If mass, speed and radius of the circular path of the particle are increased by \(100 \%\), then the…
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
\(100 \%\)
\(250 \%\)
\(300 \%\)
\(400 \%\)
Solution
Centripetal force is given as
\(F=\frac{m v^2}{r}\)
where, \(m=\) mass, \(v=\) velocity and \(r=\) radius.
\(\frac{F_2}{F_1}=\frac{m_2}{m_1} \cdot\left(\frac{v_2}{v_1}\right)^2 \cdot\left(\frac{r_1}{r_2}\right)\) ...(i)
When, mass, speed and radius of circular path of particle increases by \(100 \%\), i.e. these become double.
Hence,
\(\begin{aligned}
& m_2=2 m_1 \\
& v_2=2 v_1 \\
& r_2=2 r_1
\end{aligned}\)
\(\therefore\) From Eq. (i), we have
\(\begin{aligned}
\Rightarrow & \frac{F_2}{F_1} & =\frac{2 m_1}{m_1}\left(\frac{2 v_1}{v_1}\right)^2\left(\frac{r_1}{2 r_1}\right)=2 \times 4 \times \frac{1}{2}=4 \\
\Rightarrow \quad & F_2 & =4 F_1
\end{aligned}\)
Percentage increase in centripetal force,
\(=\frac{F_2-F_1}{F_1} \times 100=\frac{4 F_1-F_1}{F_1} \times 100=300 \%\)