Question

If mass, speed and radius of the circular path of the particle are increased by $100\%$, then the necessary force required to maintain the circular path will have to be increased by

• A. $100\%$
• B. $250\%$
• C. $300\%$
• D. $400\%$

Answer 1

Answer: (C)

Centripetal force is given as $F=\frac{m{v}^2}{r}$
where, $m=$ mass, $v=$ velocity and $r=$ radius.
$\frac{F_2}{F_1}=\frac{m_2}{m_1}\cdot {\left(\frac{v_2}{v_1}\right)}^2\cdot \left(\frac{r_1}{r_2}\right)\dots$ (i)
When, mass, speed and radius of circular path of particle increases by $100\%$, i.e. these become double.
Hence, $m_2=2m_1$
$v_2=2v_1$
$r_2=2r_1$
$\therefore$ From Eq. (i), we have
$\frac{F_2}{F_1}=\frac{2m_1}{m_1}{\left(\frac{2v_1}{v_1}\right)}^2\left(\frac{r_1}{2r_1}\right)$
$=2\times 4\times \frac{1}{2}=4$
$\Rightarrow F_2=4F_1$
Percentage increase in centripetal force,
$=\frac{F_2-F_1}{F_1}\times 100$
$=\frac{4F_1-F_1}{F_1}\times 100=300\%$