Anything that swings back and forth — a pendulum, a guitar string, a child on a swing — is oscillating. When the restoring force is proportional to displacement, the motion becomes the cleanest, most exam-friendly form: Simple Harmonic Motion (SHM). This chapter shows you the few formulas NDA repeats every year and how to apply them fast.
Why Oscillations Matter in NDA
Oscillations is a high-yield, low-effort topic in NDA General Studies Physics. Almost every paper carries one or two questions, and they are usually direct — plug values into a standard formula or recall a one-line fact about resonance or the second's pendulum.
You do not need heavy calculus here. You need clear definitions and four or five formulas at your fingertips. That is exactly what this page drills.
The good news for a student short on time is that the entire chapter rests on a single idea: a restoring force that always tries to drag the body back to its resting point. Once you truly understand that one sentence, the pendulum, the spring, the swing and the vibrating string all become the same problem wearing different clothes. Everything else in this page is simply that idea translated into formulas you can apply in seconds during the exam.
Examiners also like this topic because it connects neatly to everyday experience. You have felt a swing slow down, heard a guitar string fade, and noticed a clock keep steady time. Those familiar pictures are exactly the mental hooks that help you remember which quantity controls the motion, so keep them in mind as you read.
NDA rarely asks proofs. It asks which quantity changes when you double the length, change the mass, or move to the Moon. Train for that style.
Periodic and Oscillatory Motion
A motion that repeats itself after a fixed interval of time is called periodic motion — the hands of a clock, the Earth around the Sun.
If the body moves to and fro about a fixed mean (equilibrium) position, it is oscillatory or vibratory motion. Every oscillation is periodic, but not every periodic motion is oscillatory (planetary orbit is periodic, not to-and-fro). When the to-and-fro motion is fast and the displacements are small, as in a sounding tuning fork, we often call it vibration; when it is slow and visible, as in a swinging pendulum, we call it oscillation. Physically they are the same kind of motion.
The mean position is the point where the body would happily sit at rest if it were not disturbed. The moment you push it away, a force appears that tries to bring it back. The body overshoots the mean because of its inertia, swings to the other side, is pulled back again, and so the to-and-fro motion continues. Grasping this push-and-overshoot cycle makes the rest of the chapter feel natural rather than memorised.
Key terms
- Displacement (y): distance from the mean position at any instant.
- Amplitude (A): the maximum displacement on either side.
- Time period (T): time for one complete oscillation, measured in seconds.
- Frequency (f): number of oscillations per second; unit hertz (Hz).
Frequency and period are reciprocals: f = 1 ÷ T. Angular frequency ω = 2πf = 2π ÷ T.
What Is Simple Harmonic Motion?
Simple Harmonic Motion is the special oscillation in which the restoring force (or acceleration) is directly proportional to the displacement from the mean position and always points back towards that mean position.
Defining condition: F = −k y and a = −ω2 y.
The minus sign means the force opposes the displacement — that is what pulls the body back and keeps it oscillating.
Here k is the force constant and ω is the angular frequency. Because acceleration is largest at the extreme positions and zero at the mean, SHM has a very clean, predictable rhythm — which is why exams love it.
Why a minus sign? When the body is on the right of the mean, displacement is positive, and the force pushes it left, so the force value comes out negative. When the body is on the left, displacement is negative and the force pushes right, coming out positive. In both cases the force opposes the displacement, and the only neat way to capture that in one equation is to put a minus sign in front. This single negative sign is the heartbeat of every oscillation you will study.
Notice also that the acceleration is never constant in SHM. It changes from instant to instant because it depends on the displacement, which itself keeps changing. This is a big contrast with free fall or uniform acceleration, where the acceleration stays fixed. Many tricky NDA options try to fool you into treating SHM acceleration as constant, so always remember it varies from zero at the centre to a maximum at the ends.
Displacement, Velocity and Acceleration
The displacement of a particle in SHM can be written as a sine function of time:
y = A sin(ωt + φ)
Velocity: v = Aω cos(ωt + φ), so v = ω√(A2 − y2)
Acceleration: a = −ω2 y
Read these three relations carefully, because most numerical questions come straight from them:
- At the mean position (y = 0): velocity is maximum (v = Aω) and acceleration is zero.
- At the extreme positions (y = A): velocity is zero and acceleration is maximum (a = ω2A).
If a question says “speed is maximum,” the body is at the centre. If it says “momentarily at rest,” it is at an extreme. Spot the position first, then pick the formula.
The Simple Pendulum
A small heavy bob hung by a light, inextensible string and set swinging through a small angle performs SHM. Its time period depends only on the length and gravity — not on the mass of the bob.
T = 2π √(L ÷ g)
L = length of the pendulum, g = acceleration due to gravity.
What this tells you
- T is independent of the bob's mass and of the amplitude (for small swings).
- T is proportional to √L — to double the period you must make the string four times longer.
- On the Moon, g is smaller, so T becomes larger — the same pendulum swings slower.
It surprises many students that the bob's mass disappears from the formula. The reason is the same one that makes a feather and a stone fall together in a vacuum: a heavier bob is pulled harder by gravity, but it is also harder to accelerate because of its greater inertia. The two effects cancel exactly, leaving the period dependent only on length and gravity. This is a favourite NDA trap, so commit it firmly to memory.
The dependence on gravity explains several real-world facts. A pendulum clock that keeps perfect time in Delhi will run a little fast if carried up a high mountain, where g is slightly smaller, because the period lengthens there. Take the same clock deep into a mine and the period changes again. Whenever a question changes the location of a pendulum, your first instinct should be to ask what has happened to the value of g.
