Hooke's Law & Coupled Harmonic Oscillations
Aim: to verify Hooke's law of elasticity \(F = -kx\), determine spring stiffness by static elongation and dynamic oscillation, and analyze the normal modes, beat phenomena, and energy exchange of two coupled harmonic oscillators.
Apparatus & Oscilloscope Traces
Dual Waveforms \(x_1(t)\) and \(x_2(t)\)
Lissajous Phase Space \((x_1, x_2)\)
Observation Table
Normal Modes of Coupled Oscillators
Two identical carts of mass \(m\) moving on a frictionless horizontal rail are anchored to rigid walls by outer springs of stiffness \(k\) and coupled to each other by a central spring \(k_c\). The Newton–Lagrange equations of motion are:
$$ m \ddot{x}_1 = -k x_1 - k_c (x_1 - x_2) $$ $$ m \ddot{x}_2 = -k x_2 - k_c (x_2 - x_1) $$Writing in matrix form \(\mathbf{M}\ddot{\mathbf{x}} + \mathbf{K}\mathbf{x} = 0\), the eigenvalue problem yields two independent normal modes:
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Symmetric Mode (In-Phase, \(x_1 = x_2\)):
The middle spring remains unstretched (\(x_1 - x_2 = 0\)). Both carts oscillate together with natural frequency: $$ \omega_1 = \sqrt{\frac{k}{m}}, \qquad f_1 = \frac{1}{2\pi}\sqrt{\frac{k}{m}} $$ -
Antisymmetric Mode (Out-of-Phase, \(x_1 = -x_2\)):
The carts move in opposite directions, compressing and stretching the coupling spring by \(2x\): $$ \omega_2 = \sqrt{\frac{k + 2k_c}{m}}, \qquad f_2 = \frac{1}{2\pi}\sqrt{\frac{k + 2k_c}{m}} $$
The Phenomenon of Beats
If initial displacement is given to cart 1 alone while cart 2 is released from rest (\(x_1(0) = A, x_2(0) = 0, \dot{x}_1(0)=\dot{x}_2(0)=0\)), the general solution is an equal superposition of both normal modes:
$$ x_1(t) = A \cos\left(\frac{\omega_2 - \omega_1}{2}t\right) \cos\left(\frac{\omega_1 + \omega_2}{2}t\right) $$ $$ x_2(t) = A \sin\left(\frac{\omega_2 - \omega_1}{2}t\right) \sin\left(\frac{\omega_1 + \omega_2}{2}t\right) $$Energy flows continuously back and forth between cart 1 and cart 2. When cart 1 comes momentarily to complete rest, cart 2 oscillates with maximum amplitude \(A\), and vice versa. The beat period is:
$$ T_{\text{beat}} = \frac{2\pi}{|\omega_1 - \omega_2|} = \frac{1}{|f_1 - f_2|} $$Viva Questions
Why does the central spring not affect the symmetric mode frequency?
In the symmetric mode, both gliders have identical displacement \(x_1(t) = x_2(t)\) at every instant. The length of the coupling spring remains constant, exerting zero net force on either cart.
What is Rayleigh's effective mass correction for a real spring?
A real spring has mass \(m_s\). Because different elements of the spring move with speeds linearly varying from \(0\) at the support to \(v\) at the bob, integrating kinetic energy shows that \(\frac{1}{3}m_s\) must be added to the hanging mass: \(m_{\text{eff}} = m + \frac{1}{3}m_s\).