Question

Young's modulus of steel is $19 \times 10^{10}\, N / m ^{2}$. Express it as $x \times 10^{11} \text{dyne} / cm ^{2}$. Here dyne is the CGS unit of force. The value of $x$ is____.

Answer 1

The unit of Young's modulus is $N / m ^{2}$.
This suggests that it has dimensions of $\frac{\text { Force }}{(\text { distance })^{2}}$.
Thus, $[Y]=\frac{[F]}{L^{2}}=\frac{M L T^{-2}}{L^{2}}=M L^{-1} T^{-2} \cdot N / m ^{2}$ is in SI units.
So, $1\, N / m ^{2}=(1\, kg )(1\, m )^{-1}(1\, s )^{-2}$
and 1 dyne $/ cm ^{2}=(1\, g )(1\, cm )^{-1}(1\, s )^{-2}$
so, $\frac{1\, N / m ^{2}}{1 \text { dyne } / cm ^{2}}=\left(\frac{1\, kg }{1\, g }\right)\left(\frac{1 m }{1 cm }\right)^{-1}\left(\frac{1\, s }{1\, s }\right)^{-2}$
$=1000 \times \frac{1}{100} \times 1=10$
or $1\, N / m ^{2}=10$ dyne $/ cm ^{2}$
or $19 \times 10^{10} N / m ^{2}=19 \times 10^{11}$ dyne $/ cm ^{2}$.