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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

A small object of uniform density rolls up a curved surface with an initial velocity v'. It reaches up to a maximum height of fraction numerator 3 straight v squared over denominator 4 straight g end fraction with respect to the initial position. The object is,

  • ring

  • solid sphere

  • hollow sphere

  • disc


D.

disc

As v = square root of fraction numerator 2 gh over denominator 1 plus begin display style straight k squared over straight r squared end style end fraction end root
Given, h = fraction numerator 3 straight v squared over denominator 4 straight g end fraction
straight v squared space equals space fraction numerator 2 gh over denominator 1 space plus space begin display style straight k squared over straight r squared end style end fraction space
space space space space equals space fraction numerator 2 straight g space 3 straight v squared over denominator 4 straight g open parentheses 1 plus begin display style straight k squared over straight r squared end style close parentheses end fraction
space space space space equals space fraction numerator 6 space gv squared over denominator 4 space straight g space open parentheses 1 space plus begin display style straight u squared over straight v squared end style close parentheses end fraction
1 space equals space fraction numerator 3 over denominator 2 space open parentheses 1 plus begin display style straight k squared over straight v squared end style close parentheses end fraction
space space space 1 plus straight k squared over straight r squared space equals space 3 over 2
rightwards double arrow space straight k squared over straight r squared space equals space 3 over 2 space minus space 1 space
space space space space space space space space space space space equals space 1 half
straight k squared space equals space 1 half straight r squared space left square bracket space Equation space of space disc right square bracket
Hence, the object is disc.

As v = square root of fraction numerator 2 gh over denominator 1 plus begin display style straight k squared over straight r squared end style end fraction end root
Given, h = fraction numerator 3 straight v squared over denominator 4 straight g end fraction
straight v squared space equals space fraction numerator 2 gh over denominator 1 space plus space begin display style straight k squared over straight r squared end style end fraction space
space space space space equals space fraction numerator 2 straight g space 3 straight v squared over denominator 4 straight g open parentheses 1 plus begin display style straight k squared over straight r squared end style close parentheses end fraction
space space space space equals space fraction numerator 6 space gv squared over denominator 4 space straight g space open parentheses 1 space plus begin display style straight u squared over straight v squared end style close parentheses end fraction
1 space equals space fraction numerator 3 over denominator 2 space open parentheses 1 plus begin display style straight k squared over straight v squared end style close parentheses end fraction
space space space 1 plus straight k squared over straight r squared space equals space 3 over 2
rightwards double arrow space straight k squared over straight r squared space equals space 3 over 2 space minus space 1 space
space space space space space space space space space space space equals space 1 half
straight k squared space equals space 1 half straight r squared space left square bracket space Equation space of space disc right square bracket
Hence, the object is disc.

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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.
730 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

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 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

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