For procedures, see your mechanical laboratory manual.

Title: Moment of Inertia (M.I) of a Rotor


Aim

  1. To determine the moment of inertia of the two wheels and axle supplied by the "pendulum method".
  2. To determine the moment of inertia of the given armature using torsional oscillations.

Theory

Inertia

Inertia was best explained by Sir Isaac Newton in his first law of motion. Newton's first law of motion states that "An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force." Objects tend to "keep on doing what they're doing." In fact, it is the natural tendency of objects to resist changes in their state of motion. This tendency to resist changes in their state of motion is described as inertia.

The first law, also called the law of inertia, was pioneered by Galileo. This was quite a conceptual leap because it was not possible in Galileo's time to observe a moving object without at least some frictional forces dragging against the motion. In fact, for over a thousand years before Galileo, educated individuals believed Aristotle's formulation that, wherever there is motion, there is an external force-producing that motion.

The principle of inertia is one of the fundamental principles in classical physics that are still used today to describe the motion of objects and how they are affected by the applied forces on them.

Types of Inertia

In inertia, there isn’t just one type. Instead, you’ll find three different types of inertia including:
  1. The inertia of rest - An object stays where it is placed, and it will stay there until you or something else moves it. (i.e. Dust particles stay at rest until you shake a carpet.)
  2. The inertia of motion - An object will continue at the same speed until a force acts on it. (i.e. Body going forward when a car stops.)
  3. The inertia of direction - An object will stay moving in the same direction unless a force acts on it. (i.e. One's body movement to the side when a car makes a sharp turn.)

Moment of Inertia

The moment of inertia, otherwise known as the mass moment of inertia, angular mass, or most accurately, rotational inertia, of a rigid body, is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for the desired acceleration. It appears in the relationships for the dynamics of rotational motion. The moment of inertia must be specified concerning a chosen axis of rotation. For a point mass, the moment of inertia is just the mass times the square of the perpendicular distance to the rotation axis, I = mr². That point mass relationship becomes the basis for all other moments of inertia since any object can be built up from a collection of point masses. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.

The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass concerning distance from an axis.

Moment of inertia I is defined as the ratio of the net angular momentum L of a system to its angular velocity ω around a principal axis, that is,


If the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. If the shape of the body does not change, then its moment of inertia appears in Newton's law of motion as the ratio of an applied torque Ï„ on a body to the angular acceleration α around a principal axis, that is,


For a simple pendulum, this definition yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as I = mr².

Thus, the moment of inertia of the pendulum depends on both the mass m of a body and its geometry, or shape, as defined by the distance r to the axis of rotation.

This simple formula generalizes to define the moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses dm each multiplied by the square of its perpendicular distance r to an axis k. An arbitrary object's moment of inertia thus depends on the spatial distribution of its mass.

In general, given an object of mass m, an effective radius k can be defined, dependent on a particular axis of rotation, with such a value that its moment of inertia around the axis is I = mk². Where k is known as the radius of gyration around the axis.


References
  1. Newton's Three Laws of Motion, Stanford University.
  2. Newton's Laws - Lesson 1 - Newton's First Law of Motion, physicsclassroom.com.
  3. Examples of Inertia, yourdictionary.com.
  4. Moment of Inertia, Hyperphysics, Georgia State University.
  5. Clement, J (1982), "Students' preconceptions in introductory mechanics", American Journal of Physics vol 50, pp 66–71.
  6. Inertia, wikipedia.org.
  7. Moment of Inertia, wikipedia.org. 
  8. Paul, Burton (June 1979). Kinematics and Dynamics of Planar Machinery. Prentice-Hall.