L7 Simple harmonic motion
Describe/define simple harmonic motion (7.1)
Simple harmonic motion (SHM) is a special case of periodic motion. SHM results when the magnitude of the restoring force exerted on an object is proportional to that object’s displacement from its equilibrium position: \[\begin{equation} F_{restoring} = -k x, \end{equation}\] where \(k\) here is a constant of proportionality that could be from a spring constant (but does not have to be). The minus sign indicates that the force is in the opposite direction as the displacement \(x\) from equilibrium.
A restoring force is a force that is exerted in a direction opposite to the object’s displacement from an equilibrium position. An equilibrium position is a location at which the net force exerted on an object or system is zero.
For example, \[\begin{equation} F = -k x \end{equation}\] gives a restoring force away from an equilibrium position at \(x=0\), while \[\begin{equation} F = -k (x - l_0) \end{equation}\] gives a restoring force away from an equilibrium position at \(x=l_0\), which might represent the rest length of a spring, or the system’s rest position under an external applied constant force like gravity.
Frequency and period of simple harmonic motion (7.2)
Period \(T\) is the time it takes for the pendulum or mass-spring-oscillator or system to move through one complete cycle. Frequency \(f\) is the oscillations per unit time; while angular frequency is the radians per unit time if you consider a complete oscillation as going through \(2\pi\) radians.
For any periodic motion, including SHM, the period \(T, \unit{\second}\) is related to the angular frequency, \(\omega, \unit{\radian\per\second}\), and the frequency, \(f, \unit{\hertz}\), by the following: \[\begin{equation} T = \dfrac{2\pi}{\omega} = \dfrac{1}{f} \end{equation}\]
The period of an ideal mass-spring oscillator is given by the equation \[\begin{equation} T = 2\pi \sqrt{\dfrac{m}{k}}. \label{eq:Tmk} \end{equation}\] The period of a simple pendulum displaced by a small angle is given by the equation. \[\begin{equation} T = 2\pi \sqrt{\dfrac{l}{g}}. \label{eq:Tlg} \end{equation}\]
\[eq:Tmk,eq:Tlg\] are derived below. For now, consider \[eq:Tlg\]. In class (\[fig:measured\]) we varied the length of the large pendulum and bet a candy bar on if the functional dependence on length was linear, quadratic, exponential, logarithmic, etc... What we saw was that the period depended on \(\sqrt{l}\). We also can look at the units of the terms; \(T, \unit{\second}\), \(l, \unit{\meter}\) and \(g, \unit{\meter\per\second\squared}\) suggests that \(T\sim\sqrt{l/g}\) based on the need to cancel m and get s for \(T\).
Representing and analyzing simple harmonic motion (7.3)
For an object exhibiting SHM, the displacement of that object measured from its equilibrium position can be represented by the equations \[\begin{align} x &= A \cos{\omega t} \\ x &= A \sin{\omega t} \end{align}\] \(A\) here is the amplitude, and \(\omega\) is the (angular) frequency in rad s−1; sometimes these are also written as e.g. \(x = A\cos{2\pi f t}\) with frequency \(f\) in Hz or s. These are periodic, sinusoidal functions. Minima, maxima, and zeros of displacement, velocity, and acceleration are features of harmonic motion, and you should be able to identify where they occur. Recognizing the positions or times at which the displacement, velocity, and acceleration for SHM have extrema or zeros can help in qualitatively describing the behavior of the motion. For example, for a pendulum the position and acceleration are maximum at the extremes of the swing; while the velocity is a maximum at mid-swing.
