Simple Harmonic Motion Explained with Formula Sheet and Practice Questions

Overview

Simple harmonic motion, often shortened to SHM, is a special kind of oscillation. An object moves back and forth around an equilibrium position, and the restoring force always points toward that equilibrium. In the ideal case, the restoring force is proportional to displacement:

F = -kx

That minus sign matters. It tells you the force acts in the opposite direction to the displacement. If the mass is displaced to the right, the force points left. If displaced to the left, the force points right.

From Newton's second law, this gives the standard SHM acceleration relation:

a = -ω²x

where ω is the angular frequency. This is one of the fastest ways to identify SHM in a problem. If acceleration is proportional to displacement and opposite in direction, you are very likely dealing with simple harmonic motion.

Two of the most common SHM systems in introductory physics are:

• Mass-spring systems, especially horizontal springs and ideal vertical springs near equilibrium
• Small-angle pendulums, where the swing angle is small enough for the motion to be approximated as SHM

Students often find SHM difficult because several descriptions of the same motion are used at once:

• Force description
• Energy description
• Position, velocity, and acceleration graphs
• Trigonometric equation description

The good news is that these are not separate topics. They are different windows into the same motion. Once you connect them, many oscillation practice questions become much easier.

A useful conceptual picture is this:

• At the equilibrium position, displacement is zero, restoring force is zero, acceleration is zero, speed is maximum, and kinetic energy is maximum.
• At the turning points, displacement is maximum, restoring force is maximum, acceleration is maximum in magnitude, speed is zero, and elastic or gravitational potential energy is maximum.

This alone helps with many physics questions and answers on SHM, especially multiple-choice items that ask where energy, speed, or acceleration is greatest.

Before moving further, keep these core definitions clear:

• Amplitude A: maximum displacement from equilibrium
• Period T: time for one complete oscillation
• Frequency f: number of oscillations per second, with f = 1/T
• Angular frequency ω: related by ω = 2πf = 2π/T

If you mix up period and frequency, many later errors follow. It is worth slowing down here and making sure each symbol has a clear meaning and unit.

For broader exam prep, it also helps to connect SHM to graph interpretation. If that is a weak area, review Graphing in Physics: How to Read Position-Time, Velocity-Time, and Acceleration-Time Graphs.

Topic map

This section is your SHM formula sheet with context. Instead of memorizing isolated equations, organize them by what they describe.

1. Recognition test for SHM

Look for one of these forms:

• F = -kx
• a = -(k/m)x
• a = -ω²x

If a problem can be rewritten in one of these forms, the motion is simple harmonic.

2. Core relationships

• f = 1/T
• ω = 2πf = 2π/T
• v_max = Aω
• a_max = Aω²

These are high-value formulas because they link measurable features of motion directly.

3. Mass-spring formulas

• F = -kx
• ω = √(k/m)
• T = 2π√(m/k)
• f = (1/2π)√(k/m)

Notice the pattern: a stiffer spring means faster oscillation, while a larger mass means slower oscillation.

4. Small-angle pendulum formulas

• T = 2π√(L/g)
• f = (1/2π)√(g/L)

For the simple pendulum, the period depends on length and gravitational field strength, not on bob mass. In ideal introductory problems, it is also independent of amplitude only for small oscillations.

5. Position, velocity, and acceleration equations

A common position equation is:

x = A cos(ωt + φ)

or sometimes

x = A sin(ωt + φ)

Both are valid. The choice depends on the starting condition. From this:

• v = -Aω sin(ωt + φ) for the cosine form
• a = -Aω² cos(ωt + φ) = -ω²x

If calculus is not emphasized in your course, you may still be expected to know the shape and phase relationship:

• Velocity is a quarter-cycle out of phase with displacement
• Acceleration is half a cycle out of phase with displacement

6. Energy in SHM

For a spring system:

• Total energy: E = (1/2)kA²
• Elastic potential energy: U = (1/2)kx²
• Kinetic energy: K = E - U = (1/2)k(A² - x²)

This energy view is especially useful when a problem gives displacement and asks for speed, or gives amplitude and spring constant and asks for total mechanical energy.

