Question

A particle has acceleration (a) while moving with uniform speed v in a circle of radius r. Start with a ∝ rⁿv^m. Derive the relationship between a, r and v using dimensional analysis. 

Answer 1

Explanations & Calculations

• To deal with these questions, first, try to write the units of all the given quantities (on both sides of the equal mark) in their SI units.
• Then gather like units together & compare their indexes.

$\qquad\qquad \begin{aligned} \small a&\propto r^n\cdot v^m\\ \small [a]&= \small LT^{-2}\\ \small [r]&= \small L\\ \small [v]&= \small LT^{-1}\\ \\ \small LT^{-2}&\propto \small L^n\cdot (LT^{-1})^m\\ &\propto \small L^{(n+m)}\cdot T^{-m}\\ \\ \small L:\,\,\,\qquad1&=n+m\cdots(1)\\ \small T:\qquad-2&\propto \small -m\\ \\ \small m&= \small 2\\ \small n&= \small 1-2=-1 \end{aligned}$

• Then what we are able to write is

$\qquad\qquad \begin{aligned} \small a&\propto \small k\cdot\frac{v^2}{r}\qquad or\qquad a= k\cdot\frac{v^2}{r} \end{aligned}$

• The dimensional analysis only allows us to estimate the validity of an equation for a relationship, not fully derive the real relationship.
• For that experimental approach is needed.