Find the period of a vertical spring - block system by both methods.
Answers
The two methods are force method and energy taking method.
Force Method
Position `A`: In this position, spring is in its normal length. But net force on block is not equal to zero. Its mg is pointing downwards.
Position `B`: This is equilibrium position. Net force on block is equal to zero. Spring force `kx_(0)` is in upward position is equal to mg force in downward direction.
Position `C`: This is displaced direction.
Remaining restoring force is upward direction towards the mean position is
`F = -[k(x+x_(0))-mg] = - kx (as kx_(0) = mg)`
As, `F = - kx`
`T= 2pisqrt((m)/(k))`
Energy Method taking `h = 0` at the mean position B, whole mechanical energy in displaced position C is
`E = (1)/(2)mv^2 + (1)/(2) k (x + x_(0))^(2) - mg x`
As `E = constant (dE)/(dt) = 0`
or `(1)/(2)m(2v) (dv)/(dt) + (1)/(2)k[2(x = x_(0))] (dx)/(dt) - mg (dx)/(dt) = 0`
Substituting, `(dx)/(dt) = v`, `(dv)/(dt) = a`, `kx_(0) = mg` and `ma = F`, the above equation also changes into,
`F = - kx `
`:. T = -2pisqrt((m)/(k))`