﻿ Prove the result that the velocity v of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height h is given by v2 = 2gh/ [1 + (k2/R2) ].Using dynamical consideration (i.e. by consideration of forces and torques). Note k is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane.  from Physics System of Particles and Rotational Motion Class 11 Manipur Board

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Prove the result that the velocity of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height is given by v2 = 2gh/ [1 + (k2/R2) ].

Using dynamical consideration (i.e. by consideration of forces and torques). Note is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane.

A body rolling on an inclined plane of height h,is shown in figure below:

Let,

m
= Mass of the body

K = Radius of gyration of the body

= Translational velocity of the body

=Height of the inclined plane

g = Acceleration due to gravity

Total energy at the top of the plane, E­1= mg

Total energy at the bottom of the plane, Eb = KErot + KEtrans

=  I ω2 + mv2
But I = mk2 and ω = v /

∴ Eb =  (mk2) + mv2
= mv2 (1 +)
From the law of conservation of energy, we have

ET = E

mgh = mv2 (1 +)
∴                v =

Hence, the result.
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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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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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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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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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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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