Spring–mass oscillator Knowledge-Based Study

Study notes

1. The Ideal Spring–Mass Oscillator

   A spring–mass oscillator consists of an object of mass $m$ attached to an ideal, massless spring of force constant $k$. When the mass is displaced from its equilibrium position and released on a frictionless horizontal surface, it undergoes simple harmonic motion (SHM).

   The motion is governed by Hooke's Law, which states that the restoring force exerted by the spring is directly proportional to the displacement from equilibrium and directed opposite to it:

   $$F = -kx$$

   • Restoring Force ($F$): Always points toward the equilibrium position ($x = 0$).
   • Displacement ($x$): The position of the mass relative to equilibrium, measured in meters ($\text{m}$).
   • Spring Constant ($k$): A measure of spring stiffness, measured in newtons per meter ($\text{N/m}$).

   Mathematical Description of Motion

   Applying Newton's second law ($F = ma$) to Hooke's Law yields the differential equation of motion for an undamped spring–mass system:

   $$m\frac{d^2x}{dt^2} = -kx \implies \frac{d^2x}{dt^2} + \omega^2x = 0$$

   Here, $\omega$ represents the angular frequency of the oscillation, defined as:

   $$\omega = \sqrt{\frac{k}{m}}$$

   The position of the oscillating mass as a function of time $t$ can be expressed as:

   $$x(t) = A\cos(\omega t + \phi)$$

   where:

   • $A$: Amplitude, the maximum displacement from equilibrium.
   • $\phi$: Phase constant, determined by initial conditions at $t = 0$.
   • Velocity: $v(t) = -A\omega\sin(\omega t + \phi)$, with a maximum value of $v_{\text{max}} = A\omega$.
   • Acceleration: $a(t) = -A\omega^2\cos(\omega t + \phi) = -\omega^2x(t)$, with a maximum magnitude of $a_{\text{max}} = A\omega^2$.

   Period and Frequency Relationships

   The time required for the mass to complete one full cycle of oscillation is called the period ($T$). Because $\omega = \frac{2\pi}{T}$, substituting $\omega = \sqrt{\frac{k}{m}}$ gives:

   $$T = 2\pi\sqrt{\frac{m}{k}}$$

   The frequency ($f$), which is the number of oscillations per second, is the reciprocal of the period:

   $$f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$

   Key Conceptual Insights

   • Independence of Amplitude: For an ideal spring obeying Hooke's law, the period and frequency are strictly independent of the amplitude $A$. Doubling the release distance does not change the time per cycle.
   • Effect of Mass: Increasing the mass increases inertia, which increases the period and decreases the frequency.
   • Effect of Spring Constant: Stiffening the spring (increasing $k$) increases the restoring force, which reduces the period and increases the frequency.
   • Vertical Spring–Mass Systems: A vertical oscillator experiences a constant gravitational force that merely shifts the equilibrium position downward by $\Delta y = \frac{mg}{k}$. The oscillation about this new equilibrium follows the exact same period formula, $T = 2\pi\sqrt{\frac{m}{k}}$.

   Energy Conservation in Simple Harmonic Motion

   In the absence of non-conservative forces such as friction or air resistance, the total mechanical energy $E$ of the system remains constant:

   $$E = K + U = \text{constant}$$

   The components of mechanical energy are:

   • Elastic Potential Energy: $U = \frac{1}{2}kx^2$
   • Kinetic Energy: $K = \frac{1}{2}mv^2$

   The total mechanical energy can be expressed in terms of amplitude $A$ or maximum velocity $v_{\text{max}}$:

   $$E = \frac{1}{2}kA^2 = \frac{1}{2}mv_{\text{max}}^2$$

   Energy Distribution Across Extremes and Equilibrium

   • At the extremes ($x = \pm A$): The speed is zero ($v = 0$), so kinetic energy is zero ($K = 0$) and all energy is stored as elastic potential energy ($U = E = \frac{1}{2}kA^2$).
   • At equilibrium ($x = 0$): Displacement is zero, so potential energy vanishes ($U = 0$) and all energy is kinetic ($K = E = \frac{1}{2}mv_{\text{max}}^2$).
   • At an intermediate position $x$: Solving $E = \frac{1}{2}kx^2 + \frac{1}{2}mv^2$ for speed yields:

   $$v = \pm \omega \sqrt{A^2 - x^2}$$