Vertical Spring - Position locations and Shifted equilibrium

Vertical Spring–Mass System

Natural Length, Mean Position and Extreme Position of a Vertical Spring

Any Position

The spring obeys Hooke’s Law:

\[F_{\text{spring}} = -kx\] \[ma = -kx\] \[a = -\frac{k}{m}x\] \[a = -\omega^2 x\] \[\boxed{\omega = \sqrt{\frac{k}{m}}\frac{k}{m}}\]

Mean Position

At equilibrium:

\[F_{\text{spring}} = F_{\text{gravity}}\] \[kx_{\text{mean}} = mg\] \[\boxed{x_{\text{mean}}=\frac{mg}{k}}\]

Extreme Position

Using conservation of energy:

\[W_{\text{gravity}}=PE_{\text{spring}} +KE\] \[mgx_{\text{extreme}}=\frac{1}{2}kx_{\text{extreme}}^2 + 0\] \[\boxed{x_{\text{extreme}}=\frac{2mg}{k}}\]

Key Idea: The mass oscillates vertically about the mean position. At the equilibrium position, spring force balances gravity. The extreme position corresponds to maximum extension where velocity becomes zero momentarily.

Vertical Spring SHM with Shifted Equilibrium

Simple Harmonic Motion of a Vertical Spring–Mass System

Forces on the Mass

Net restoring force:

\[F_{\text{net}}=-\left(F_{\text{spring}}-F_{\text{gravity}}\right)\] \[F_{\text{net}}=-(kx - mg)\]

At equilibrium:

\[mg = kx_{\text{mean}}\] \[F_{\text{net}}=-(kx - kx_{\text{mean}})\] \[ma = -k(x - x_{\text{mean}})\]

Using shifted coordinate:

\[X = x - x_{\text{mean}}\]

Equation of SHM

\[a = -\frac{k}{m}X\] \[a = -\omega^2 X\] \[\boxed{\omega=\sqrt{\dfrac{k}{m}}\dfrac{k}{m}}\]

Velocity at Mean Position

Using conservation of energy:

\[PE_{\text{spring}} + KE_{\text{mean}} = U_i - U_f\] \[\frac{1}{2}kx_{\text{mean}}^{2} + \frac{1}{2}mv_{\text{mean}}^{2} = mgx_{\text{mean}}\] \[\frac{1}{2}k\left(\frac{mg}{k}\right)^2 + \frac{1}{2}mv_{\text{mean}}^{2}= mg\left(\frac{mg}{k}\right)\] \[\frac{1}{2}mv_{\text{mean}}^{2}= \frac{1}{2}\frac{m^{2}g^{2}}{k}\] \[v_{\text{mean}} = g\sqrt{\frac{m}{k}}\frac{m}{k}\] \[\boxed{v_{\max}=g\sqrt{\frac{m}{k}}\frac{m}{k}}\]

Important Concept: The vertical spring performs Simple Harmonic Motion about a shifted equilibrium position. Gravity changes only the equilibrium position of oscillation and does not change the angular frequency: Thus, the motion remains SHM with a new origin located at the mean position. \[\\ \] Frequency of PE and KE curves: Also what will be the frequency of the PE and KE ? Hint: substitute x = A sin wt and v = Aw cost wt in the PE and KE formulas and find out. You should be able to see that the frequency will be 2w (twice the frequncy of the SHM).