Question

A circular railway track of radius r is banked at angle so that a train moving with speed v can safely go round the track. A student writes: $\tan \theta =\frac{rg}{v^2}$. Why this relation is not correct?
(i) Equality of dimensions does not guarantee correctness of the relation.
(ii) Dimensionally correct relation may not be numerically correct
(iii) The relation is dimensionally incorrect.

• A. (i) and (ii)
• B. (ii) and (iii)
• C. (iii) and (i)
• D. (i), (ii) and (iii)

Answer 1

Answer: (A)

(i) and (ii)

$\left[\tan \theta \right]=\frac{\left[rg\right]}{\left[v^2\right]}=\frac{L\left(L{T}^{-2}\right)}{(L{T}^{-1}{)}^2}=1=\left[M^0{L}^0{T}^0\right]$.
Hence formula is dimensionally correct. But the relation is not correct because of (i) and (ii).
The correct relation is $\tan \theta =\frac{v^2}{rg}$.