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Physics Crash Course: Simple Harmonic Motion (S.H.M.) Notes

Physics Crash Course: Simple Harmonic Motion (S.H.M.)

This lecture covers the fundamental concepts of periodic, oscillatory, and simple harmonic motion (S.H.M.). It derives kinematic equations for displacement, velocity, and acceleration, analyzes their graphs, and demonstrates problem-solving using standard NEET past-year questions.

NEET
Physics
English · 6 topics
Checked against the video on 23 Oct 2026

Made by AI from the YouTube video SHM IN ONE SHOT || Simple Harmonic Motion || NEET Physics Crash Course (opens YouTube in a new tab) by Competition Wallah, then checked against it. All credit for the teaching goes to the creator. Tap a topic’s time to confirm a fact in the video.

Physics Crash Course: Simple Harmonic Motion (S.H.M.)

English6 topics · 5 MCQs · 13 flashcardsWatch the video (opens YouTube in a new tab)

Checked against the video

In short

This lecture covers the fundamental concepts of periodic, oscillatory, and simple harmonic motion (S.H.M.). It derives kinematic equations for displacement, velocity, and acceleration, analyzes their graphs, and demonstrates problem-solving using standard NEET past-year questions.

Introduction and Crash Course Announcements

The Umeed batch crash course for NEET is scheduled to conclude around 13th to 14th August, ensuring completion about one month before the exam.
The NEET test series will implement the new exam pattern starting from test NT-18.
A free comprehensive revision series focused on high-yield question practice on the PW app starts on 17th July.
Today's session covers the complete theoretical foundation and problem-solving for Simple Harmonic Motion (S.H.M.).

Periodic Motion vs Oscillatory Motion

Periodic motion is defined as motion that repeats itself after a fixed or regular interval of time.
Examples of periodic motion include uniform circular motion, orbital motion of planets around the Sun, rotation of Earth (time period of 24 hours), and an insect climbing and falling periodically.
An oscillatory (or harmonic) motion is a specific type of periodic motion characterized by to-and-fro, back-and-forth, or up-and-down movement about a fixed point.
The fixed point about which an oscillation occurs is called the mean position or equilibrium position, where the net force acting on the particle is zero (F_net = 0).
Must knowAll oscillatory motions (with no energy loss) are periodic, but not all periodic motions are oscillatory (for example, uniform circular motion is periodic but not oscillatory).

Condition for S.H.M. and Restoring Force

For any oscillatory or harmonic motion, the restoring force is given by F = -k xⁿ, where n must be an odd integer (1, 3, 5, 7, ...).
Must knowSimple Harmonic Motion (S.H.M.) is the simplest form of oscillatory motion, occurring along a straight line when n = 1, giving the restoring force condition F = -k x.
The restoring force is always directed towards the mean position, acting in the direction opposite to displacement from the mean position.
The restoring force constant k is defined in terms of particle mass m and angular frequency ω as k = m ω².
Because the restoring force depends on displacement (x), the acceleration in S.H.M. is not constant but variable.

Kinematic Parameters: Amplitude, Time Period, Velocity, and Acceleration

Amplitude (A) is the maximum displacement of the oscillating particle from the mean position (x = 0 to x = ±A).
The total distance covered by an oscillating particle in one complete time period (T) is 4A, while the net displacement is zero.
The acceleration of a particle in S.H.M. at displacement x is given by a = -ω² x.
Acceleration has a minimum magnitude of 0 at the mean position (x = 0) and reaches a maximum magnitude of |a_max| = ω² A at the extreme positions (x = ±A).
The velocity of a particle at any displacement x is given by v = ±ω √(A² - x²).
Must knowVelocity reaches its maximum magnitude |v_max| = ω A at the mean position (x = 0), and is 0 at the extreme positions (x = ±A).

Parameter	Mean Position (x = 0)	Extreme Position (x = ±A)
Net Force (F)	0 (Minimum)	k A (Maximum)
Acceleration (a)	0 (Minimum)	ω² A (Maximum)
Velocity (v)	ω A (Maximum)	0 (Minimum)

Differential Equation and Standard Equations of S.H.M.

Combining a = d² x / dt² and a = -ω² x gives the standard differential equation of S.H.M.: d² x / dt² + ω² x = 0.
The general solution for displacement along the x-axis is x = A sin(ωt + φ), where (ωt + φ) represents the phase and φ is the initial phase (epoch).
The value of initial phase φ lies in the range 0 ≤ φ < 2π and depends on the initial position and initial direction of velocity at t = 0.
If motion begins from the mean position moving towards the positive extreme, x = A sin(ωt).
If motion begins from the positive extreme position moving towards the mean, x = A cos(ωt).
Must knowThe phase of velocity leads displacement by π/2 (90°), the phase of acceleration leads velocity by π/2 (90°), and the phase difference between acceleration and displacement is π (180°).

Time Taken Between Positions in S.H.M.

For a particle starting from the mean position (x = 0 at t = 0) moving towards +A:
The time taken to travel from x = 0 to x = A/2 is t = T/12.
The time taken to travel from x = A/2 to x = A is t = T/6.
The time taken to cover the second half of the amplitude is greater because the particle slows down as it approaches the extreme position.
Must knowBecause T/6 = 2 × (T/12), the time taken from A/2 to A is twice the time taken from 0 to A/2.

Key terms

Periodic Motion: Motion that repeats itself at regular, fixed intervals of time.
Oscillatory Motion: To-and-fro or back-and-forth periodic motion about a fixed equilibrium position.
Simple Harmonic Motion (S.H.M.): The simplest form of oscillatory motion along a straight line where acceleration is directly proportional to displacement and directed towards the mean position.
Mean Position: The equilibrium point where the net restoring force on the particle is zero.
Extreme Position: The turning point of maximum displacement where particle velocity momentarily becomes zero.
Initial Phase (Epoch): The constant phase angle φ indicating the initial state of motion and position at time t = 0.

Formulas

F = -k x: Restoring force condition for S.H.M.
k = m ω²: Relationship between force constant, mass, and angular frequency
a = -ω² x: Acceleration as a function of displacement in S.H.M.
v = ±ω √(A² - x²): Velocity as a function of displacement in S.H.M.
d² x / dt² + ω² x = 0: Standard linear differential equation of S.H.M.
T = 2π / ω: Time period of oscillation
T = 2π √((x2² - x1²) / (v1² - v2²)): Time period derived from two positions and their corresponding velocities

Exam tips and traps

Always verify whether a question asks for distance or displacement over one time period; in one full period T, total distance is 4A while displacement is zero.
Do not assume the time to travel from 0 to A/2 equals the time from A/2 to A; speed decreases away from the mean, making t(0 → A/2) = T/12 and t(A/2 → A) = T/6.

Quick revision

Read these the night before the exam.

All oscillatory motions are periodic, but not all periodic motions are oscillatory.
The mean position is strictly defined by F_net = 0.
S.H.M. takes place strictly along a straight line (one dimension).
In S.H.M., acceleration is directly proportional to displacement and opposite in sign: a = -ω² x.
Maximum velocity occurs at the mean position and equals ω A.
Maximum acceleration occurs at the extreme positions and equals ω² A.
The phase difference between velocity and acceleration in S.H.M. is π/2 radians (90°).
The phase difference between displacement and acceleration in S.H.M. is π radians (180°).
Average velocity over one complete cycle of S.H.M. is identically zero.
The time taken to travel from the mean position to half the amplitude (0 to A/2) is T/12.

These notes are made by AI from the video. Use the timestamps to check any important fact against the video before you rely on it.

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