7. Quick comparison table

At x = 0: force = 0, acceleration = 0, speed maximum, kinetic energy maximum.

At x = ±A: force maximum, acceleration maximum, speed = 0, potential energy maximum.

8. Worked example: spring motion physics

Question: A 0.50 kg mass is attached to a spring with spring constant 200 N/m. It oscillates with amplitude 0.10 m. Find the angular frequency, period, maximum speed, and total energy.

Step 1: Angular frequency
ω = √(k/m) = √(200/0.50) = √400 = 20 rad/s

Step 2: Period
T = 2π/ω = 2π/20 = 0.314 s

Step 3: Maximum speed
v_max = Aω = 0.10 × 20 = 2.0 m/s

Step 4: Total energy
E = (1/2)kA² = 0.5 × 200 × (0.10)² = 1.0 J

Answer: ω = 20 rad/s, T = 0.314 s, v_max = 2.0 m/s, E = 1.0 J.

That is a classic step by step physics solution. Notice how one set of known values, m, k, and A, unlocks most of the important quantities.

Related subtopics

SHM is easiest to learn when you see how it connects to neighboring topics rather than treating it as a standalone chapter.

Force and equilibrium

Many SHM setups begin with force analysis. In a horizontal spring problem, the restoring force comes directly from Hooke's law. In a vertical spring problem, the system oscillates around a shifted equilibrium where spring force balances weight. In pendulum motion, the restoring effect comes from a component of gravity.

If force breakdown is still shaky, review Free-Body Diagram Guide: Rules, Examples, and Practice Questions.

Graphs of oscillation

Displacement, velocity, and acceleration graphs in SHM are sinusoidal and phase-shifted. A student may know the formula but still lose marks by misreading a graph. Practice identifying:

• When displacement is zero
• When speed is greatest
• When acceleration changes sign
• How the slope of the displacement-time graph relates to velocity

Graph literacy often matters as much as formula recall. The linked graphing guide above is a strong companion resource.

Energy methods

Some oscillation questions are easiest with energy rather than kinematics. For example, if a spring-mass system is at displacement x, use total energy and spring potential energy to find kinetic energy and then speed. This avoids writing a full time-dependent equation.

Energy-based SHM questions also train a useful exam habit: choose the shortest correct method rather than the most familiar one.

Pendulum approximations

The simple pendulum formula T = 2π√(L/g) works for small angles. At larger amplitudes, real motion is periodic but no longer perfectly simple harmonic. In most school and introductory college questions, the approximation is stated or implied. Read the wording carefully.

Damping and driving forces

Real oscillations often lose energy to friction or air resistance. That is damped motion. Some systems are also pushed by an external periodic force, creating driven oscillations and sometimes resonance. These ideas are often taught after ideal SHM because they build on the same language of frequency, period, and energy exchange.

This is one reason SHM is worth revisiting. It is not just a single chapter. It becomes a foundation for waves, alternating current, mechanical resonance, and later mathematical modeling.

Measurement and data handling

Lab versions of SHM add uncertainty and practical variation. If you measure spring period or pendulum length, your result is only as good as your timing method, significant figures, and graph fit. For lab work, pair this topic with Measurement Uncertainty and Significant Figures in Physics Labs and Physics Constants List: SI Values, Units, and Where They Are Used.

Practice questions

Use these short questions to check understanding.

1. Concept check: In SHM, where is the speed greatest?
Answer: At equilibrium, where x = 0.

2. Concept check: Where is acceleration greatest in magnitude?
Answer: At the turning points, where |x| = A.

3. Formula check: A spring-mass system has m = 2.0 kg and k = 50 N/m. Find the period.
Solution: T = 2π√(m/k) = 2π√(2.0/50) = 2π√(0.04) = 2π(0.20) = 1.26 s.

