A famous relation in physics relates ‘moving mass’ m to the ‘rest mass’ m_oof 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 :

$straight m space equals space fraction numerator straight m subscript straight o over denominator left parenthesis square root of 1 minus straight v squared end root right parenthesis to the power of bevelled 1 half end exponent end fraction$

Guess where to put the missing c.

Solution

Given the relation,
$straight m space equals space fraction numerator straight m subscript straight o over denominator left parenthesis square root of 1 minus straight v squared end root right parenthesis to the power of bevelled 1 half end exponent end fraction$
Dimension of m = M¹ L⁰ T⁰Dimension of m_o = M¹ L⁰ T⁰Dimension of v = M⁰ L¹ T^–1Dimension of v² = M⁰ L² T^–2Dimension of c = M⁰ L¹ T^{^–1}

The given formula will be dimensionally correct only when the dimension of L.H.S is the same as that of R.H.S.
This is possible when the factor in the denominator is dimensionless. This is only possible if $straight nu squared space is space divided space by space straight c squared.$
Hence the correct relation is
$straight m space equals space straight m subscript straight o over open parentheses 1 minus begin display style straight v squared over straight c squared end style close parentheses to the power of bevelled 1 half end exponent$

Some More Questions From Units and Measurement Chapter

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Units And Measurement

Some More Questions From Units and Measurement Chapter

Give four examples of physical quantities.

Define unit.

What are the two main required characteristics of a unit?

Can we use same units for different physical quantities?

Can we use different units to measure same physical quantity?

To measure a physical quantity X, the unit U is repeatedly used for n times. What is the magnitude of physical quantity?

Does the numerical factor in the quantitative measurement depend on the system of unit used?

Is one unit sufficient to measure a physical quantity?

Can a physical quantity have two or more units? Give example.

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