simple kit: to study the relationship
between the Angular Velocity and Moment of Inertia for a rotating body
Basic Concept:
In the case of a rotational motion, in
the absence of an external Torque, the total Angular Momentum of the system
remains constant. This is called the principle of Conservation of Angular
Momentum. Angular Momentum of a rotating body can be expressed as a product of
its Moment of Inertia and Angular Velocity. That is,
L = Iω
Where, L is the
Angular Momentum
I is the Moment of Inertia of the body
ω = Angular Velocity
The principle of Conservation of Angular Velocity requires
that ,
Initial Angular Momentum= Final Angular Momentum
i.e. L= L’
i.e. I’ω’ =
Iω
Hence, when the Moment of Inertia of a rotating body is decreased,
its Angular Velocity must be increased by equal number of times so as to
conserve the Angular Momentum of the system and vice-versa.
The Model
The model
consists of an acrylic pipe through which
a string has been passed. One end
of the string is connected to a ball and the other end of the thread is
connected to a weight.
Keeping the acrylic tube vertical, rotate the ball in a horizontal circle.
In that case, the Centripetal Force necessary to keep the ball in circular motion will be
supplied by the Weight of the body. Now, pull the string downward to make the radius
of orbit of the rotating ball shorter.
This decreases the Moment of Inertia of the rotating ball. However, such a
Force (directed radially inward) produces no external Torque on the system
allowing the Law of Conservation of Angular Momentum to apply. Hence, as the
radius of the rotating ball is made shorter decreasing its Moment of Inertia,
the ball rotates faster and faster with increasing Angular Velocity to conserve
its Angular Momentum.
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