Oscillations: common doubts, answered
The questions students ask most often about Oscillations, each with a short answer. For the full chapter, read the Oscillations notes.
About the chapter
How do you check whether a given function represents SHM?
Differentiate it twice. If the result is a = −ω²x, with a positive constant ω², the motion is simple harmonic about the point where x = 0. Functions like sin ωt − cos ωt pass, since they combine into one sine curve, but sin ωt + sin 2ωt or e^(−ωt) do not.
Periodic and oscillatory motion
Read this section in the notes →What is the difference between periodic and oscillatory motion?
Periodic motion is any motion that repeats itself at regular intervals of time. Oscillatory motion is periodic motion that goes to and fro about a fixed mean position. Every oscillation is periodic, but not every periodic motion is an oscillation: the Earth's orbit around the Sun repeats each year but does not swing back and forth about a mean position.
Is every oscillation simple harmonic motion?
No. SHM is a special oscillation in which the restoring force is proportional to displacement and directed towards the mean position, giving a sine or cosine variation with time. A ball bouncing on the floor repeats its motion but is not SHM, since gravity is constant rather than proportional to displacement. A function like sin² ωt is periodic but not SHM either.
Period, frequency and displacement
Read this section in the notes →What is the difference between frequency and angular frequency?
Frequency ν is the number of complete oscillations per second, measured in hertz. Angular frequency ω is 2π times that, ω = 2πν, measured in radians per second. They differ by a factor of 2π, so writing T = 1/ω instead of T = 1/ν or T = 2π/ω is a common slip.
Simple harmonic motion: amplitude, phase and angular frequency
Read this section in the notes →What is the phase of an oscillation?
In x = A cos(ωt + φ), the whole angle ωt + φ is the phase, and it tells you where in its cycle the particle is at time t, both its position and its direction of motion. The constant φ is the phase at t = 0 and depends on how the motion started. Since ω is in rad/s, ωt comes out in radians, so φ must also be expressed in radians before the two are added.
Does a larger amplitude make the period of SHM longer?
No. In SHM the period depends only on the system's properties, T = 2π√(m/k) for a spring, not on the amplitude. A larger swing means a longer path but also a larger restoring force and higher speeds, and the two effects cancel exactly. This independence from amplitude is a defining feature of simple harmonic motion.
SHM as the shadow of uniform circular motion
Read this section in the notes →Velocity and acceleration in SHM
Read this section in the notes →Where is the speed maximum and where is acceleration maximum in SHM?
Speed is greatest, ωA, at the mean position and zero at the extremes. Acceleration is the opposite: zero at the mean position and greatest, ω²A, at the extremes, because a = −ω²x is proportional to displacement. So the particle moves fastest exactly where it is not accelerating at all.
Is acceleration zero where velocity is zero in SHM?
No. At the extreme positions the velocity is zero but the acceleration is at its largest, pulling the particle back towards the mean position. If acceleration were zero there too, the particle would stay at the extreme forever. Zero velocity simply means the particle is turning around.
How do you find velocity at a given displacement in SHM?
Use v = ω√(A² − x²). This comes from combining the displacement and velocity equations to remove time, or from energy conservation. It gives the maximum speed ωA at x = 0 and zero speed at x = ±A. The sign of v depends on whether the particle is moving towards or away from the mean position.
Force law for SHM: F = −kx
Read this section in the notes →Why is there a minus sign in F = −kx?
The minus sign shows that the restoring force always points opposite to the displacement, back towards the mean position. Pull the mass to the right and the force pulls left; push it left and the force pushes right. Without this sign the force would push the body farther away, and there would be no oscillation.
What is the spring constant when a mass is between two springs?
If a block sits between two identical springs fixed to walls on either side, both springs push it back towards the centre when it is displaced, so their forces add. The effective spring constant is 2k, giving T = 2π√(m/2k). Using k alone would make the period √2 times too long.
Does the mass of a spring-block system affect its period?
Yes. The period is T = 2π√(m/k), so a heavier block oscillates more slowly and a stiffer spring makes it oscillate faster. This contrasts with a simple pendulum, whose period does not depend on the bob's mass. The difference arises because the spring force does not depend on mass, while the pendulum's restoring force is proportional to it.
Energy in SHM
Read this section in the notes →Is total energy conserved in SHM?
Yes, in ideal SHM without friction. The energy changes back and forth between kinetic energy K = ½k(A² − x²) and potential energy U = ½kx², but their sum stays fixed at E = ½kA². All the energy is kinetic at the mean position and all potential at the extremes.
What is the period of kinetic energy in SHM?
Kinetic energy repeats with half the period of the displacement, T/2. In one complete cycle the particle passes through the mean position twice, and each time its kinetic energy peaks. Since energy depends on the square of velocity, the direction of motion does not matter, so the energy pattern repeats twice as often as the motion itself.
At what displacement are kinetic and potential energy equal in SHM?
At x = A/√2, about 0.71 times the amplitude. Setting ½kx² equal to ½k(A² − x²) gives 2x² = A². Each energy is then half the total. Many students guess A/2, but at that point the potential energy is only a quarter of the total.
The simple pendulum
Read this section in the notes →Does a heavier bob make a pendulum swing more slowly?
No. For a simple pendulum T = 2π√(L/g), so the period depends only on the length and on g, not on the mass of the bob. A heavier bob feels a larger gravitational pull, but it also has proportionally more inertia, so the two effects cancel exactly.
Why does a pendulum's motion count as SHM only for small angles?
The restoring torque on a pendulum is proportional to sin θ, not θ. Only for small angles is sin θ approximately equal to θ in radians, so the torque becomes proportional to displacement and the motion is simple harmonic. At large swings this approximation fails and the period grows slightly longer.
What happens to a pendulum's period on the Moon?
It gets longer. Since T = 2π√(L/g) and g on the Moon is about one-sixth of its value on Earth, the period becomes about √6, or roughly 2.45, times longer. A pendulum clock calibrated on Earth would run slow on the Moon.
How can a pendulum be used to measure g?
Measure the length L and the period T of a simple pendulum, then use g = 4π²L/T². Timing many oscillations together and dividing reduces the error in T. Keeping the swing small ensures the motion is close to simple harmonic so the formula holds.
Lumi is not affiliated with or endorsed by NCERT. The official NCERT textbooks are free to read and download from NCERT's own website, ncert.nic.in. These notes and simulations are original work by Lumi (Aikolumi Software Pvt Ltd), © 2026, shared under CC BY-NC 4.0: copy, print, share and adapt them for any non-commercial use, with credit to Lumi and a link to lumineet.com.