System of Particles and Rotational Motion

Physics Part I

Physics

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Two discs of moments of inertia *I*_{1} and *I*_{2} about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds ω_{1} and ω_{2} are brought into contact face to face with their axes of rotation coincident. (a) What is the angular speed of the two-disc system? (b) Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take ω_{1} ≠ ω_{2}.

(a)** **Moment of inertia of disc I= I_{1 }

Angular speed of disc I = ωMoment of inertia of disc II = I

Angular speed of disc II = ω

Angular momentum of disc I, L

Angular momentum of disc II, L

Total initial angular momentum

When the two discs are joined together, their moments of inertia get added up.

Moment of inertia of the system of two discs,

Let ω be the angular speed of the system.

Total final angular momentum,

Using the law of conservation of angular momentum, we have

∴ ω =

(b) Kinetic energy of disc I,

Kinetic energy of disc II,

Total initial kinetic energy,

When the discs are joined, their moments of inertia get added up.

Moment of inertia of the system,

Angular speed of the system = ω

Final kinetic energy is given by,

= (

=

∴

Solving the equation, we get

=

All the quantities on RHS are positive

Therefore,

When the two discs come in contact with each other, there is a frictional force between the two. Hence there would be a loss of Kinetic Energy.

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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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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 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.

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

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