Simple harmonic motion — definition, restoring force F = −kx

HOW do we DERIVE the motion from scratch?

We start from only $F=-kx$ and reach $x(t)$ with no memorized answer.

Step 1 — Write Newton's law. $m\frac{d^2x}{dt^2} = -kx.$ Why this step? Force = mass × acceleration, and acceleration is the second time-derivative of position.

Step 2 — Define $\omega^2 = k/m$ to clean it up. $\frac{d^2x}{dt^2} = -\omega^2 x.$ Why this step? This is now a pure math question: "what function equals minus a constant times itself after two derivatives?"

Step 3 — Guess a function whose 2nd derivative flips its sign. Try $x = A\cos(\omega t + \phi)$ . $\frac{dx}{dt} = -A\omega\sin(\omega t+\phi),\qquad \frac{d^2x}{dt^2} = -A\omega^2\cos(\omega t+\phi) = -\omega^2 x.\ \checkmark$ Why this step? Sine and cosine are the only smooth functions that return to themselves (up to sign) after differentiating twice — so they must describe SHM.

Step 4 — Read off the physics.

$A$ = amplitude (max displacement), set by initial conditions.
$\phi$ = phase constant, set by where it started.
Velocity: $v = -A\omega\sin(\omega t+\phi)$ , max speed $v_{\max}=A\omega$ .
Acceleration: $a = -A\omega^2\cos(\omega t+\phi)$ , max $a_{\max}=A\omega^2$ .

Figure — Simple harmonic motion — definition, restoring force F = −kx

Worked Examples

Common Mistakes (Steel-manned)

Quick Recall

Recall Self-test (hide and answer)

What two conditions define SHM? → restoring force toward equilibrium and $\propto$ displacement.
Where is speed max / acceleration max? → speed max at $x=0$ ; acceleration max at $x=\pm A$ .
Does $T$ depend on $A$ ? → No.
Derive $\omega$ from $F=-kx$ and $F=ma$ . → $\omega=\sqrt{k/m}$ .

Recall Feynman: explain to a 12-year-old

Imagine a ball sitting in the bottom of a smooth bowl. If you nudge it, it rolls back. The further you push it up the side, the harder the bowl pushes it back home — twice as far, twice the push. Because the push grows so neatly, the ball rocks back and forth in a perfectly steady rhythm, like a clock. A tiny nudge and a big nudge both take the same time to go back and forth — that steady rhythm is "simple harmonic motion." A spring with a weight does exactly the same dance.

Connections

Energy in SHM — where $\tfrac12 kA^2$ comes from
Simple Pendulum — small-angle approximation gives $F=-(mg/L)x$
Hooke's Law — the origin of $F=-kx$
Damped Oscillations — add a velocity term $-bv$
Uniform Circular Motion — SHM is its shadow (projection)
Taylor Series — why every valley bottom is a parabola

What is the defining condition for SHM?

Restoring force (or acceleration) is directed toward equilibrium and proportional to displacement:

F=-kx

a=-\omega^2x

What does the minus sign in

F=-kx

mean?

Force is opposite in direction to the displacement (restoring), not a fixed negative value.

Formula for angular frequency of a mass–spring SHM?

\omega=\sqrt{k/m}

, from matching

a=-\frac{k}{m}x

with

a=-\omega^2x

Does the period of SHM depend on amplitude?

No.

T=2\pi\sqrt{m/k}

contains no amplitude (isochronism).

Where in the cycle is speed maximum and where is it zero?

Max speed at

x=0

(

v_{max}=A\omega

); zero speed at

x=\pm A

Where is acceleration maximum?

At the extremes

x=\pm A

(

a_{max}=A\omega^2

), where force is greatest.

Expression for speed at displacement

x

v=\omega\sqrt{A^2-x^2}

(from energy conservation).

Total mechanical energy of an SHM oscillator?

E=\tfrac12 kA^2=\tfrac12 mv^2+\tfrac12 kx^2

(constant).

Why is SHM so common in nature?

Near any stable equilibrium

U\approx\tfrac12kx^2

(parabola), so

F=-dU/dx=-kx

Is a function

x=A\cos(\omega t+\phi)

a valid SHM solution? Show why.

Yes; its second derivative is

-\omega^2x

, satisfying

a=-\omega^2x

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Simple Harmonic Motion ka core idea bilkul simple hai: jab koi cheez equilibrium (home position) se hilti hai, to ek "wapas khinchne wala" force lagta hai jise restoring force kehte hain. Aur khaas baat — yeh force displacement ke exactly proportional hota hai: $F=-kx$ . Minus sign ka matlab number negative nahi hai, balki yeh batata hai ki force hamesha opposite direction mein, yaani wapas centre ki taraf, lagta hai. Jitna door jaaoge, utna zyada zor se wapas khinchega.

Ab Newton ka law lagao: $ma=-kx$ , isse $a=-\frac{k}{m}x$ milta hai. Isse compare karke $\omega=\sqrt{k/m}$ aata hai, aur $T=2\pi\sqrt{m/k}$ . Yahan dekhne wali sabse important cheez — time period amplitude pe depend nahi karta. Chahe thoda kheencho ya zyada, ek pura swing utna hi time lega. Iska reason yeh hai ki zyada kheechne pe force bhi zyada, speed bhi zyada — dono cancel ho jaate hain. Isko isochronism kehte hain, aur yahi clocks aur pendulums ke peeche ka raaz hai.

Motion ki shape sine/cosine hoti hai: $x=A\cos(\omega t+\phi)$ , kyunki sirf sin aur cos hi aise functions hain jinka double derivative khud ka ulta sign deta hai. Energy ki baat karo to total energy constant rehti hai: $E=\tfrac12 kA^2$ . Centre ( $x=0$ ) pe speed maximum, force zero; ends ( $x=\pm A$ ) pe speed zero par force aur acceleration maximum. Yeh rule yaad rakho — "Far = Force, Fast = Free" — aur har SHM question aasaani se nikal jaayega.

Important: har back-and-forth motion SHM nahi hoti. Test sirf ek hai — kya force linear aur opposing hai ( $F\propto -x$ )? Agar haan, to SHM. Bade angle wala pendulum ya bouncing ball oscillate to karte hain par SHM nahi, kyunki unka force linear nahi hota.

Simple harmonic motion — definition, restoring force F = −kx

WHY does SHM exist at all?

WHAT is SHM? (definition)

HOW do we DERIVE the motion from scratch?

Worked Examples

Common Mistakes (Steel-manned)

Quick Recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Oscillations & Waves

Connections