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System of Particles and Rotational Motion

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Physics Part I

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CBSE Gujarat Board Haryana Board

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Class 10 Class 12
(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 child’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?


Initial angular velocity, ω1= 40 rev/min 

Final angular velocity = ω

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

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

The two moments of inertia are related as, 

                         I2 = open parentheses 2 over 5 close parentheses I1
Since no external force acts on the boy, the angular momentum L is a constant. 

Hence, for the two situations, 

            I2ω2  =  I1 ω

                ω2 = open parentheses straight I subscript 1 over straight I subscript 2 close parentheses ω

                     equals space open square brackets fraction numerator straight I subscript 1 over denominator begin display style bevelled 2 over 5 end style space straight I subscript 1 end fraction close square brackets space straight x space 40 space
equals space open parentheses 5 over 2 close parentheses space straight x space 40 space
equals space 100 space rev divided by min 


Final K.E. = 2.5 Initial K.E.

Final kinetic rotation, EF = (1/2) I2 ω2

Initial kinetic rotation, EI =  (1/2) I1 ω12 

    straight E subscript straight f over straight E subscript straight i =open parentheses 1 half close parentheses I2 ω2/ open parentheses 1 half close parentheses I1 ω1

         =open parentheses 2 over 5 close parentheses I1 (100)2 / I1 (40)

         = 2.5 


EF = 2.5 E

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

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. 

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. 

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. 

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. 

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.