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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 the area of the triangle contained between the vectors a and b is one half of the magnitude of a × b.

Consider two vectors OK = vector |a| and OM = vector |b|, inclined at an angle θ, as shown in the following figure.


                           

In space increment space OMN comma space using space the space relation comma space

sin space straight theta space equals space MN over OM space equals space fraction numerator MN over denominator open vertical bar straight b with rightwards harpoon with barb upwards on top close vertical bar end fraction

space space MN space equals space open vertical bar straight b with rightwards harpoon with barb upwards on top close vertical bar space sin space straight theta

open vertical bar straight a with rightwards harpoon with barb upwards on top space straight x space straight a with rightwards harpoon with barb upwards on top close vertical bar space equals space open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar space open vertical bar straight b with rightwards harpoon with barb upwards on top close vertical bar space sin space straight theta space

space space space space space space space space space space space space space space space equals space OK space straight x space MN space straight x space 2 over 2

space space space space space space space space space space space space space space space equals space 2 space straight x space Area space of space increment space OMK space

therefore space space Area space of space increment space OMK space equals space 1 half space open vertical bar straight a with rightwards harpoon with barb upwards on top space straight x space straight b with rightwards harpoon with barb upwards on top close vertical bar 

Area of the triangle contained between the vectors a and b is one-half of the magnitude of a x b


362 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 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

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