Exercises: Energy in oscillatory motion

A mass attached to a spring oscillates horizontally without friction between positions $x = -A$ and $x = +A$. The figure shows the oscillator with these positions marked.

1. a) At which position or positions is the elastic potential energy maximum?
2. b) At which position is the kinetic energy maximum?
3. c) At which positions is the kinetic energy zero? Justify your answers using formulas $E_p = \frac{1}{2} \times k \times x^2$ and $E_c = \frac{1}{2} \times m \times v^2$.

The vibration motor of a mobile phone stores a mechanical energy of $2\ \text{J}$ while oscillating.

1. a) What is its maximum kinetic energy? At which position is it reached?
2. b) What is its maximum elastic potential energy? At which positions is it reached?
3. c) What relationship exists between these two values and the mechanical energy? Justify your answer.

A trolley of $m = 0.5\ \text{kg}$ attached to a spring passes through the equilibrium position with a velocity of $2\ \text{m/s}$.

1. a) Calculate its kinetic energy at that point.
2. b) What is its elastic potential energy at that point? And its mechanical energy? Justify your answer.

In a Newton's cradle (the steel ball pendulum seen in offices), a ball of $m = 0.3\ \text{kg}$ is raised $h = 0.2\ \text{m}$ above its lowest point before being released (take $g = 10\ \text{m/s}^2$). The figure shows the raised ball, about to be released.

1. a) Calculate its gravitational potential energy relative to the lowest point, $E_p = m \times g \times h$.
2. b) What will this energy be converted into when the ball is released? What will the kinetic energy be when passing through the lowest point?

A gymnast jumps on a trampoline and oscillates vertically, without losing contact with the mat, between a highest point and a lowest point.

1. a) At which point of the oscillation is her gravitational potential energy maximum?
2. b) At which point is the elastic potential energy stored by the mat maximum?
3. c) Where is her kinetic energy maximum? Justify all three answers by energy exchange.

A mass-spring oscillator oscillates with amplitude $A = 0.1\ \text{m}$. The position where kinetic energy and elastic potential energy are equal is to be found.

1. a) Set up the condition $E_c = E_p$ taking into account that $E_c + E_p = \frac{1}{2} \times k \times A^2$, and show that at this point $E_p = \frac{1}{4} \times k \times A^2$.
2. b) Rearrange for displacement and calculate its numerical value.
3. c) Check that the result is less than the amplitude: is it reasonable?

The steel ball of a wrecking crane hangs from a cable of $l = 2\ \text{m}$ and is released from rest when the cable forms an angle of $60^\circ$ with the vertical (take $g = 9.8\ \text{m/s}^2$). The height it descends relative to the lowest point is $h = l \times (1 - \cos 60^\circ)$. The situation is shown in the figure.

1. a) Calculate that height $h$.
2. b) Apply conservation of energy, $m \times g \times h = \frac{1}{2} \times m \times v^2$, and calculate the speed of the ball when passing through the lowest point.
3. c) Does this speed depend on the mass of the ball? Justify your answer.

The spring of a pinball launcher has a spring constant $k = 100\ \text{N/m}$. The figure shows how elastic potential energy increases with compression.

1. a) Calculate the stored elastic potential energy when compressed by $0.1\ \text{m}$ and when compressed by $0.2\ \text{m}$.
2. b) What additional energy must be supplied to go from one compression to the other?
3. c) Why is it not correct to say that "double the compression means double the energy"? Reason using the formula.

During the spin cycle, the drum of a washing machine vibrates with a mechanical energy of $1.44\ \text{J}$. The shock absorbers supporting it behave like a spring with an effective constant of $k = 200\ \text{N/m}$.

1. a) Rearrange formula $E_m = \frac{1}{2} \times k \times A^2$ for amplitude.
2. b) Calculate the amplitude of vibration, in metres, and express it in centimetres as well.

The hammer of an electric bell oscillates with amplitude $A = 5\ \text{cm}$ driven by a spring of constant $k = 800\ \text{N/m}$; its mass is $m = 0.1\ \text{kg}$.

1. a) Express the amplitude in metres and calculate the mechanical energy of the oscillator.
2. b) Calculate the maximum kinetic energy of the hammer.
3. c) Calculate the maximum speed of the hammer.

A person swings on a vine of length $l = 2\ \text{m}$ and passes through the lowest point at $3\ \text{m/s}$ (take $g = 10\ \text{m/s}^2$). The figure shows the swing.

1. a) Calculate the maximum height reached above the lowest point, applying $\frac{1}{2} \times m \times v^2 = m \times g \times h$.
2. b) The height and the maximum angle $\theta$ with the vertical satisfy $h = l \times (1 - \cos\theta)$. Calculate the value of $\cos\theta$ and deduce the maximum angle reached by the vine.

