72. Damped Oscillator A class looked at the oscillation of a...

72. Damped Oscillator A class looked at the oscillation of a mass on a spring. They observed for 10 seconds and found its oscillation was well fit by assuming that the mass's motion was governed by Newton's Second Law with the spring force $F_{x}^{\text {sping }}=-k \Delta x$, where $\Delta x$ represents the stretch or squeeze of the spring and $k$ is the spring constant. However, it was also clear that this was not an adequate representation for times on the order of 10 minutes, since by that time the mass had stopped oscillating. (a) Suppose the mass was started at an initial position $x_{1}$ with a velocity $0 .$ Write down the solution, $x(t)$ and $v_{x}(t)$, for the equations of motion of the mass using only the spring force. What is the total energy of the oscillating mass? (b) Assume that there is also a velocity-dependent force (the damping force) that the spring exerts on an object when it is moving. Let's make the simplest assumption that the dynamic-spring force is linear in the velocity, $F_{x}^{\text {spring-dyn }}=-\gamma v_{x}$. Assume further that over one period of oscillation the velocity-dependent piece is small. Therefore, take $x(t)$ and $v_{x}(t)$ to be given by the oscillation without

Key Concepts

Damped Harmonic Oscillator

A damped harmonic oscillator includes a frictional or resistive force that opposes the motion and is often proportional to the velocity. This results in an additional term in the differential equation of motion, causing the amplitude of oscillation to gradually decrease with time. The presence of the damping force leads to energy dissipation, so the system’s total mechanical energy decreases as work is done against the damping force.

Weak Damping Approximation

The weak damping approximation applies when the damping force is small compared to the restoring force, allowing the system to undergo many oscillations before the amplitude decays significantly. In this regime, the oscillations are nearly sinusoidal, and the damping can be treated as a small perturbation. This approximation simplifies the analysis by allowing the use of the undamped oscillatory solution as a starting point, with an exponential decay factor to account for the gradual loss of amplitude.

Simple Harmonic Motion

Simple harmonic motion describes the oscillatory motion of a system governed by a restoring force proportional to displacement, as given by Newton's Second Law. In an ideal mass-spring system without damping, the motion is sinusoidal, with the solution expressed in terms of sine and cosine functions. The natural frequency of these oscillations is determined by the system parameters, specifically the spring constant and the mass.

Energy Conservation in Oscillations

In an undamped oscillator, the total mechanical energy, which is the sum of the kinetic and potential energies, remains constant over time. This energy is exchanged back and forth between kinetic energy, associated with the mass's velocity, and potential energy, stored in the spring. This principle explains why the amplitude of the oscillation does not decrease in the absence of dissipative forces.

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