# Obtain an expression for the frequency of vertical oscillations ofa loaded spring.

1
by TulipBahl325

2015-09-22T01:37:21+05:30

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Mass  m  is attached to the upper end of a mass less spring with a constant of k.
The lower end of spring is attached to a firm surface of a rigid body.

under the weight, the spring gets compressed a little, the displacement being α.  The mass stays in equilibrium and stationary.  Let us call the displacement from this equilibrium position = x, positive upwards.

m g = k α      =>  α = m g / k

Now if the mass is displaced a little from this equilibrium, then it will oscillated in SHM.  Let X be the displacement of the mass from its unextended length.  Let x be  the displacement from the equilibrium position.
X = x - α

Forces acting on the mass:   - m g downwards. and  - k x  (negative as when x is positive, the force is downwards).
F = m a = m d²X / dt² = - m g - k X
m d²x/ dt² =  - m g - k x + k α  = - k x  = - ω²  x

so angular frequency = ω = √(k/m)
frequency = f = ω/2π  = 1/2π √(k/m)

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