Question

The radius of curvature of the centre of mass of the bike is

A. $\dfrac{l}{\tan \theta }\sqrt{4+{{\tan }^{2}}\theta }$
B. $\dfrac{l}{\tan \theta }\sqrt{2+{{\tan }^{2}}\theta }$
C. $\dfrac{l}{2\tan \theta }\sqrt{4+{{\tan }^{2}}\theta }$
D. $\dfrac{l}{2\tan \theta }\sqrt{2+{{\tan }^{2}}\theta }$

Answer 1

Hint: We are actually given all the necessary conditions that would be required so as to find the radius of curvature of the centre of mass. First you could find the tangent of the angle in the figure. Then you could make use of Pythagorean theorem and later do necessary substitutions to get the required quantity in the form given in the options.

Complete answer:
In the question, we are given a figure with the alignment of the wheels of a bike and also its center of mass marked. Let us assume that the bike doesn’t slip. We are supposed to find the radius of curvature of the centre of mass of the bike by using these conditions.
From the given figure, we see that the tangent of the angle $\theta $ could be given by,
$\tan \theta =\dfrac{l}{{{R}_{1}}}$
$\Rightarrow {{R}_{1}}=\dfrac{l}{\tan \theta }$ ……………………………………………. (1)
Now, we could simply use the Pythagorean theorem to find the radius of curvature.
By Pythagorean theorem, we know that the squares of the hypotenuse is equal to the sum of the squares of the other two sides of the triangle. That is,
${{R}_{CM}}^{2}={{\left( \dfrac{l}{2} \right)}^{2}}+{{R}_{1}}^{2}$
Substituting (1), we get,
${{R}_{CM}}^{2}={{\left( \dfrac{l}{2} \right)}^{2}}+{{\left( \dfrac{l}{\tan \theta } \right)}^{2}}$
$\Rightarrow {{R}_{CM}}=\sqrt{\dfrac{{{l}^{2}}}{4{{\tan }^{2}}\theta }\left( {{\tan }^{2}}\theta +4 \right)}$
$\therefore {{R}_{CM}}=\dfrac{l}{2\tan \theta }\sqrt{4+{{\tan }^{2}}\theta }$
Therefore, we found the radius of curvature of the centre of curvature to be,
${{R}_{CM}}=\dfrac{l}{2\tan \theta }\sqrt{4+{{\tan }^{2}}\theta }$

So the correct answer is option C.

Note:
We have assumed the weight of the tires to be the same as we have chosen the midpoint joining these tires to be the point at which the center of mass of the bike resides. The rest of the conditions are made based on the assumption that the bike doesn’t skid on the road.