Simple Harmonic Motion

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.

1. a) Define, applying them to the swing, the concepts of displacement, amplitude, period and frequency.
2. 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.

1. a) Calculate the frequency of the heartbeat, in $\text{Hz}$.
2. b) Calculate its period, in seconds: how much time passes between two consecutive beats?

The mains voltage in Mexico oscillates with a frequency of $50\ \text{Hz}$. Calculate its angular frequency $\omega = 2 \times \pi \times f$, in $\text{rad/s}$.

The displacement of a mass oscillating on a horizontal spring is given, in SI units, by $x = 0.1 \times \sen(20 \times \pi \times t)$.

1. a) What are the amplitude $A$ and the angular frequency $\omega$?
2. b) Calculate the period $T$ and the frequency $f$ of the motion, remembering that $T = \frac{2 \times \pi}{\omega}$ and $f = \frac{1}{T}$.

An oscillator describes simple harmonic motion with equation $x = 0.08 \times \sen(\pi \times t)$, in SI units.

1. a) Calculate the displacement at the instant $t = 0.5\ \text{s}$.
2. b) What is the maximum value that the displacement can take? At what times is it reached?

Two identical oscillators vibrate according to the equations, in SI units, $x_1 = 0.05 \times \sen(3 \times \pi \times t)$ and $x_2 = 0.05 \times \sen(3 \times \pi \times t + \frac{\pi}{2})$.

1. a) What quantities do the two motions have in common?
2. b) Calculate the initial displacement (at $t = 0$) of each and explain how they differ.

The piston rod of a combustion engine vibrates according to the equation $x = 0.04 \times \sen(10 \times \pi \times t + \frac{\pi}{2})$, in SI units.

1. a) State the value of the amplitude $A$, the angular frequency $\omega$ and the initial phase $\varphi_0$.
2. b) Calculate the period $T$ and the frequency $f$.
3. c) Calculate the displacement at $t = 0.1\ \text{s}$ and interpret its sign.

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 $\varphi_0 = 0$.

1. a) Calculate the angular frequency $\omega$.
2. b) Write the complete equation of displacement, $x = A \times \sen(\omega \times t + \varphi_0)$, expressing the amplitude in meters.

A wave energy converter buoy oscillates vertically with the waves according to the equation $x = 0.1 \times \sen(2 \times \pi \times t)$, in SI units.

1. a) Calculate the period and frequency of the oscillation.
2. b) Calculate the displacement at instant $t = \frac{1}{12}\ \text{s}$.
3. c) At what instant does it reach its maximum height for the first time?

A point at the center 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 $v_{max} = A \times \omega$.

1. a) Calculate the angular frequency $\omega$ of the vibration.
2. b) Express the amplitude in meters and calculate the maximum speed of the point on the string.

On a frictionless air track, a trolley of $m = 0.25\ \text{kg}$ is attached to a spring of constant $k = 400\ \text{N/m}$. The trolley is pulled $10\ \text{cm}$ from the equilibrium position and released from rest at instant $t = 0$. The setup is shown in the figure.

1. a) Calculate the angular frequency, $\omega = \sqrt{\frac{k}{m}}$, and the period of oscillation.
2. b) Taking $x(t) = A\sin(\omega t + \varphi_0)$, explain why the initial phase is $\varphi_0 = \frac{\pi}{2}$: what is the value of the displacement at $t = 0$?
3. c) Write the complete equation of displacement, in SI units.
4. d) Calculate the maximum velocity of the trolley, $v_{max} = A \times \omega$.

The recording needle of a seismograph oscillates according to the equation $x = 0.06 \times \sen(8 \times \pi \times t)$, in SI units. The velocity in simple harmonic motion is $v = A \times \omega \times \cos(\omega \times t + \varphi_0)$. The graph of displacement against time is shown in the figure.

1. a) Calculate the first instant at which the displacement is $x = 0.03\ \text{m}$ (half the amplitude).
2. b) Calculate the velocity of the needle at that instant.
3. c) At what position is the velocity maximum and what is its value?

The period of a mass-spring oscillator is $T = 2 \times \pi \times \sqrt{\frac{m}{k}}$. 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 $T = 2 \times \pi \times \sqrt{\frac{l}{g}}$. 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.

On the International Space Station, weighing scales cannot be used to measure the mass of astronauts. Instead, the astronaut sits on a seat attached to a spring of constant $k = 1000\ \text{N/m}$ and the period of oscillation of the system, $T = 2 \times \pi \times \sqrt{\frac{m}{k}}$, is measured.

1. a) If the measured period is $T = 1.6\ \text{s}$, rearrange the formula for mass and calculate the astronaut's mass (ignore that of the seat).
2. b) Explain why this method works in weightlessness and weighing scales do not.