A second's pendulum has a time period of exactly 2 seconds (one second to swing each way). Its length on Earth is about 0.99 m ≈ 1 metre.
Spring-Mass Oscillator
A mass m attached to a spring of stiffness (force constant) k oscillates when pulled and released. Here, unlike the pendulum, the mass does matter and g does not.
T = 2π √(m ÷ k)
Stiffer spring (large k) → faster oscillation (smaller T). Heavier mass → slower oscillation (larger T).
Combining springs
- Springs in series: the combination is softer; effective constant 1÷k = 1÷k1 + 1÷k2.
- Springs in parallel: the combination is stiffer; effective constant k = k1 + k2.
Students often write the pendulum formula with mass or the spring formula with g. Keep them separate: pendulum → L and g; spring → m and k.
Energy in SHM
During oscillation, energy keeps converting between kinetic and potential, but the total mechanical energy stays constant (ignoring friction).
Total energy E = ½ m ω2 A2 (constant).
Kinetic energy KE = ½ m ω2 (A2 − y2)
Potential energy PE = ½ m ω2 y2
- At the mean position: all energy is kinetic, PE = 0.
- At the extreme position: all energy is potential, KE = 0.
- Total energy is proportional to A2 — double the amplitude and the energy becomes four times larger.
Picture the energy as constantly sloshing between two buckets. As the body races through the centre, almost all of the energy is in the kinetic bucket and the body is moving fastest. As it climbs towards an extreme, the restoring force does negative work, the kinetic bucket empties into the potential bucket, and the body slows to a momentary stop. Then the flow reverses. At every single instant the two buckets together hold the same total, which is why we say energy is conserved in ideal SHM.
This is also why a real swing eventually stops without a push. Air resistance and friction quietly leak energy out of the system, so the total slowly shrinks and the amplitude with it. In the idealised SHM of your textbook we ignore that leak, but it is worth remembering that the constant-energy picture is the friction-free version of reality.
If asked “where are KE and PE equal?”, set the two equal: this happens at y = A ÷ √2, about 71% of the way out.
Free, Damped, Forced Oscillations and Resonance
NDA loves the conceptual difference between these four ideas, so memorise them as a set.
- Free oscillation: a body left to vibrate on its own swings at its natural frequency with constant amplitude (idealised, no friction).
- Damped oscillation: real oscillations lose energy to friction and air resistance, so the amplitude gradually decreases and finally stops.
- Forced oscillation: an external periodic force drives the body, which then vibrates at the driving frequency, not its own.
- Resonance: when the driving frequency equals the body's natural frequency, the amplitude becomes very large.
Resonance examples: a soldier's marching step matching a bridge's natural frequency (so troops break step on bridges), a radio tuned to a station, and a wine glass shattering at the right sound pitch.
Worked Example
A simple pendulum of length 1 m is taken to a place where g = 9.8 m/s2. Find its time period. (Take π = 3.14.)
So a 1 m pendulum on Earth swings with a period of about 2 seconds — that is why a one-metre pendulum is the classic second's pendulum.
Carry √(1÷9.8) ≈ 0.32 in memory. Many pendulum MCQs reduce to multiplying 6.28 by a small square root, so practising that arithmetic saves time.
Common Mistakes to Avoid
- Assuming a heavier pendulum bob swings slower — mass does not affect the pendulum's period.
- Forgetting that the time period formula holds only for small angular displacements (roughly under 15°).
- Mixing up maximum velocity (at mean) with maximum acceleration (at extreme).
- Writing total energy as proportional to A instead of A2.
On the Moon a clock pendulum runs slow, not fast. Lower g means larger T, so each “tick” takes longer and the clock loses time.
Previous-Year Style Question
Q. The length of a simple pendulum is increased to four times its original value. By what factor does its time period change?
Answer: Since T ∝ √L, making L four times larger multiplies the period by √4 = 2. The new pendulum swings with twice the original time period.
These “by what factor” questions are pure proportionality. Identify the formula, see which quantity is under the square root, and apply the root to the change.
Quick Revision
- SHM condition: F = −ky, a = −ω2y; ω = 2π÷T.
- Pendulum: T = 2π√(L÷g) — mass-independent; second's pendulum ≈ 1 m, T = 2 s.
- Spring: T = 2π√(m÷k) — depends on mass and stiffness, not g.
- Velocity max at mean, acceleration max at extreme.
- Total energy E = ½mω2A2, proportional to A2.
- Resonance: driving frequency = natural frequency → huge amplitude.
Frequently asked questions
Is every oscillation a simple harmonic motion?
No. Oscillation only means to-and-fro motion about a mean position. It becomes SHM only when the restoring force is directly proportional to the displacement and directed back towards the mean position.
Does the mass of a pendulum bob affect its time period?
No. The time period of a simple pendulum, T = 2π√(L÷g), depends only on length and gravity. A heavier bob of the same length has exactly the same period.
What is a second's pendulum?
A second's pendulum has a time period of exactly 2 seconds, taking one second for each swing. On Earth its length is about 0.99 metre, which is close to one metre.
Why do soldiers break step while crossing a bridge?
If the rhythm of their marching matches the bridge's natural frequency, resonance can build dangerously large vibrations. Breaking step avoids a single driving frequency and keeps the amplitude small.
Where is the velocity maximum in SHM?
Velocity is maximum at the mean (equilibrium) position, where v = Aω, and zero at the extreme positions. Acceleration behaves oppositely, being maximum at the extremes and zero at the mean.
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