Units and Measurements

A standard vernier caliper divides its sliding scale into $N = 10$ parts that together span 9 main scale divisions (MSD), so that:

The least count is just the gap this creates between one main scale division and one vernier division:

Dimensional analysis uses the same kind of matching. To find how the time period of a simple pendulum depends on its length $L$ and gravity $g$, assume a power-law form and match dimensions on both sides:

Matching powers of $T$ gives $-2b = 1$, so $b = -\tfrac{1}{2}$; matching powers of $L$ gives $a + b = 0$, so $a = \tfrac{1}{2}$:

That last derivation found the shape of the pendulum formula, the square root, the ratio of $L$ to $g$, but it could not find the constant $k = 2\pi$. Dimensional analysis can never recover a pure number, because a pure number has no dimensions to match against. It’s why every dimensionally-derived formula in this course carries an unknown constant that has to come from somewhere else, usually a full derivation or an experiment.

The method has sharper limits too. It cannot handle a quantity that depends on the sum of two differently-scaled terms, like $s = ut + \tfrac{1}{2}at^2$: both terms already have the same dimension, so dimensional analysis alone could never have told you there should be a $\tfrac{1}{2}$ in front of the second one, or even that the equation has two terms rather than one. And it breaks down completely for any relation built from a trigonometric, exponential, or logarithmic function, because the argument of $\sin\theta$ or $e^x$ has to be dimensionless in the first place, which dimensional analysis simply assumes rather than proves.