An antique grandfather clock works with a pendulum whose period must be exactly $T = 2.0\ \text{s}$: each complete oscillation advances a tooth of the mechanism (take $g = 9.8\ \text{m/s}^2$). The pendulum is shown in the figure.

1. a) Rearrange formula $T = 2 \times \pi \times \sqrt{\frac{l}{g}}$ for length.
2. b) Calculate the required length of the pendulum.

In a workshop, a spring with spring constant $k = 250\ \text{N/m}$ is tested by attaching a mass of $0.4\ \text{kg}$ to it and making it oscillate.

1. a) Calculate the period of oscillation, $T = 2 \times \pi \times \sqrt{\frac{m}{k}}$.
2. b) Calculate the frequency, in $\text{Hz}$.

A pendulum of length $l = 1.5\ \text{m}$ is part of an experiment that will be installed on a lunar base. Gravity on the surface of the Moon is $g_L = 1.6\ \text{m/s}^2$ and on Earth $g_T = 9.8\ \text{m/s}^2$.

1. a) Calculate the period of the pendulum on Earth.
2. b) Calculate the period on the Moon.
3. c) How many times slower does it oscillate on the Moon? Calculate the ratio of both periods.

The Foucault pendulum installed in the Panthéon in Paris has a length of $l = 67\ \text{m}$ (take $g = 9.8\ \text{m/s}^2$).

1. a) Calculate its period of oscillation.
2. b) Calculate how many complete oscillations it performs in one hour.
3. c) Foucault used a sphere of $28\ \text{kg}$. Justify, using the period formula, why the mass of the sphere does not affect the measurement and why it is advantageous to make the pendulum as long as possible to observe the phenomenon easily.

A pendulum with a period of $T = 2.00\ \text{s}$ is to be designed for an experiment operating on the surface of Mars, where $g_M = 3.71\ \text{m/s}^2$.

1. a) Calculate the required length of the pendulum on Mars.
2. b) If that same pendulum were brought back to Earth ($g_T = 9.8\ \text{m/s}^2$), what would its period be?
3. c) Geologists use precision pendulums to measure $g$ at different points on Earth and detect subsurface anomalies. Explain, using the formula, why the period of a pendulum reveals the value of $g$ and which quantities need to be measured most carefully.

In the laboratory, two oscillators are set up using the same spring, with spring constant $k = 100\ \text{N/m}$: oscillator A has a mass of $0.1\ \text{kg}$ and oscillator B has a mass of $0.4\ \text{kg}$. The figure shows both setups.

1. a) Calculate the period of each oscillator and their ratio $\frac{T_B}{T_A}$.
2. b) Without repeating the calculations, use formula $T = 2 \times \pi \times \sqrt{\frac{m}{k}}$ to reason the relationship between that ratio and the ratio of masses $\frac{m_B}{m_A}$.
3. c) What spring constant should the spring of oscillator B have so that it oscillates with the same period as A? Justify your answer.

A mass of $m = 0.5\ \text{kg}$ is suspended from a vertical spring of constant $k = 200\ \text{N/m}$ (take $g = 9.8\ \text{m/s}^2$). The setup is shown in the figure.

1. a) Calculate how much the spring stretches to reach the equilibrium position, where the elastic force $k \times x_0$ equals the weight $m \times g$.
2. b) The mass is pulled slightly from that position and released. Calculate the period of the vertical oscillation, $T = 2 \times \pi \times \sqrt{\frac{m}{k}}$.
3. c) Explain why gravity does not modify the period compared to the same oscillator arranged horizontally: what is the only thing gravity changes in vertical 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 once every $2.0\ \text{s}$. The figure shows how the amplitude of the forced oscillation depends on the frequency of the pushes.

1. a) Calculate the natural period and natural angular frequency of the bridge.
2. b) Calculate the frequency of the wind pushes and compare it with the natural frequency of the bridge.
3. c) Explain what resonance is and why it is dangerous in this situation, even if each push is weak.
4. d) Propose a design modification that avoids the problem and justify it by indicating which physical quantity changes.

A group of students measures gravity using a pendulum. The measured length is $l = 1.20\ \text{m}$ and they time $20$ complete oscillations in $44.0\ \text{s}$.

1. a) Calculate the period of the pendulum.
2. b) Rearrange formula $T = 2 \times \pi \times \sqrt{\frac{l}{g}}$ for $g$ and calculate its experimental value.
3. c) Compare with the accepted value $g = 9.81\ \text{m/s}^2$ and explain why $20$ oscillations are timed rather than a single one to find the period.