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(C) The condition for the mean position in simple harmonic motion is that the net restoring force must be zero.Given the force equation: $F = -kx + F_

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about $x = F_0/k$ with $\omega = \sqrt{k/m}$

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(C) The condition for the mean position in simple harmonic motion is that the net restoring force must be zero.Given the force equation: $F = -kx + F_0$.At the mean position,$F = 0$,so $0 = -kx + F_0$,which gives $x = F_0/k$.To find the angular frequency $\omega$,we rewrite the force equation in terms of the displacement from the mean position $x' = x - F_0/k$. Then $x = x' + F_0/k$.Substituting this into the force equation: $F = -k(x' + F_0/k) + F_0 = -kx' - F_0 + F_0 = -kx'$.Comparing this with the standard $SHM$ equation $F = -m\omega^2 x'$,we get $m\omega^2 = k$,which implies $\omega = \sqrt{k/m}$.Therefore,the particle oscillates about $x = F_0/k$ with an angular frequency $\omega = \sqrt{k/m}$.

$A$ particle of mass $m$ is under the influence of a force $F$ which varies with the displacement $x$ according to the relation $F = -kx + F_0$,where $k$ and $F_0$ are constants. The particle,when disturbed,will oscillate:

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