﻿ (a) Prove the theorem of perpendicular axes. (Hint: Square of the distance of a point (x, y) in the x–y plane from an axis through the origin perpendicular to the plane is (x2 + y2).  from Physics System of Particles and Rotational Motion Class 11 Manipur Board

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System of Particles and Rotational Motion

Physics Part I

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(a) Prove the theorem of perpendicular axes.

(Hint: Square of the distance of a point (x, y) in the x–y plane from an axis through the origin perpendicular to the plane is (x+ y2).

The theorem of perpendicular axes states that the moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with the perpendicular axis and lying in the plane of the body.

A physical body with centre O and a point mass m,in the xyplane at (xy) is shown in the following figure below.

Moment of inertia about x-axis, Ix = mx

Moment of inertia about y-axis, Iy = my

Moment of inertia about z-axis, Iz = m(x2 + y2)1/2

Therefore,

Ix + Iy = mx2 + my

= m(x2 + y2

=

I
x + Iy = I

Hence, the theorem is proved.

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What is the significance of defining the center of mass of a system?

The motion of n particle system can be reduced to one particle motion.

An equivalent single point object would enable us to discuss the overall motion of the system.
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Define centre of mass.

Centre of mass of a body or a system of bodies is a point at which the entire mass of the body or system is supposed to be concentrated.
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What is the need of centre of mass?

Newton’s second law of motion is strictly applicable to point masses only. To apply the Newton's law of motion to rigid bodies, the concept of centre of mass is introduced.

The concept of centre of mass of a system enables us to discuss overall motion of the system by replacing the system by an equivalent single point object.
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Is it necessary that there should be matter at the centre of mass of system?

No, it is not necessary that there be matter at the centre of mass of the system.

For e.g., if two equal point masses are separated by certain distance, the centre of mass lies at the mid point of two point masses and there is no mass at that point.
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Is it necessary for centre of mass to lie within the body?

No, centre of mass needs not to lie within the body. It is not necessary that the total mass of the system be actually present at the centre.

The position of the centre of mass is calculated using the usual Newtonian type of equations of motion.
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