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Simple harmonic motion in spring-mass systems review

Overview of key terms, equations, and skills for the simple harmonic motion of spring-mass systems, including comparing vertical and horizontal springs. 


EquationSymbol breakdownMeaning in words
T, start subscript, s, end subscript, equals, 2, pi, square root of, start fraction, m, divided by, k, end fraction, end square rootT, start subscript, s, end subscript is the period of the spring, m is the mass, and k is the spring constant.The period of a spring-mass system is proportional to the square root of the mass and inversely proportional to the square root of the spring constant.

How to analyze vertical and horizontal spring-mass systems

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, 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.
Figure 1. To the left of this image is the resting position of the spring and to the right is the displaced equilibrium position of the spring when the mass is attached. A vertical spring mass system oscillates around this equilibrium position of y, equals, 0.
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:
Figure 2. The forces on the spring-mass system in figure 1.
Then, we can use Newton's second law to write an equation for the net force on the block:
ΣF=ma=FsFg=kdmg\begin{aligned}\Sigma F &= ma \\ \\ &=F_s - F_g \\ \\ &= kd - mg \end{aligned}
The block in figure 1 is not accelerating, so our equation simplifies to:
k, d, minus, m, g, equals, 0

Common mistakes and misconceptions

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.

Learn more

For deeper explanations of spring-mass systems see the video on period dependence for a mass on a spring.
To check your understanding and work toward mastering these concepts, check out our exercises:

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