A tuning fork, an engine piston, the needle of a seismograph, a swing: all of them oscillate about an equilibrium position and always return to it. If the motion is sufficiently "clean" —the force that causes it is proportional to the displacement and acts in the opposite direction— a single type of mathematical equation describes it precisely: simple harmonic motion (SHM). This unit formalises with derivatives what you saw in a descriptive way in 4th ESO.
If you attach a marker to the end of a mass oscillating on a spring and pull a strip of paper beneath it at a constant rate, the tip draws a smooth, periodic curve identical to a sine function: how can we describe, with mathematical precision, the position, the rhythm and the reach of that back-and-forth motion at every instant?
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A swing oscillates freely after a push. Timing with a stopwatch, it completes $15$ full oscillations in $30\ \text{s}$. The figure shows the swing in three positions…
A swing oscillates freely after a push. Timing with a stopwatch, it completes $15$ full oscillations in $30\ \text{s}$. The figure shows the swing in three positions during its oscillation.
a) Define, applying them to the swing, the concepts of displacement, amplitude, period and frequency.
b) Calculate the period $T$ and the frequency $f$ of the oscillation.
An electrocardiogram records a heart rate of $75$ beats per minute, perfectly regular. The figure represents the recorded signal. a) Calculate the frequency of the…
An electrocardiogram records a heart rate of $75$ beats per minute, perfectly regular. The figure represents the recorded signal.
a) Calculate the frequency of the heartbeat, in $\text{Hz}$.
b) Calculate its period, in seconds: how much time passes between two consecutive beats?
The displacement of a mass oscillating on a horizontal spring is given, in SI units, by $x = 0,1 \cdot \sen(20 \cdot \pi \cdot t)$. a) What are the amplitude $A$ and the…
The displacement of a mass oscillating on a horizontal spring is given, in SI units, by $x = 0,1 \cdot \sen(20 \cdot \pi \cdot t)$.
a) What are the amplitude $A$ and the angular frequency $\omega$?
b) Calculate the period $T$ and the frequency $f$ of the motion, remembering that $T = \frac{2 \cdot \pi}{\omega}$ and $f = \frac{1}{T}$.
Two identical oscillators vibrate according to the equations, in SI units, $x_1 = 0,05 \cdot \sen(3 \cdot \pi \cdot t)$ and…
Two identical oscillators vibrate according to the equations, in SI units, $x_1 = 0,05 \cdot \sen(3 \cdot \pi \cdot t)$ and $x_2 = 0,05 \cdot \sen(3 \cdot \pi \cdot t + \frac{\pi}{2})$.
a) What quantities do the two motions have in common?
b) Calculate the initial displacement (at $t = 0$) of each and explain how they differ.
The needle of a sewing machine moves up and down in simple harmonic motion with an amplitude of $6\ \text{cm}$ and a period of $0,5\ \text{s}$. At the initial instant,…
The needle of a sewing machine moves up and down in simple harmonic motion with an amplitude of $6\ \text{cm}$ and a period of $0,5\ \text{s}$. At the initial instant, the needle passes through the equilibrium position moving in the positive direction, so that $\varphi_0 = 0$.
a) Calculate the angular frequency $\omega$.
b) Write the complete equation of displacement, $x = A \cdot \sen(\omega \cdot t + \varphi_0)$, expressing the amplitude in metres.
Velocity and acceleration: the derivatives of x(t)
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A point at the centre of a guitar string vibrates with simple harmonic motion of amplitude $A = 2\ \text{mm}$ and frequency $f = 440\ \text{Hz}$ (the concert pitch 'A').…
A point at the centre of a guitar string vibrates with simple harmonic motion of amplitude $A = 2\ \text{mm}$ and frequency $f = 440\ \text{Hz}$ (the concert pitch 'A'). The maximum velocity of an oscillator is $v_{max} = A \cdot \omega$.
a) Calculate the angular frequency $\omega$ of the vibration.
b) Express the amplitude in metres and calculate the maximum speed of the point on the string.
