A particle of mass ' \(m\) ' performs uniform circular motion of radius ' \(r\) ' with linear speed ' \(v\) ' under the application of force ' \(F\) '. If \(\mathrm{m}, \mathrm{v}\) and \(\mathrm{r}\) are all increased by \(50 \%\), the necessary change in force required to maintain the particle in uniform circular motion is

Given that, Mass \(=\mathrm{m}\), Velocity \(=\mathrm{v}\), Radius \(=\mathrm{r}\)
We know that, The force is
\(\text F =\frac{\text{ mv} ^2}{\text r }\quad\ldots(1)\)
Now, the increment is \(50 \%\) in mass, velocity and radius
\(\text m=\text m+\frac{\text m}{2},\text v=\text v+\frac{\text v}{2}\) and \(\text r=\text r+\frac{\text r}{2}\).
Now, new force is
\(F^{\prime}=\frac{\left(m+\frac{m}{2}\right)\left(v+\frac{v}{2}\right)^2}{\left(r+\frac{r}{2}\right)}=\frac{\frac{3}{2} m \times \frac{9}{4} v^2}{\frac{3}{2} r}=F \times \frac{9}{4}\)
Now, the \% change in force
\(\frac{F^{\prime}-F}{F} \times 100=\left(\frac{9}{4}-1\right) \times 100=125 \%\)