If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Oscillations and waves (Essentials) - Class 11th>Unit 2

Lesson 2: Mathematics of simple harmonic motion

# Introduction to simple harmonic motion review

Overview of key terms, equations, and skills for simple harmonic motion, including how to analyze the force, displacement, velocity, and acceleration of an oscillator.

## Key terms

TermMeaning
Oscillatory motionRepeated back and forth movement over the same path about an equilibrium position, such as a mass on a spring or pendulum.
Restoring forceA force acting opposite to displacement to bring the system back to equilibrium, which is its rest position. The force magnitude depends only on displacement, such as in Hooke’s law.
Simple harmonic motion (SHM)Oscillatory motion where the net force on the system is a restoring force.

## Equations

EquationSymbolsMeaning in words
$|{F}_{s}|=k|x|$${F}_{s}$ is spring force, $k$ is the spring constant, and $x$ is displacementThe magnitude of the spring force is directly proportional to the spring constant and the magnitude of displacement
$x\left(t\right)=A\mathrm{cos}\left(2\pi ft\right)$$x$ is displacement as a function of time, $A$ is amplitude, $f$ is frequency, and $t$ is timeDisplacement as a function of time is proportional to amplitude and the cosine of $2\pi$ times frequency and time

## Force, displacement, velocity, and acceleration for an oscillator

Simple harmonic motion is governed by a restorative force. For a spring-mass system, such as a block attached to a spring, the spring force is responsible for the oscillation (see Figure 1).
${F}_{s}=-kx$
Since the restoring force is proportional to displacement from equilibrium, both the magnitude of the restoring force and the acceleration is the greatest at the maximum points of displacement. The negative sign tells us that the force and acceleration are in the opposite direction from displacement.
$\begin{array}{rl}F& =ma\\ \\ -kx& =ma\\ \\ a& =-\frac{k}{m}x\end{array}$
The mass's displacement, velocity, and acceleration over time can be visualized in the graphs below (Figure 2-4).

## Analyzing graphs: Period and frequency

We can graph the movement of an oscillating object as a function of time. Frequency $f$ and period $T$ are independent of amplitude $A$. We can find the period $T$ by taking any two analogous points on the graph and calculating the time between them. It’s often easiest to measure the time between consecutive maximum or minimum points of displacement. Once the period is known, the frequency can be found using $f=\frac{1}{T}$.
Finding displacement and velocity
Distance and displacement can be found from the position vs. time graph for simple harmonic motion. Velocity and speed can be found from the slope of a position vs. time graph for simple harmonic motion.

## Common mistakes and misconceptions

Sometimes people confuse period and frequency. These quantities are the inverse of each other. If we can find one, we can also find the other through the relationship:
$f=\frac{1}{T}$
This means that if the frequency is large, the period is small, and vice versa.

For deeper explanations of simple harmonic motion, see our videos:
To check your understanding and work toward mastering these concepts, check out our exercises:

## Want to join the conversation?

• Why we are interested in finding HM?
• in my perspective, the mathematical model used in analyzing simple harmonic motion is fairly common,you can google the equation of simple harmonic motion and you will find that it's actually a solution of differential eqaution of SHM ( which is also described by Sal). this kind of modeling can also be used in predicting the kinematics of basketball bouncing and in my research case- violin bowing technques, where restoring forces are responsible for similar motions. I guess it's just a way of analyzing the diverse kinematics of nature
• can the displacement, velocity, and acceleration be at their greatest magnitudes at the same time?
• Nope. Because the velocity is always greatest when the displacement is 0, and when the displacement is 0, acceleration must also be 0.
• what is the SI unit of frequency
• It's hertz, which is 1/sec, since frequency is cycles/second = 1/s
(1 vote)
• How to find maximum acceleration if mass, displacement, and period is given?
• What happens at equilibrium in SHM?
• At the point of equilibrium, the spring does not exert any force on the block. At this point, the block is traveling at its maximum velocity because all the elastic potential energy stored in the spring is converted into the block's kinetic energy and the acceleration is zero at this point.
• Can you explain the velocity vs. time paragraph for figure 1? I get why it starts at zero, but shouldn't it go from negative to positive, so that the first loop is below the x-axis, instead of positive to negative, like it is?
(1 vote)
• the oscillator starts going in the -x direction, so naturally even the velocity will be negative, since from the start, the displacement is negative
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 19, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.khanacademy.org/science/oscillations-and-waves-essentials/x9db3ed27fc69f96d:how-are-tall-buildings-protected-from-earthquakes/x9db3ed27fc69f96d:mathematics-of-simple-harmonic-motion/a/introduction-to-simple-harmonic-motion-review", "fetch_time": 1719328571000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-26/segments/1718198866143.18/warc/CC-MAIN-20240625135622-20240625165622-00360.warc.gz", "warc_record_offset": 737671707, "warc_record_length": 73145, "token_count": 1158, "char_count": 5252, "score": 4.4375, "int_score": 4, "crawl": "CC-MAIN-2024-26", "snapshot_type": "latest", "language": "en", "language_score": 0.8709908127784729, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00026-of-00064.parquet"}