Moment of Inertia, General Form

chinmay_kothekar

sir my concepts of centre of mass and moment of inertia are not clear so please explain me the origin of these concepts and related topics.

Admin

Center of Mass is the point in a system which behaves as though the entire mass of system is concentrated there and its motion is same if the resultant of all forces acting on the system were applied directly to it.

Mathematically:

Co-ordinates of Center of mass:

Where is total mass

You can read more about it at http://www.goiit.com/chapters/tutorial/physics/center_mass.htm

Will request soem expert to throw seom light too.. cheers

~forum administrator

ac

The concept of centre of mass has been rightly described by admin.
Consider a ball thrown vertically up in the air with a spin provided to it. A point on the surface of the ball will follow a complicated path in space owing to the spin. But the centre of the ball will follow a sraight vertical path. Hence, it is easier to describe the motion of the ball in terms of its centre of mass than in terms of point particles distributed over the entire mass of the ball.

Moment of Inertia:
Linear acceleration of a body is proportional to force applied to it and described as:
F = ma, where m is the mass of the body.
Similarly, when we talk about a body in rotational motion about an axis,
its angular acceleration is proportional to the total torque on the body.

Consider a body rotating about a fixed axis:
Then, radial acceleration of a particle at distance r from the axis is
v²/r =

² r

Radial force = m

²r. ----- (1)

Tangential acceleration = dv/dt
Tangenial force = m dv/dt = mr d

/dt = mr

. ------ (2)

Torque due to (1) about the axis is zero as it intersects the axis.
Torque due to (2) is mr²

.

Summing over all particles,
Total torque = _[i]

m_ir_i²

= I

,

where I = _[i]

m_ir_i² is the moment of inertia of the body about the axis of rotation.
Moment of inertia depends upon the choice of the axis of rotation.

Hope that clears your concept of moment of inertia.

neeraj

I just wanted to add few things about moment of inertia

Moment of inertia is defined with respect to a specific rotation axis. The moment of inertia of a point mass with respect to an axis is defined as the product of the mass times the distance from the axis squared. The moment of inertia of any extended object is built up from that basic definition. The general form of the moment of inertia involves an integral.

Moment of Inertia, General Form

Since the moment of inertia of an ordinary object involves a continuous distribution of mass at a continually varying distance from any rotation axis, the calculation of moments of inertia generally involves calculus, the discipline of mathematics which can handle such continuous variables. Since the moment of inertia of a point mass is defined by

then the moment of inertia contribution by an infinitesmal mass element dm has the same form. This kind of mass element is called a differential element of mass and its moment of inertia is given by

Note that the differential element of moment of inertia dI must always be defined with respect to a specific rotation axis. The sum over all these mass elements is called an integral over the mass.

Usually, the mass element dm will be expressed in terms of the geometry of the object, so that the integration can be carried out over the object as a whole (for example, over a long uniform rod).

Having called this a general form, it is probably appropriate to point out that it is a general form only for axes which may be called "principal axes", a term which includes all axes of symmetry of objects. The concept of moment of inertia for general objects about arbitrary axes is a much more complicated subject. The moment of inertia in such cases takes the form of a mathematical tensor quantity which requires nine components to completely define it.

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Moment of Inertia, General Form