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Show that the oscillation of the loaded sping is Simple Harmonic

Asked by Agnes Sara 14th February 2011, 12:00 AM

Dear student

A particle is said to be simple harmonic  motion if its acceleration is directly proportional to the displacement from a fixed point and is always directed towards that point.

Horizontal and vertical oscillations of a loaded spring are some examples of simple harmonic motion.

The block-spring system is a linear simple harmonic oscillator .All oscillating systems like diving board,violin string have some element of springiness, k(spring constant) and some element of inertia,m.

Simple Harmonic Motion -horizontal Oscillations of Spring:

Consider a mass(m) attached to an end of a spiral spring(whcih obey's Hooke's law) whose other end is fixed to a support as shown in figure.

The body is placed on a smooth horizontal surface. Let the body be displaced through a distance x towards right and released. It oscillates about its mean position, The restoring force acts in the opposite direction and it is proportional to the displacement.

$\displaystyle\therefore$ Restoring force F = -kx

From Newton's second law, F= ma

$\displaystyle\Rightarrow$    ma = -kx

$\displaystyle\Rightarrow$    a = $\displaystyle\frac{{-{k}}}{{m}}$ x

Comparing with the equation of simple harmonic motion a = -$\displaystyle\omega$ 2x

$\displaystyle\Rightarrow{\omega}^{{2}}=\frac{{k}}{{m}}$

$\displaystyle\Rightarrow\omega=\sqrt{{\frac{{k}}{{m}}}}$

But time period T = $\displaystyle\frac{{{2}\pi}}{\omega}$

Time period T = $\displaystyle{2}\pi\sqrt{{\frac{{{m}}}{{{k}}}}}$

Frequency $\displaystyle\eta$ = $\displaystyle\frac{{1}}{{T}}=\frac{{1}}{{{2}\pi}}\sqrt{{\frac{{{k}}}{{{m}}}}}$

Hope this helps.
Regards
Team
Topperlearning
Answered by Expert 15th February 2011, 11:44 AM

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