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A calorie is a unit of heat energy and its value is 4.18 J where 1 $J = 1 k g - m^{- 2} / s^{2}$ . Suppose we use a new system of units in which unit of mass equals $α$ kg, the unit of length equals $β$ m and the unit of the time is $γ \sec$ . Then the value of a calorie in the new system of units is :

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$4.18 \frac{^{α β^{2}}}{γ^{2}}$

$4.18 \frac{γ^{2}}{^{α β^{2}}}$

$4.18 \frac{^{γ^{2}}}{α}$

$4.18 \frac{β^{2}}{^{α γ^{2}}}$

answer is A.

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Detailed Solution

$1 J = (1 k g) {(1 m)}^{2} {(1 \sec)}^{- 2}$

$1 x = (α k g) {(β m)}^{2} {(γ \sec)}^{- 2}$

$∴ \frac{1 J}{1 x} = (\frac{1}{α}) {(\frac{1}{β})}^{2} {(γ)}^{2} = \frac{γ^{2}}{α β^{2}}$

$∴ 1 J = \frac{γ_{}^{2}}{α β^{2}} x$

$∵ u n i t becomes \frac{γ^{2}}{α β^{2}} times$

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