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?

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 time $t = \frac{1}{12}\ \text{s}$.
3. c) At what time does it reach its maximum height for the first time?

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 $f = \frac{1}{T}$.

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.

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?

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

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.

The period of a mass-spring oscillator is $T = 2 \times \pi \times \sqrt{\frac{m}{k}}$. In the lab, 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 lobby 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, bathroom 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 bathroom scales do not.

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 $\text{Hz}$.

In the lab, 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 the formula $T = 2 \times \pi \times \sqrt{\frac{m}{k}}$ to reason out 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?

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.

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 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.

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.