A particle of mass $m$ is under an influence of a force $\mathbf{F}=\left(-k \mathbf{x}+\mathbf{F}_0\right)…
A particle of mass $m$ is under an influence of a force $\mathbf{F}=\left(-k \mathbf{x}+\mathbf{F}_0\right) \mathrm{N}$. The particle when disturbed will oscillate
about $x=0$ with $\omega=\sqrt{\frac{k}{m}}$
about $x=0$ with $\omega=\sqrt{\frac{m}{k}}$
about $x=\frac{F_0}{k}$ with $\omega=\sqrt{\frac{k}{m}}$
about $x=\frac{F_0}{k}$ with $\omega \neq \sqrt{\frac{k}{m}}$
Solution
Given that,
Mass of particle $=m$
Restoring force, $\mathbf{F}=\left(-k \mathbf{x}+\mathbf{F}_0\right) \mathrm{N}$
$\begin{aligned} & =-k\left(x-\frac{\mathbf{F}_0}{k}\right) \\ & =-k y \quad\left[\text { Take, } x-\frac{\mathbf{F}_0}{k}=y\right]\end{aligned}$
Which is SHM.
We know that, at the mean position,
$y=0$
$x-\frac{\mathbf{F}_0}{k}=0 \Rightarrow x=\frac{\mathbf{F}_0}{k}$
$\therefore \quad F=m a \Rightarrow m a=-k y$
$m\left(-\omega^2 y\right)=-k y \Rightarrow \omega=\sqrt{\frac{k}{m}}$
Frequency of SHM, $\omega=\sqrt{\frac{k}{m}}$