4. Pendulum check: A simple pendulum has length 1.0 m. Take g = 9.8 m/s². Find the period.
Solution: T = 2π√(L/g) = 2π√(1.0/9.8) ≈ 2.01 s.

5. Energy check: A spring with k = 80 N/m oscillates with amplitude 0.15 m. Find total energy.
Solution: E = (1/2)kA² = 0.5 × 80 × (0.15)² = 0.90 J.

If you struggle with setup more than algebra, work through How to Solve Physics Word Problems Step by Step. That approach transfers well to oscillation questions.

How to use this hub

This article works best as a return-to reference, not just a one-time read. Here is a practical way to use it for physics homework help and physics exam prep.

Start with recognition

Before reaching for formulas, ask: does the restoring force point toward equilibrium and scale with displacement? If yes, the problem likely belongs to SHM. This simple question prevents using the wrong chapter's method.

Choose the right route

Most SHM problems fall into one of four routes:

• Route 1: Period or frequency using spring or pendulum formulas
• Route 2: Force and acceleration using F = -kx or a = -ω²x
• Route 3: Energy using total energy and potential energy
• Route 4: Time equations and graphs using x, v, and a relationships

Identify the route first, then write the smallest set of equations needed.

Build a one-page SHM formula sheet

For revision, put these on one page:

• F = -kx
• a = -ω²x
• ω = 2πf = 2π/T
• T = 2π√(m/k)
• T = 2π√(L/g)
• v_max = Aω
• a_max = Aω²
• E = (1/2)kA²

Next to each formula, add one note about when to use it. This matters more than having a longer sheet. If you need help organizing formula revision across courses, see GCSE Physics Equation Sheet Explained by Topic or AP Physics 1 Formula Sheet Explained: What Each Equation Means and When to Use It.

Practice with mixed questions

Do not study only one type at a time. Mix:

• Concept questions about turning points and equilibrium
• Numeric spring calculations
• Pendulum equations
• Graph interpretation
• Energy-based speed questions

Mixed practice is closer to real exams, where the method is not announced in advance.

Use internal links as a study path

If SHM appears during wider exam revision, combine this hub with:

• College Physics Midterm Study Guide: What to Review First
• A-Level Physics Revision Checklist by Topic and Exam Season

That helps place oscillations in the larger mechanics and waves sequence.

Common mistakes to avoid

• Using the pendulum formula for large-angle motion without checking assumptions
• Confusing amplitude with total distance traveled
• Forgetting that acceleration is zero at equilibrium
• Using v = Aω as speed everywhere instead of maximum speed only
• Mixing up frequency, angular frequency, and period
• Ignoring units, especially radians per second for ω

A calm, repeatable workflow usually beats memorizing more formulas.

When to revisit

Come back to this hub whenever oscillations show up in a new form. SHM is a topic that becomes more useful over time because later chapters keep reusing its core ideas.

Revisit this article when:

• You start a unit on waves and need the meaning of period, frequency, and phase again
• You move from force-based questions to energy-based questions
• You begin pendulums after learning springs
• You are reviewing graphs before a test
• You need a compact SHM formula sheet the night before an exam
• You meet damping, resonance, or driven oscillations and want the ideal model first

Best next actions:

1. Copy the core formulas into your own notes.
2. Solve one spring problem and one pendulum problem from memory.
3. Sketch displacement, velocity, and acceleration against time on the same axes.
4. Explain out loud why acceleration and displacement always have opposite signs in SHM.
5. Return here after your next homework set and add the formulas you actually used most.

If you treat SHM as a topic map rather than a list of equations, it becomes much easier to retain. That is the main purpose of this hub: not just to explain simple harmonic motion once, but to give you a clear reference for the next time spring motion physics, pendulum equations, or oscillation practice questions appear in your course.