Q. 154.7( 32 Votes )

# A famous relation in physics relates ‘moving mass’ m to the ‘rest mass’ m_{o} of a particle in terms of its speed v and the speed of light, c. (This relation first arose as a consequence of special relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant c. He writes:

Guess where to put the missing c.

Answer :

The dimension of the quantities involved are:

m = M^{1}L^{0}T^{0}

m_{0} = M^{1}L^{0}T^{0}

v = M^{0}L^{1}T^{-1}

c = M^{0}L^{1}T^{-1}

For a correct dimensional equation, the dimensions of LHS and RHS should be equal. We can see that dimensions of m and m_{0} are equal thus the remaining part of the equation should be constant.

should be unitless.

1 is dimensionless so should be dimensionless.

On subtraction, dimension of the 2 quantities should be same. So, dimension of 1 and v^{2} should be same. Therefore, v^{2} should be dimensionless and therefore v.

We can see that dimension of v and c are same. So, replacing v by v/c will make it dimensionless and the given equation dimensionally correct.

Hence, correct relation will be,

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PREVIOUSA book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion:A. y = a sin 2 t/TB. y = a sin vtC. y = (a/T) sin t/aD. (a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.NEXTThe unit of length convenient on the atomic scale is known as an angstrom and is denoted by Å: 1 Å = 10–10 m. The size of a hydrogen atom is about 0.5 Å. What is the total atomic volume in m3 of a mole of hydrogen atoms?

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