Question

A calorie is a unit of heat or energy and it equals about $4.2\, J$ where $1\, J = 1 \,kg\, m^{2} s^{-2}$. Suppose we employ a system of units in which the unit of mass equals $\alpha\, kg ,$ the unit of length equals $\beta\, m$, the unit of time is $\gamma \, s ,$ then the magnitude of calorie in the new units is $4.2 . \alpha^{x} \cdot \beta^{y} \cdot \gamma^{2}$ such that

• A. $x=1, y=2, z=-2$
• B. $x=-1, y=-2, z=+2$
• C. $x=1, y=-2, z=+2$
• D. $x=-1, y=2, z=+2$

Answer 1

Answer: (B)

$1$ calorie $=4.2\, kg\, m ^{2} s ^{-2}$
As $\,n_{1} u_{1}=n_{2} u_{2}$
$4.2\, kg m ^{2} s ^{-2}=\left(n_{2}\right)(\alpha\, kg )(\beta\, m )^{2}(\gamma\, s )^{-2}$
$\Rightarrow 4.2\, kg m ^{2} s ^{-2}=\left(n_{2}\right)\left(\alpha \cdot \beta^{2} \cdot \gamma^{-2}\right) kg \,\,m ^{2} s ^{-2}$
or $n_{2}=4.2 \alpha^{-1} \cdot \beta^{-2} \cdot \gamma^{2}$
$\Rightarrow x=-1, y=-2, z=+2$