System of Particles and Rotational Motion

Physics Part I

Physics

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Prove the result that the velocityÂ *vÂ *of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a heightÂ *hÂ *is given byÂ *v*^{2}Â = 2*gh*/ [1 + (*k*^{2}/*R*^{2}) ].

Using dynamical consideration (i.e. by consideration of forces and torques). NoteÂ*kÂ *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.Â

Using dynamical consideration (i.e. by consideration of forces and torques). NoteÂ

A body rolling on an inclined plane of heightÂ

m

Total energy at the top of the plane,Â

Total energy at the bottom of the plane,Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â

ButÂ

âˆ´Â

Â Â Â =

From the law of conservation of energy, we have

âˆ´ Â Â Â Â Â Â Â Â

Hence, the result.Â

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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.Â

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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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What is the need of centre of mass?

Newtonâ€™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.Â

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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.Â

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

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