One end of a massless spring of spring constant k and natural length l 0 is fixed while the other end is connected to a small object of mass m lying on a frictionless table. The spring remains horizon

Question

One end of a massless spring of spring constant k and natural length l 0 is fixed while the other end is connected to a small object of mass m lying on a frictionless table. The spring remains horizontal on the table. If the object is made to rotate at an angular velocity ω about an axis passing trough fixed end, then the elongation of the spring will be

TestHub · Accepted Answer

The particle is moving in a horizontal circle, so it is accelerated towards the centre with magnitude $\frac{v^{2}}{r}$ . The horizontal force on the particle is due to the spring and is given by $k x$ , where $x$ is the elongation and $k$ is the spring constant.

$k x = \frac{m v^{2}}{r} = m ω^{2} r = m ω^{2} (l_{0} + x)$

$\Rightarrow (k - m ω^{2}) x = m ω^{2} l_{0}$

$\Rightarrow x = \frac{m ω^{2} l_{0}}{k - m ω^{2}}$

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