A block of mass m is connected to a spring, the other end of...

A block of mass $m$ is connected to a spring, the other end of which is fixed. There is also a viscous damping mechanism. The following observations have been made on this system: (1) If the block is pushed horizontally with a force equal to $m g$, the static compression of the spring is equal to $h$. (2) The viscous resistive force is equal to $m g$ if the block moves with a certain known speed $u$. (a) For this complete system (including both spring and damper) write the differential equation governing horizontal oscillations of the mass in terms of $m, g, h$, and $u$. Answer the following for the case that $u=3 \sqrt{g h}$ : (b) What is the angular frequency of the damped oscillations? (c) After what time, expressed as a multiple of $\sqrt{h / g}$, is the energy down by a factor $1 / e ?$ (d) What is the $Q$ of this oscillator? (c) This oscillator, initially in its rest position, is suddenly set into motion at $t=0$ by a bullet of negligible mass but nonnegligible momentum traveling in the positive $x$ direction. Find the value of the phase angle $\delta$ in the equation $x=A e^{-y+2} \cos (\omega t-\delta)$ that describes the subsequent motion, and sketch $x$ versus $t$ for the first few cycles. (f) If the oscillator is driven with a force $m g \cos \omega t$, where $\omega=\sqrt{2 g / h}$, what is the amplitude of the steady-state response?

A block of mass $m$ is connected to a spring, the other end of which is fixed. There is also a viscous damping mechanism. The following observations have been made on this system:
(1) If the block is pushed horizontally with a force equal to $m g$, the static compression of the spring is equal to $h$.
(2) The viscous resistive force is equal to $m g$ if the block moves with a certain known speed $u$.
(a) For this complete system (including both spring and damper) write the differential equation governing horizontal oscillations of the mass in terms of $m, g, h$, and $u$. Answer the following for the case that $u=3 \sqrt{g h}$ :
(b) What is the angular frequency of the damped oscillations?
(c) After what time, expressed as a multiple of $\sqrt{h / g}$, is the energy down by a factor $1 / e ?$
(d) What is the $Q$ of this oscillator?
(c) This oscillator, initially in its rest position, is suddenly set into motion at $t=0$ by a bullet of negligible mass but nonnegligible momentum traveling in the positive $x$ direction. Find the value of the phase angle $\delta$ in the equation $x=A e^{-y+2} \cos (\omega t-\delta)$ that describes the subsequent motion, and sketch $x$ versus $t$ for the first few cycles.
(f) If the oscillator is driven with a force $m g \cos \omega t$, where $\omega=\sqrt{2 g / h}$, what is the amplitude of the steady-state response?

Key Concepts

Forced Response in Driven Oscillators

When an oscillator is subjected to an external periodic driving force, it eventually reaches a steady-state response where the oscillations occur at the driving frequency. The amplitude and phase of this steady-state motion depend on the frequency of the driving force relative to the oscillator’s natural frequency, as well as on the damping present in the system. Understanding this concept is vital for analyzing resonance phenomena and the response of systems under periodic excitation.

Quality Factor

The quality factor, Q, represents a measure of the oscillator’s damping or 'sharpness' of resonance. It is defined as the ratio of the stored energy to the energy lost per cycle and determines the rate at which the oscillations decay. A high Q indicates low damping and sustained oscillations, while a low Q signifies rapidly decaying motion.

Transient Response and Phase Angle

The transient response of an oscillator involves the system’s behavior as it adjusts from initial conditions, including any impulses or sudden forces applied. Determining the phase angle in the resultant sinusoidal motion is crucial because it expresses the offset between the system’s displacement and its driving force, reflecting the system’s response to initial conditions or applied impulses.

Energy Decay in Damped Oscillations

In damped oscillatory systems, energy is continuously lost, most often exponentially, due to frictional or resistive forces. The energy decay rate or time constant quantifies how quickly the oscillation’s energy diminishes, which is paramount when analyzing the long-term behavior or determining when the system’s energy falls to a specific fraction of its initial value.

Damped Harmonic Oscillator

This concept involves the study of systems in which a mass is attached to a spring and experiences a damping force, typically proportional to its velocity. The motion of such systems is described by a second order linear differential equation that captures both the restoring force of the spring and the energy dissipated by the damper, laying the foundation for understanding the time evolution of oscillatory systems under non-conservative forces.

Viscous Damping

Viscous damping refers to a resistive force that is linearly proportional to the velocity of the moving object. It is characterized by a damping coefficient, which quantifies how strongly the system resists motion. Understanding the damping coefficient is essential for modeling energy dissipation and for predicting the rate at which oscillations are attenuated over time.

Spring Constant Determination

For a mass–spring system in static equilibrium, the compression or elongation of the spring due to a known force allows the determination of the spring constant, typically via Hooke’s law. This relationship, expressed as F = kx, is crucial in converting measured displacements under applied forces into an effective stiffness, a parameter needed to fully characterize the oscillator’s dynamics.

Damped Natural Frequency

The damped natural frequency is the frequency at which a damped oscillator oscillates after the effects of damping have been incorporated. It is lower than the undamped natural frequency and is calculated by modifying the ideal frequency to account for the damping coefficient; this concept is key to understanding how non-conservative forces alter the temporal characteristics of an oscillator.

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