Chapter 13 ยท Introduction

Chapter 13 โ€” Oscillations

A swing, a guitar string, your heartbeat, the quartz crystal in a watch: motions that repeat themselves. This chapter finds the one pattern behind most of them โ€” simple harmonic motion โ€” and uses it to time springs and pendulums.

Push a child on a swing and let go: the swing comes back to you, again and again, in very nearly the same time whether the swing is large or small. In 1583 a young Galileo, bored in the cathedral at Pisa, timed a swinging lamp against his pulse and noticed the same thing. That observation became the pendulum clock, the first clock good enough to keep time to a minute a day โ€” and repeating motions have kept our time ever since: the balance wheel of a mechanical watch, the quartz crystal vibrating $32\,768$ times a second in a digital one, and the caesium atoms that define the second itself.

This chapter studies such repeating motions. You will start by describing periodic and oscillatory motion with a period and a frequency, and then meet the simplest and most important oscillation of all, simple harmonic motion (SHM), in which the displacement varies as a cosine of time. You will see that SHM is just the shadow of uniform circular motion, find its velocity, acceleration and energy, and discover the force law that produces it: a restoring force proportional to the displacement. Finally you will apply that law to the two systems every physics laboratory owns โ€” a block on a spring and a simple pendulum โ€” and use the pendulum to measure $g$.

Periodic motionโ†’$T,\ \nu=\dfrac1T$โ†’$x=A\cos(\omega t+\phi)$โ†’Reference circleโ†’$a=-\omega^2x$โ†’$F=-kx$โ†’$E=\tfrac12kA^2$โ†’$T=2\pi\sqrt{\dfrac mk}$โ†’$T=2\pi\sqrt{\dfrac lg}$
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • distinguish periodic, oscillatory and simple harmonic motion, and find the period and frequency of a periodic motion (including sums such as $\sin\omega t+\cos\omega t$);
  • describe SHM by $x=A\cos(\omega t+\phi)$, identify amplitude, angular frequency, phase and phase constant, and find $A$ and $\phi$ from the starting conditions;
  • show that the projection of uniform circular motion on a diameter is SHM, and use the reference circle to read off positions, velocities and times;
  • use $v=-\omega A\sin(\omega t+\phi)$, $v=\pm\omega\sqrt{A^2-x^2}$ and $a=-\omega^2x$, and explain the phase relations between $x$, $v$ and $a$;
  • recognise SHM from the force law $F=-kx$, with $\omega=\sqrt{k/m}$, and use the energies $K=\tfrac12k(A^2-x^2)$, $U=\tfrac12kx^2$, $E=\tfrac12kA^2$;
  • find the period of a block on a spring, horizontal or vertical, and of a simple pendulum, $T=2\pi\sqrt{l/g}$, and measure $g$ from a $T^2$โ€“$l$ graph;
  • tackle Board, JEE Main, JEE Advanced and NEET-style problems on oscillations (with springs in series and parallel, and โ€” for JEE Advanced โ€” damped and forced oscillations and resonance, as exam extensions).
Teaser โ€” The Pen Recorder

A block on a spring carries a pen that rests on a strip of paper. The paper is pulled steadily to the left, so the pen draws the block's position against time. Before you press Play, guess: what shape will the trace be? Then change the amplitude. Does a bigger swing take longer?

4.0 cm
spring constant $k$โ€”
displacement nowโ€”
measured periodโ€”
oscillations completedโ€”

Sections in this chapter

13.1

Periodic Motion and Simple Harmonic Motion

Period and frequency, oscillations, the SHM equation $x=A\cos(\omega t+\phi)$, phase, and SHM as the shadow of uniform circular motion.

$x=A\cos(\omega t+\phi)$
Open section โ†’13.2

Velocity, Acceleration, Force Law and Energy in SHM

How $v$ and $a$ change through a cycle, the restoring force $F=-kx$, and the steady exchange of kinetic and potential energy.

$a=-\omega^2x$
Open section โ†’13.3

The Springโ€“Block Oscillator and the Simple Pendulum

Blocks on springs (horizontal and vertical), the simple pendulum and its small-angle period, and measuring $g$.

$T=2\pi\sqrt{\dfrac lg}$
Open section โ†’
โœ… Before you start
You will need uniform circular motion โ€” angular speed $\omega=\dfrac{2\pi}{T}$, frequency and centripetal acceleration $\omega^2r$ (Section 3.4); Newton's second law (Section 4.1); the spring force $F=-kx$ and its potential energy $\tfrac12kx^2$, and conservation of mechanical energy (Section 5.2); Hooke's law for real materials (Section 8.1); and torque $\tau=I\alpha$ about a fixed axis (Section 6.3) for the pendulum. From mathematics you need radian measure and the graphs of $\sin$ and $\cos$, the identity $\cos(A+B)=\cos A\cos B-\sin A\sin B$, and velocity and acceleration as rates of change (Sections 2.1โ€“2.2).
โ† 12.3 Equipartition of Energy, Specific Heats and Mean Free Path13.1 Periodic Motion and Simple Harmonic Motion โ†’