System of Particles and Rotational Motion

Physics Part I

Physics

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(a) A child stands at the centre of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of 40 rev/min. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to 2/5 times the initial value? Assume that the turntable rotates without friction.

(b) Show that the childâ€™s new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?

Â

(a)Â

Initial angular velocity, Ï‰Final angular velocity = Ï‰

The moment of inertia of the boy with stretched hands =Â

The moment of inertia of the boy with folded hands =Â

The two moments of inertia are related as,Â

Since no external force acts on the boy, the angular momentumÂ

Hence, for the two situations,Â

Â Â Â Â Â Â Â Â Ï‰

Â Â Â Â Â Â Â Â Â Â Â Â Â

(b)

Final kinetic rotation,Â

Initial kinetic rotation,Â

Â Â Â Â =Â

Â Â Â Â Â =Â

Â Â Â Â Â = 2.5Â

Therefore,Â

The increase in the rotational kinetic energy is attributed to the internal energy of the boy.Â

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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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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.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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