Effect of a Constant External Fo on a Harmonic Oscillator suppose the same sprin

Question

Effect of a Constant External Fo on a Harmonic Oscillator suppose the same spring shown in Figure 3.21 is held in a vertical position, supporting the same mass m (Fig. 32.5). The total force acting is now given by adding the weight mg to the restoring force, (3.2.20) mg where the positive direction is down. This equation could be written F kr mg by defining to be X X., as previously. However it is more convenient to define the vari ablex in a erent way namely, as the displacement from the new equilibrium position New equilibrium position Figure 3.2.5 The vertical case for the harmonic oscillator.

Explanation / Answer

when the spring mass system is placed in vertical position.

there are two forces acting on it

spring force Fk = -k(X-Xe) upwards

gravitational force Fg = mg downwards

at equillibrium these two forces are equal in magnitude

mg = k(X-Xe)

=> X-Xe = mg/k

where X-Xe is the extension in the spring

so the new position is Xe+ mg/k

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