Differential equation for simple harmonic motion
The position as a function of time for an object exhibiting SHM is a solution of the second order linear constant coefficient ordinary differential equation derived from the application of Newton’s second law. \[\begin{equation} \dfrac{d^2 x}{dt^2} = -\omega^2 x \end{equation}\]
One method of solution is the method of assumed solutions, also known as the method of undetermined coefficients, which is an educated guess about the form of the solutions. Based on watching the pendulum go we assume solutions of the form \[\begin{equation} x = A \cos{(\omega t + \phi)} \end{equation}\]
Substituting into the differential equation gives solutions for \(\omega\). For example, for a mass-spring oscillator \[\begin{align} m \dfrac{d^2 x}{dt^2} &= - k x \\ - m A \omega^2 \cos^2{(\omega t + \phi)} &= -k A \cos{(\omega t + \phi)} \\ \omega^2 &= \dfrac{k}{m} \\ \omega &= \sqrt{\dfrac{k}{m}}. \end{align}\] Similarly, for a simple pendulum with small angle approximations \[\begin{align} m \dfrac{d^2 x}{dt^2} &= mg\sin{\theta} \approx mg \dfrac{x}{l} \\ \omega^2 &= \dfrac{g}{l}\\ \omega &= \sqrt{\dfrac{g}{l}}. \end{align}\] \(\omega\) here is the natural frequency of a system, the frequency at which the system will oscillate when it is displaced from its equilibrium position. Sometimes it is written as \(\omega_0\) or \(\omega_n\).
For simple harmonic motion of linear systems, the period and natural frequency do not depend on the amplitude of motion (there is no \(A\) in the results for \(\omega\)).
Application of chain rule in periodic functions
The acceleration of an object exhibiting SHM is related to the object’s angular frequency and position. \[\begin{equation} a = -\omega^2 x \end{equation}\] It can also be shown that the maximum velocity and acceleration of an object exhibiting SHM are related to the angular frequency of the object’s motion. \[\begin{align} v_{max} &= A \omega \\ a_{max} &= A \omega^2 \end{align}\]
Resonance
In the presence of a sinusoidal external force, a system may exhibit resonance. Resonance occurs when an external force is exerted at the natural frequency of an oscillating system. Resonance increases the amplitude of oscillating motion. In an undamped or lightly damped system the amplitude can increase to very very large values.
Energy of simple harmonic oscillators (7.4)
The total energy of a system exhibiting SHM is the sum of the system’s kinetic and potential energies. \[\begin{equation} E_{total} = U + K \end{equation}\]
Conservation of energy indicates that the total energy of a system exhibiting SHM is constant. The kinetic energy of a system exhibiting SHM is at a maximum when the system’s potential energy is at a minimum, and vice versa; the potential energy of a system exhibiting SHM is at a maximum when the system’s kinetic energy is at a minimum. The minimum kinetic energy of a system exhibiting SHM is zero.
Changing the amplitude of a system exhibiting SHM will change the maximum potential energy of the system and, therefore, the total energy of the system. Relevant equation for a spring–object system: \[\begin{equation} E_{total} = \dfrac{1}{2} k A^2 \end{equation}\]
Simple and physical pendulums (7.5)
A physical pendulum is a rigid body that undergoes oscillations about a fixed axis.
For small amplitudes of motion, the period of a physical pendulum is derived from the application of Newton’s second law in rotational form. \[\begin{equation} T = 2\pi \sqrt{\dfrac{I}{mgd}} \end{equation}\]
When displaced from equilibrium, the gravitational force exerted on a physical pendulum’s center of mass provides a restoring torque. \[\begin{equation} \tau = -mgd \sin{\theta} \end{equation}\]
For small amplitudes of motion, the small angle approximation can be applied to the restoring torque. \[\begin{align} \sin{\theta} &\approx \theta \\ \tau \approx -mgd\theta = I \alpha \end{align}\]
The small-angle approximation and Newton’s second law in rotational form yield a second order differential equation that describes SHM: \[\begin{equation} \dfrac{d^2\theta}{dt^2} = -\omega^2 \theta \end{equation}\]
A simple pendulum is a special case of physical pendulums in which the hanging object can be modeled as a point mass at a distance, l, from the pivot point. \[\begin{equation} T = 2\pi \sqrt{\dfrac{l}{g}} \end{equation}\]
A torsion pendulum is a case of SHM where the restoring torque is proportional to the angular displacement of a rotating system. For example, a horizontal disk that is suspended from a wire attached to its center of mass may undergo rotational oscillations about the wire in the horizontal plane. \[\begin{equation} I \alpha = - k \Delta\theta \end{equation}\]
See also
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