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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 particle of mass m is projected at an angle θ with the x–axis with an initial velocity v0 in the x–y plane as shown in the figure. At a time t<vosin/g, the angular momentum of the particle is

  • negative mgv subscript straight o space straight t squared space cos space straight theta space bold j with bold hat on top
  • mgv subscript 0 straight t space cos space straight theta space bold k with bold hat on top
  • negative 1 half space mgv subscript 0 straight t squared space cos space straight theta bold space bold k with bold hat on top bold space
  • 1 half space mgv subscript straight o space straight t squared space cos space straight theta space bold i with bold hat on top

C.

negative 1 half space mgv subscript 0 straight t squared space cos space straight theta bold space bold k with bold hat on top bold space

The angular momentum  of the projectile is given by,
L = m (r x v)
straight L space equals space straight m space left square bracket space straight v subscript 0 space cos space straight theta space straight t space bold i with hat on top space plus space left parenthesis straight v subscript 0 space sin space straight theta space minus space gt right parenthesis bold j with hat on top right square bracket
space space space space space space space space space space space space space straight x space left square bracket space straight v subscript straight o space cos space straight theta space straight t space bold i with hat on top space plus space left parenthesis straight v subscript 0 space sin space straight theta minus space gt right parenthesis bold j with hat on top right square bracket
equals space mv subscript 0 space cos space θt space open square brackets negative 1 half gt close square brackets bold space bold k with bold hat on top
bold space bold equals space minus 1 half space mgv subscript 0 space straight t squared space cos space straight theta bold space bold k with bold hat on top

The angular momentum  of the projectile is given by,
L = m (r x v)
straight L space equals space straight m space left square bracket space straight v subscript 0 space cos space straight theta space straight t space bold i with hat on top space plus space left parenthesis straight v subscript 0 space sin space straight theta space minus space gt right parenthesis bold j with hat on top right square bracket
space space space space space space space space space space space space space straight x space left square bracket space straight v subscript straight o space cos space straight theta space straight t space bold i with hat on top space plus space left parenthesis straight v subscript 0 space sin space straight theta minus space gt right parenthesis bold j with hat on top right square bracket
equals space mv subscript 0 space cos space θt space open square brackets negative 1 half gt close square brackets bold space bold k with bold hat on top
bold space bold equals space minus 1 half space mgv subscript 0 space straight t squared space cos space straight theta bold space bold k with bold hat on top

821 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

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