Modulus of Elasticity and Torque Dimensions Shortcut

 

❓ Question

In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimensions

$$[M^aL^bT^c]$$

If

$$b=-3$$

find the value of $c$.

---

✍️ Solution

Modulus of elasticity is

$$Y=\frac{\text{Stress}}{\text{Strain}}$$

Since strain is dimensionless,

$$[Y]=[\text{Stress}]$$
$$[Y]=\frac{[\text{Force}]}{[\text{Area}]}$$
$$[Y]=\frac{[MLT^{-2}]}{[L^2]}$$
$$\boxed{[Y]=[ML^{-1}T^{-2}]}$$

Now, torque is

$$\tau=F\times r$$

Therefore,

$$[\tau]=[MLT^{-2}][L]$$
$$\boxed{[\tau]=[ML^2T^{-2}]}$$

The required measured quantity is

$$\frac{Y}{\mathrm{\tau }}$$

Thus,

$$\left[\frac{Y}{\tau}\right] = \frac{[ML^{-1}T^{-2}]}{[ML^2T^{-2}]}$$
$$=[M^0L^{-3}T^0]$$

Comparing with

$$[M^aL^bT^c]$$

we get

$$a=0,\qquad b=-3,\qquad c=0$$

✅ Final Answer

$$\boxed{c=0}$$