Energy Conservation and Maximum Speed in Oscillators

All flashcards

How does amplitude affect a spring oscillator’s maximum speed?

For fixed mass and spring stiffness, maximum speed is proportional to amplitude: $v_{\max}=X\sqrt{k/m}$.

What is a spring oscillator’s speed at displacement $x$?

For amplitude $X$, $v=\pm\sqrt{k/m}\sqrt{X^2-x^2}$; the sign indicates direction.

What is the maximum tangential speed of a small-angle pendulum?

For length $L$ and angular amplitude $\theta_{\max}$ in radians, $v_{\max}=\sqrt{gL}\,\theta_{\max}$.

What is the amplitude of an oscillator?

The magnitude of its greatest displacement from equilibrium.

How does a small-angle pendulum’s energy change during a swing?

Gravitational potential energy decreases toward the lowest point as kinetic energy increases.

Quiz questions

1. A spring oscillator has amplitude $X$ and is at displacement $x$. Which expression gives its instantaneous velocity?Answer: $v=\pm\sqrt{k/m}\sqrt{X^2-x^2}$
2. What is the maximum speed of a mass $m$ oscillating on a spring of stiffness $k$ with amplitude $X$?Answer: $v_{\max}=X\sqrt{k/m}$
3. For a small-angle pendulum of length $L$ and maximum angular displacement $\theta_{\max}$, what is its maximum angular speed?Answer: $\omega_{\max}=\sqrt{g/L}\,\theta_{\max}$
4. If an oscillator’s displacement is $y(t)=a\sin(\omega t)$, which expression gives its velocity?Answer: $v(t)=a\omega\cos(\omega t)$
5. How are angular frequency $\omega$ and period $T$ related?Answer: $\omega=2\pi/T$