Simple harmonic motion in spring-mass systems review

Both vertical and horizontal spring-mass systems without friction oscillate identically around an equilibrium position if their masses and springs are the same.

For vertical springs however, we need to remember that gravity stretches or compresses the spring beyond its natural length to the equilibrium position. After we find the displaced position, we can set that as $y=0$y, equals, 0 and treat the vertical spring just as we would a horizontal spring. Figure 1 below shows the resting position of a vertical spring and the equilibrium position of the spring-mass system after it has stretched a distance $d$d.

We can use a free body diagram to analyze the vertical motion of a spring mass system. We would represent the forces on the block in figure 1 as follows:

Then, we can use Newton's second law to write an equation for the net force on the block:

The block in figure 1 is not accelerating, so our equation simplifies to:

Sometimes people think that the period of a spring-mass oscillator depends on the amplitude. Increasing the amplitude means the mass travels more distance for one cycle. However, increasing the amplitude also increases the restoring force. The increase in force proportionally increases the acceleration of the mass, so the mass moves through a greater distance in the same amount of time. Thus, increasing the amplitude has no net effect on the period of the oscillation.

For deeper explanations of spring-mass systems see the video on period dependence for a mass on a spring.

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