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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
Show that a. (b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors, ab and c

A parallelepiped with origin O and sides a, b, and c is shown in the following figure. 


                

Volume space of space the space given space parallelopiped space equals space abc space

OC with rightwards harpoon with barb upwards on top space space equals space straight a with rightwards harpoon with barb upwards on top space
stack OB space with rightwards harpoon with barb upwards on top space equals space straight b with rightwards harpoon with barb upwards on top
OC with rightwards harpoon with barb upwards on top space space equals space straight c with rightwards harpoon with barb upwards on top

Let space straight n with hat on top space be space straight a space unit space vector space perpendicular space to space both space straight b space and space straight c. space

Hence comma space straight n with hat on top space and space straight a space have space the space same space direction. space

therefore space straight b with rightwards harpoon with barb upwards on top space straight x space straight c with rightwards harpoon with barb upwards on top space equals space bc space Sin space straight theta space straight n with hat on top

space space space space space space space space space space space space space space space space equals space bc space straight n with hat on top space

space space space space space space space space space space space space space space space space equals space straight a with rightwards harpoon with barb upwards on top. space open parentheses straight b with rightwards harpoon with barb upwards on top space straight x space straight c with rightwards harpoon with barb upwards on top space close parentheses

space space space space space space space space space space space space space space space space equals straight a. space left parenthesis bc space straight n with hat on top space right parenthesis

space space space space space space space space space space space space space space space space equals space abc space Cosθ space straight n with hat on top

space space space space space space space space space space space space space space space space equals space abc space cos space 0 space

space space space space space space space space space space space space space space space space equals space abc

space space space space space space space space space space space space space space space space equals space Volume space of space the space parallelopiped space

 Hence proved. 
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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. 
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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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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. 
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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. 
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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