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### Root Mean square value of AC

• We know that time average value of AC over one cycle is zero and it can be proved easily
• Instantaneous current I and time average of AC over half cycle could be positive for one half cycle and negative for another half cycle but quantity i2 would always remain positive
• So time average of quantity i2 is

This is known as the mean square current
• The square root of mean square current is called root mean square current or rms current.
Thus,

thus ,the rms value of AC is .707i0 of the peak value of alternating current
• Similarly rms value of alternating voltage or emf is

• If we allow the AC current represented by i=i0sin(ωt+φ) to pass through a resistor of resistance R,the power dissipated due to flow of current would be
P=i2R
• Since magnitude of current changes with time ,the power dissipation in circuit also changes
• The average Power dissipated over one complete current cycle would be

If we pass direct current of magnitude irms through the resistor ,the power dissipate or rate of production of heat in this case would be
P=(irms)2R
• Thus rms value of AC is that value of steady current which would dissipate the same amount of power in a given resistance in a given tine as would gave been dissipated by alternating current
• This is why rms value of AC is also known as virtual value of current

### What is center of mass?

• Consider a body consisting of large number of particles whose mass is equal to the total mass of all the particles. When such a body undergoes a translational motion the displacement is produced in each and every particle of the body with respect to their original position.
• If this body is executing motion under the effect of some external forces acting on it then it has been found that there is a point in the system , where if whole mass of the system is supposed to be concentrated and the nature the motion executed by the system remains unaltered when force acting on the system is directly applied to this point. Such a point of the system is called centre of mass of the system.
• Hence for any system Centre of mass is the point where whole mass of the system can be supposed to be concentrated and motion of the system can be defined in terms of the centre of mass.
• Consider a stationary frame of refrance where a body of mass M is situated. This body is made up of n number of particles. Let mi be the mass and ri be the pisition vector of i'th particle of the body.
• Let C be any point in the body whose position vector with respect to origin O of the frame of refrance is Rc and position vector of point C w.r.t. i'th particle is rci as shown below in the figure.

• From triangle OCP
ri=Rc+rci                               (1)
multiplying both sides of equation 1 bt mi we get
miri=miRc+mirci
taking summation of above equation for n particles we get

If for a body

then point C is known as the centre of mass of the body.
• Hence a point in a body w.r.t. which the sum of the product of mass of the particle and their position vector is equal to zero is equal to zero is known as centre of mass of the body.