What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

Define simple harmonic motion.

A periodic motion in which the displacement varies as \(x=A\cos(\omega t+\phi)\) and the restoring force is proportional to and directed opposite to the displacement.

Relate angular frequency to period and frequency.

\(\omega=\dfrac{2\pi}{T}=2\pi\nu\).

How is SHM related to uniform circular motion?

SHM is the projection of uniform circular motion onto a diameter of the reference circle.

Write the velocity and acceleration in SHM.

\(v=-\omega A\sin(\omega t+\phi)\) and \(a=-\omega^{2}x\); acceleration is maximum at the extremes and zero at the mean position.

State the force law for SHM.

The force is \(F=-kx\), proportional to displacement and directed toward the mean position, with \(k=m\omega^{2}\).

Write the kinetic and potential energies in SHM.

\(K=\tfrac12 m\omega^{2}A^{2}\sin^{2}(\omega t+\phi)\) and \(U=\tfrac12 kA^{2}\cos^{2}(\omega t+\phi)\); both vary with period \(T/2\).

Show that the total energy in SHM is constant.

Adding kinetic and potential energy gives \(E=\tfrac12 kA^{2}\), independent of time.

Write the period of a simple pendulum.

\(T=2\pi\sqrt{\dfrac{L}{g}}\), valid for small oscillations, independent of mass and amplitude.

Why is the motion of a simple pendulum SHM only for small angles?

Because the restoring torque is proportional to \(\sin\theta\), which approximates to \(\theta\) only for small angles, giving a linear restoring effect.

Where are speed and acceleration maximum in SHM?

Speed is maximum at the mean position (\(x=0\)); acceleration is maximum at the extreme positions (\(x=\pm A\)).

Oscillations

Class 11 Physics · Physics Part II · Chapter 6

163 members in the CBSE community

View all

CBSE Class 11 · Physics

Ch 8–14 formulas and table values, by page

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
20-mark block: Ch 8–12
10-mark block: Oscillations and Waves
Common traps, corrected
7 chapters: every solved-example result
Sound, standing waves and beats covered

Page 1 free about 8 min read, Full kit about 55 min read

₹159See the full kit

CBSE Class 11 · Physics

Every formula of Ch 1–7, with page numbers

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
23-mark block: Ch 1–4
17-mark block: Ch 5–7
Common traps, corrected
7 chapters: every solved-example result
Every unit and dimension, listed

Page 1 free about 8 min read, Full kit about 55 min read

₹159See the full kit

View all 44 amazing Marks Booster kits →

Summary

A motion that repeats itself at regular intervals is periodic, and a to-and-fro motion about a mean position is oscillatory; every oscillation is periodic though not every periodic motion is oscillatory. The period \(T\) is the time for one cycle and the frequency \(\nu=1/T\) is measured in hertz. The simplest oscillation is simple harmonic motion (SHM), in which the displacement varies sinusoidally as \(x(t)=A\cos(\omega t+\phi)\), where \(A\) is the amplitude, \(\omega\) the angular frequency and \(\phi\) the phase constant. SHM is the projection of uniform circular motion on a diameter. Velocity \(v=-\omega A\sin(\omega t+\phi)\) and acceleration \(a=-\omega^{2}x\) are derived; acceleration is always directed toward the mean position and proportional to displacement. The force law \(F=-kx\) defines a linear harmonic oscillator with \(\omega=\sqrt{k/m}\). The kinetic and potential energies vary with time but their sum, the total energy \(E=\tfrac12 kA^{2}\), is constant. A simple pendulum executes SHM for small angles with period \(T=2\pi\sqrt{L/g}\), independent of mass and amplitude. These ideas underlie springs, pendulums and the vibrations of strings and air columns, and they form the basis for studying damped and forced oscillations and resonance.

Periodic and oscillatory motion; period and frequencySimple harmonic motionSHM and uniform circular motionVelocity and acceleration in SHMForce law for SHMEnergy in simple harmonic motionThe simple pendulum

Key terms

Periodic motion

Motion that repeats itself at regular intervals of time.

Simple harmonic motion

Oscillation in which displacement varies sinusoidally with time and the restoring force is proportional to displacement, \(F=-kx\).

Amplitude

The magnitude of the maximum displacement of a particle from its mean position in SHM.

Angular frequency

The quantity \(\omega=2\pi/T=2\pi\nu\) relating the period of SHM to its phase rate.

Phase

The argument \((\omega t+\phi)\) of the sinusoidal function that determines the state of the oscillator at any instant.

Restoring force

The force in SHM directed toward the mean position, proportional to and opposite to the displacement.

Extra questions & answers

Try one from this chapter

In simple harmonic motion the restoring force is:

Flashcards & interactive practice