The recording needle of a seismograph oscillates according to the equation $x = 0,06 \cdot \sen(8 \cdot \pi \cdot t)$, in SI units. The velocity in simple harmonic…
The recording needle of a seismograph oscillates according to the equation $x = 0,06 \cdot \sen(8 \cdot \pi \cdot t)$, in SI units. The velocity in simple harmonic motion is $v = A \cdot \omega \cdot \cos(\omega \cdot t + \varphi_0)$. The graph of displacement against time is shown in the figure.
a) Calculate the first instant at which the displacement is $x = 0,03\ \text{m}$ (half the amplitude).
b) Calculate the velocity of the needle at that instant.
c) At what position is the velocity maximum and what is its value?
The dynamics of SHM: springs, pendulums and resonance
When you pull a body attached to a spring $2\ \text{cm}$ away, you feel a slight resistance towards the centre, but if you pull it $4\ \text{cm}$ the force tugging at your hand immediately doubles: how does this restoring force determine the exact time the system takes to complete each swing?
This restoring force is what produces SHM: applying Newton's second law ($F=m\cdot a$) to $F=-k\cdot x$ and to $a=-\omega^2\cdot x$, we find that the angular frequency of the mass–spring oscillator does not depend on the amplitude, only on $k$ and the mass $m$:
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The period of a mass-spring oscillator is $T = 2 \cdot \pi \cdot \sqrt{\frac{m}{k}}$. In the laboratory, a $m = 0,25\ \text{kg}$ block oscillates attached to a spring of…
The period of a mass-spring oscillator is $T = 2 \cdot \pi \cdot \sqrt{\frac{m}{k}}$. In the laboratory, a $m = 0,25\ \text{kg}$ block oscillates attached to a spring of spring constant $k = 100\ \text{N/m}$. Calculate the period of the oscillation.
The period of a simple pendulum is $T = 2 \cdot \pi \cdot \sqrt{\frac{l}{g}}$. A lamp in a museum entrance hall hangs from a cable $l = 1\ \text{m}$ long and oscillates…
The period of a simple pendulum is $T = 2 \cdot \pi \cdot \sqrt{\frac{l}{g}}$. A lamp in a museum entrance hall hangs from a cable $l = 1\ \text{m}$ long and oscillates slightly after a gust of wind (take $g = 9,8\ \text{m/s}^2$). Calculate its period of oscillation.
A suspension bridge has a natural frequency of oscillation of $0,5\ \text{Hz}$. The wind, passing around the pillars, generates vortices that push the bridge laterally…
A suspension bridge has a natural frequency of oscillation of $0,5\ \text{Hz}$. The wind, passing around the pillars, generates vortices that push the bridge laterally once every $2,0\ \text{s}$. The figure shows how the amplitude of the forced oscillation depends on the frequency of the pushes.
a) Calculate the natural period and natural angular frequency of the bridge.
b) Calculate the frequency of the wind pushes and compare it with the natural frequency of the bridge.
c) Explain what resonance is and why it is dangerous in this situation, even if each push is weak.
d) Propose a design modification that avoids the problem and justify it by indicating which physical quantity changes.
To estimate $g$ with a pendulum, set up $T^2=(4\pi^2/g)L$. Use small oscillations, measure from the point of suspension to the centre of the mass, and time ten periods to reduce the effect of reaction time. Keep the stand stable and the surroundings clear; share out and rotate the tasks of measuring, recording and cross-checking.
Check the residuals and any non-zero intercept: they could indicate errors in the length or in the timing. The uncertainty in the gradient does not automatically include all systematic errors. Document the raw data, method, sources and limitations, and state whether you used a simulation instead of an experiment.
In the PAU, organise each solution like this: diagram and data in SI units; model and conditions; justified physical law; algebraic working; substitution with units; rounded result and a check of sense and order of magnitude. The official Madrid and Catalonia marking criteria reward reasoning and diagrams; memorising a formula and giving only a number leaves the answer incomplete.
The diagram links the main ideas:
Let's practice! 8 medium exercises in this section