A real oscillator with friction loses $10\ \%$ of its mechanical energy in each cycle. It starts with an initial energy $E_0 = 2\ \text{J}$. The figure shows how the amplitude of oscillation decreases with time.

1. a) Calculate the remaining mechanical energy after $5$ cycles.
2. b) Given that $E_m = \frac{1}{2} \times k \times A^2$, that is, energy is proportional to the square of the amplitude, calculate the factor by which the amplitude has been reduced after those $5$ cycles.
3. c) What has the lost energy been transformed into? Explain why the period of oscillation, however, hardly changes even though the amplitude decreases (hint: on which quantities does $T$ depend?).

A laboratory demonstration pendulum has a length $l = 1\ \text{m}$ and is released from rest forming $60^\circ$ with the vertical (take $g = 9.8\ \text{m/s}^2$). Recall that the height above the lowest point is $h = l \times (1 - \cos\theta)$. The figure shows the pendulum in the three positions considered.

1. a) Calculate the maximum speed it reaches when passing through the lowest point.
2. b) Calculate its speed when the string forms $30^\circ$ with the vertical, applying conservation of energy between the points at $60^\circ$ and $30^\circ$.
3. c) Compare the two speeds: why is the speed at $30^\circ$ more than half of the maximum speed even though the angle is exactly half?

An oscillator of $m = 1\ \text{kg}$ and $k = 100\ \text{N/m}$ oscillates first with amplitude $A = 0.1\ \text{m}$ and then with amplitude $A' = 0.2\ \text{m}$.

1. a) Calculate the mechanical energy in each case and their ratio $\frac{E_m'}{E_m}$.
2. b) Calculate the maximum speed in each case and their ratio $\frac{v_{max}'}{v_{max}}$.
3. c) Generalise the result: if the amplitude is multiplied by a factor $n$, by what factor are the mechanical energy and the maximum speed multiplied? Justify your answer using the formulas.

In a harmonic oscillator, it is measured that when the displacement is $x = \frac{A}{2}$, the kinetic energy is $6\ \text{J}$.

1. a) Show that at this point the elastic potential energy is one quarter of the mechanical energy, that is, $E_p = \frac{E_m}{4}$.
2. b) From the measurement, calculate the mechanical energy of the oscillator.
3. c) Calculate the kinetic energy when passing through equilibrium and the potential energy at the extreme points.

A steel sphere of $m = 2\ \text{kg}$ hangs from a string of $l = 2.5\ \text{m}$ and is released from rest forming $30^\circ$ with the vertical (take $g = 9.8\ \text{m/s}^2$; recall $h = l \times (1 - \cos\theta)$).

1. a) Calculate the mechanical energy of the pendulum relative to the lowest point of the trajectory.
2. b) Calculate the maximum speed when passing through the lowest point.
3. c) If the sphere had double the mass and were released from the same angle, how would the mechanical energy and the maximum speed change? Justify your answer using the formulas and explain why one of them does not change.

Unit challenge. Is mechanical energy conserved in a real oscillator? Analyse the behaviour of a homemade pendulum by applying simple harmonic motion (SHM) and the energy balance.

1. Setup and data collection: Hang a small, dense object (such as keys) from a thin thread ($L \approx 0.70\text{ m}$). Displace it by a small angle ($\theta \le 10^\circ$) to ensure the SHM regime. Measure the time for $10$ full oscillations using your smartphone and repeat $5$ times to determine the period with its uncertainty ($T \pm \Delta T$). Record the mass using a kitchen scale ($m \pm \Delta m$). Also record the initial angle $\theta_0$ and calculate the amplitude $A = L\theta_0$ (with $\theta_0$ in radians).

2. Kinematics of SHM: Calculate the angular frequency $\omega = 2\pi/T$ and write the displacement equation $x(t) = A\cos(\omega t)$. Compare the measured period with the theoretical one ($T = 2\pi\sqrt{L/g}$, with $g = 9.80\text{ m/s}^2$), justifying any differences.

3. Energy and graph: Calculate the maximum kinetic energy, $E_{c,\text{max}} = \frac{1}{2}m\omega^2 A^2$, and compare it with the initial potential energy. Create a table with $E_c(t)$, $E_p(t)$, and $E_m(t)$ at $8$ instants of one oscillation and plot their curves on the same graph.

4. Final report: Measure (for example, by recording a video) the initial amplitude and the amplitude after $10$ oscillations, estimate the loss of mechanical energy and submit a reasoned proposal explaining the observed losses due to friction.

Assessment criteria:

• Rigour in obtaining experimental measurements and uncertainties.

• Correctness in the SHM equations and energy calculations.

• Technical quality of the table and comparative graph.

• Critical analysis of the degradation of mechanical energy.