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System of Particles and Rotational Motion

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CBSE Gujarat Board Haryana Board

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Class 10 Class 12
Two discs of moments of inertia I1 and I2 about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds ω1 and ω2 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 ω1 ≠ ω2.

(a) Moment of inertia of disc I= I

Angular speed of disc I = ω

Moment of inertia of disc II = I

Angular speed of disc II = ω

Angular momentum of disc I, L1 = I1ω

Angular momentum of disc II, L2 = I2ω

Total initial angular momentum Li = I1ω1 + I2ω

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

Moment of inertia of the system of two discs, I = I1 + I

Let ω be the angular speed of the system.

Total final angular momentum, LT = (I1 + I2ω

Using the law of conservation of angular momentum, we have

Li = L

I1ω1 + I2ω2 = (I1 + I2)ω

∴              ω =

(b) Kinetic energy of disc I, E1 =  I1ω1
Kinetic energy of disc II, E2=  I2ω2
Total initial kinetic energy, Ei =I1ω12 + I2ω22

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

Moment of inertia of the system, I = I1 + I

Angular speed of the system = ω

Final kinetic energy is given by,

E= I1 + I2) ω2

=  ( I1 + I2

=

∴  Ei - Ef
Solving the equation, we get

=

All the quantities on RHS are positive

Therefore,

Ei - Ef > 0

i.e.,       Ei > E

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.
439 Views

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.
1449 Views

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.
943 Views

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.
794 Views

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.
730 Views

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.
911 Views