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System of Particles and Rotational Motion

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Physics Part I

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Physics

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CBSE Gujarat Board Haryana Board

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Class 10 Class 12

From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc, about a perpendicular axis, passing through the centre?

  • 13 MR2/32

  • 11MR2/32

  • 9MR2/32

  • 15MR2/32


A.

13 MR2/32

Illustrating the above figure,


As given in the above fig.,

Moment of Inertia of the disc, I = Iremain + I(R/2)

rightwards double arrow space straight I subscript remain space equals space straight I space minus space straight I subscript straight R divided by 2 end subscript

Now, using the values, we get

equals space MR squared over 2 space minus space open square brackets fraction numerator begin display style straight M over 4 end style open parentheses begin display style straight R over 2 end style close parentheses squared over denominator 2 end fraction plus straight M over 4 open parentheses straight R over 2 close parentheses squared close square brackets
equals space MR squared over 2 space minus open square brackets MR squared over 32 plus MR squared over 16 close square brackets
equals MR squared over 2 minus open square brackets fraction numerator MR squared space plus space 2 MR squared over denominator 32 end fraction close square brackets
equals space MR squared over 2 minus fraction numerator 3 MR squared over denominator 32 end fraction
equals space fraction numerator 16 space MR squared minus 3 MR squared over denominator 32 end fraction
That is,

Iremainfraction numerator 13 space MR squared over denominator 32 end fraction

Illustrating the above figure,


As given in the above fig.,

Moment of Inertia of the disc, I = Iremain + I(R/2)

rightwards double arrow space straight I subscript remain space equals space straight I space minus space straight I subscript straight R divided by 2 end subscript

Now, using the values, we get

equals space MR squared over 2 space minus space open square brackets fraction numerator begin display style straight M over 4 end style open parentheses begin display style straight R over 2 end style close parentheses squared over denominator 2 end fraction plus straight M over 4 open parentheses straight R over 2 close parentheses squared close square brackets
equals space MR squared over 2 space minus open square brackets MR squared over 32 plus MR squared over 16 close square brackets
equals MR squared over 2 minus open square brackets fraction numerator MR squared space plus space 2 MR squared over denominator 32 end fraction close square brackets
equals space MR squared over 2 minus fraction numerator 3 MR squared over denominator 32 end fraction
equals space fraction numerator 16 space MR squared minus 3 MR squared over denominator 32 end fraction
That is,

Iremainfraction numerator 13 space MR squared over denominator 32 end fraction
10122 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

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

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

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 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