Simple harmonic motion is defined by an acceleration that is always

Proportional to displacement and directed opposite to it. The correct answer is proportional to displacement and directed opposite to it. This restoring relationship, a = -kx, is what defines SHM. Read more: GenSci0054 — Simple Harmonic Motion and Oscillations.

Simple Harmonic Motion and Oscillations

A swinging pendulum. A vibrating guitar string. A mass bouncing on a spring. All three repeat the same back-and-forth motion.

Physics calls this simple harmonic motion, or SHM. It is one of the most tested motion patterns in competitive exams.

A simple gravity pendulum: a mass on a string, swinging under gravity. Chetvorno, Public Domain, via Wikimedia Commons.

MCQ Questions

GenSci0054

a = −ω0x

The defining equation of SHM: acceleration is proportional to displacement, and always points the opposite way.

Mean Position

Speed max

Acceleration is zero here, at the centre of the swing

Extreme Position

Speed zero

Acceleration is maximum here, at the edge of the swing

Restoring Force

Always inward

Constantly pulls the particle back toward the mean position

Time Period

T = 2π/ω

Time for one complete oscillation, fixed for a given system

The exam angle: exams love asking where speed and acceleration are at their maximum or minimum. Speed peaks where acceleration is lowest, and vice versa — they’re always opposite.

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📑 Contents

1What Makes Motion "Simple Harmonic"
2Time Period and Real Examples
3Test Yourself
4Energy in SHM
5Exam Point of View

🏛️ Must Know

What Makes Motion "Simple Harmonic"

Definition Simple harmonic motion is a back-and-forth motion where acceleration is always proportional to displacement, and points back toward the mean position.
The Restoring Force A restoring force constantly pulls the oscillating object back toward its centre, or mean position. This force is what keeps the motion going.
Mean Position At the mean position, displacement is zero. Speed is at its maximum here, and acceleration is at its minimum, zero.
Extreme Position At the extreme ends of the swing, speed drops to zero. Acceleration is at its maximum here, pulling the object back inward.
The Governing Equation Acceleration equals a negative constant times displacement, written as a = −ω²x. The negative sign shows it always opposes displacement.

📘 Good to Know

Time Period and Real Examples

Time Period Time period T is the time for one complete oscillation. It equals 2π divided by angular frequency ω.
Amplitude Amplitude is the maximum displacement from the mean position, marking how far the object swings out on either side.
Simple Pendulum A simple pendulum, a mass on a string, executes SHM for small swing angles. Its time period depends only on its length and gravity, not its mass.
Spring-Mass System A mass on a spring also executes SHM. Its time period depends on the mass and the spring's stiffness.
Everyday Examples A swing, a tuning fork's prongs, and a vibrating guitar string all approximate simple harmonic motion.

Test Yourself

🌟 Great to Know

Energy in SHM

Kinetic Energy Kinetic energy is maximum at the mean position, where speed peaks, and zero at the extreme positions.
Potential Energy Potential energy is the reverse: zero at the mean position, and maximum at the extreme positions.
Total Energy The total energy, kinetic plus potential, stays constant throughout the motion, assuming no friction or air resistance.
Speed-Acceleration Trade-Off Speed and acceleration are always at opposite extremes of each other. When one peaks, the other bottoms out at zero.
Recognising the Equation Any relation of the form a = −kx, with a negative constant and a linear term, describes SHM. A squared term, like a = kx², does not.

📝 Exam Point of View

NDA & NA (II) 2016 GAT — Three SHM Questions

Question NDA & NA (II) 2016, General Ability Test, Q105: which statement about the acceleration of a particle in SHM is true? The correct answer is that acceleration is minimum when speed is maximum.
Why Acceleration is proportional to displacement, which is zero at the mean position, exactly where speed peaks. So acceleration is at its lowest exactly when speed is highest.
Link See the full question, NDA & NA (II) 2016 GAT, Q105.
Question NDA & NA (II) 2016 GAT Q106: which of four particles, given their displacement-acceleration relation, is executing SHM? The correct answer is a_x = −3x.
Why SHM needs acceleration proportional to displacement with a negative sign, matching a = −kx exactly. Only this option has both the correct linear form and the correct opposite-direction sign.
Link See the full question, NDA & NA (II) 2016 GAT, Q106.
Question NDA & NA (II) 2016 GAT Q110: a particle in SHM with 2 cm amplitude has equal speed and acceleration magnitudes at 1 cm from the mean position. The correct answer for the time period is 2π/√3 seconds.
Why Setting the SHM speed and acceleration formulas equal at this point and solving gives angular frequency √3 radians per second, and time period follows directly from T = 2π/ω.
Link See the full question, NDA & NA (II) 2016 GAT, Q110.