Question

The relation $\tan \theta =\dfrac{{{v}^{2}}}{rg}$ provides the angle of banking of the cyclist going round the curve. Where $v$ be the speed of the cyclist, $r$ be the radius of the curve, and $g$ be the acceleration due to gravity. Which of the following statements about this equation tend to be true?
A. it is both dimensionally and also numerically correct.
B. it is neither dimensionally nor numerically correct.
C. it is dimensionally correct not numerically.
D. it is numerically correct but not dimensionally correct.

Answer 1

Hint: The banking angle is defined as the angle at which the vehicle has been inclined. The inclination occurs at the longitudinal and horizontal axis. This information will help you in finding the answer.

Complete step-by-step answer:
The equation of the banking angle is given as,
$\tan \theta =\dfrac{{{v}^{2}}}{rg}$
Where $v$ be the speed of the cyclist, $r$ be the radius of the curve, and $g$ be the acceleration due to gravity.
That is the equation is numerically correct.
Now let us check its dimensional aspects.
As we all know, the angle is not having any kind of dimension. Hence it is dimensionless.
$\left[ \theta \right]=\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}} \right]$
The velocity can be given in the dimensional format as,
$\left[ v \right]=\left[ L{{T}^{-1}} \right]$
Radius can be dimensionally expressed as,
$\left[ r \right]=\left[ L \right]$
And the acceleration due to gravity has been explained by the dimensional formula,
$\left[ g \right]=\left[ L{{T}^{-2}} \right]$
Substituting all these in the equation at the right hand side will give,
$\left[ \dfrac{{{v}^{2}}}{rg} \right]=\left[ \dfrac{{{\left( L{{T}^{-1}} \right)}^{2}}}{\left( L \right)\left( L{{T}^{-2}} \right)} \right]=\dfrac{{{L}^{2}}{{T}^{-2}}}{{{L}^{2}}{{T}^{-2}}}={{L}^{0}}{{T}^{0}}$
This tells that the right hand side is also a dimensionless quantity. The left hand side is already found to be dimensionless. Hence the equation is dimensionally also correct. The final status is that the equation is numerically as well as dimensionally correct. The answer is given as option A.

So, the correct answer is “Option A”.

Note: Banking of roads is referred to as the technique in which the edges are raised on the curved roads above the inner edge to give the necessary centripetal force to the vehicles because of that they can take a safe turn. Banking angle is the angle at which the road is banked. The banking angle does not depend on the mass of the vehicle.