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.
$\frac{F_{2}}{F_{1}}=\frac{m_{2}}{m_{1}} \cdot\left(\frac{v_{2}}{v_{1}}\right)^{2} \cdot\left(\frac{r_{1}}{r_{2}}\right) \ldots$ (i)
When, mass, speed and radius of circular path of particle increases by $100 \%$, i.e. these become double.
Hence, $m_{2}=2 m_{1}$
$v_{2}=2 v_{1}$
$r_{2}=2 r_{1}$
$\therefore $ From Eq. (i), we have
$\frac{F_{2}}{F_{1}}=\frac{2 m_{1}}{m_{1}}\left(\frac{2 v_{1}}{v_{1}}\right)^{2}\left(\frac{r_{1}}{2 r_{1}}\right)$
$=2 \times 4 \times \frac{1}{2}=4$
$\Rightarrow F_{2}=4 F_{1}$
Percentage increase in centripetal force,
$=\frac{F_{2}-F_{1}}{F_{1}} \times 100$
$=\frac{4 F_{1}-F_{1}}{F_{1}} \times 100=300 \%$