A simple pendulum has a length ( I, of 1 m. In free vibration the amplitude of its swings falls

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A simple pendulum has a length ( I, of 1 m. In free...

A simple pendulum has a length ( $I$, of $1 \mathrm{~m}$. In free vibration the amplitude of its swings falls off by a factor $e$ in 50 swings. The pendulum is set into forced vibration by moving its point of suspension horizontally in SHM with an amplitude of $1 \mathrm{~mm}$. (a) Show that if the horizontal displacement of the pendulum bob is $x$, and the horizontal displacement of the support is $\xi$, the equation of motion of the bob for small oscillations is $$ \frac{d^{2} x}{d t^{2}}+\gamma \frac{d x}{d t}+\frac{g}{l} x=\frac{g}{l} \xi $$ Solve this equation for steady-state motion, if $\xi=\xi_{0} \cos \omega t .$ (Put $\left.\omega_{0}^{2}=g / l .\right)$ (b) At exact resonance, what is the amplitude of the motion of the pendulum bob? (First, use the given information to find $Q .$ ) (c) At what angular frequencies is the amplitude half of its resonant value?

A simple pendulum has a length ( $I$, of $1 \mathrm{~m}$. In free vibration the amplitude of its swings falls off by a factor $e$ in 50 swings. The pendulum is set into forced vibration by moving its point of suspension horizontally in SHM with an amplitude of $1 \mathrm{~mm}$.
(a) Show that if the horizontal displacement of the pendulum bob is $x$, and the horizontal displacement of the support is $\xi$, the equation of motion of the bob for small oscillations is
$$
\frac{d^{2} x}{d t^{2}}+\gamma \frac{d x}{d t}+\frac{g}{l} x=\frac{g}{l} \xi
$$
Solve this equation for steady-state motion, if $\xi=\xi_{0} \cos \omega t .$ (Put $\left.\omega_{0}^{2}=g / l .\right)$
(b) At exact resonance, what is the amplitude of the motion of the pendulum bob? (First, use the given information to find $Q .$ )
(c) At what angular frequencies is the amplitude half of its resonant value?

Key Concepts

Simple Pendulum

A simple pendulum consists of a mass suspended by a light, inextensible string or rod, free to oscillate under the influence of gravity. For small angular displacements, the motion is well approximated by simple harmonic motion, with the natural frequency given by ?(g/l), where g is the gravitational acceleration and l is the length of the pendulum.

Damped Harmonic Motion

Damped harmonic motion refers to oscillatory systems where energy is gradually lost over time, often due to friction or air resistance. This energy loss results in a decay of the oscillation amplitude, typically modeled by a damping coefficient. The system’s response features both transient and steady-state components, with the transient part diminishing over time.

Forced Vibrations

Forced vibrations occur when an external periodic force drives a system. In this context, even if the system is damped, the drive causes the system to eventually reach a steady-state oscillatory motion in which the oscillation frequency matches that of the driving force. The resulting amplitude depends on factors such as the natural frequency, the damping coefficient, and the driving frequency.

Resonance

Resonance is a phenomenon in which a system subject to forced vibrations exhibits a maximum amplitude when the driving frequency is equal to the system's natural frequency. In lightly damped systems, resonance can lead to large amplitude responses, and it is characterized by a sharp peak in the amplitude versus frequency response curve.

Quality Factor (Q)

The quality factor, Q, is a dimensionless parameter that measures the sharpness of the resonance peak in a damped oscillatory system. It is defined as the ratio of the stored energy to the energy dissipated per cycle. A high Q indicates weak damping and a narrow, high resonance peak, while a low Q corresponds to strong damping and a broader resonance response.

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