Question

Show dimensionally that the expression, $Y=\frac{MgL}{\pi r^{2}l}$ is dimensionally correct, where Y is Young's modulus of the material of wire, L is length of wire, Mg is the weight applied on the wire and l is the increase in the length of the wire.

Answer 1

Young's modulus $Y=$ stress/strain. Stress =Force/area and strain =increase in length/ original length So strain is dimensionless. Thus, Y will have dimension equal to stress.

The dimension of LHS $=\left[Y\right]=\left[M{L}^{-1}{T}^{-2}\right]$ and

The dimension of RHS $=\frac{\left[M\right]\left[L{T}^{-2}\right]\left[L\right]}{\left[L^2\right]\left[L\right]}$ ($\pi$ is a dimensionless quantity)

$=\left[M{L}^{-1}{T}^{-2}\right]$

As the dimension of LHS is same as the dimension of RHS so the given equation is dimensionally correct.