Combining x = $A\cos$($\omega$t) and v = -$A\omega\sin$($\omega$t) with the $identity \sin$^2
Oscillations - OCR A-Level Physics
- Combining $x = A\cos(\omega t)$ and $v = -A\omega\sin(\omega t)$ with the identity $\sin^2 + \cos^2 = 1$ gives:
- $v^{2} = \omega^2(A^{2} - x^{2})$, so $v = \pm\omega\sqrt{A^2 - x^2}$.
- This shows that velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹. depends only on the currentThe rate of flow of chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C).. Measured in amperes (A). displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). x, not on time directly.
- At $x = 0$: $v = \pm A\omega$ (maximum speed).
- At $x = \pm A$: $v = 0$ (momentarily at rest at the turning points).
- A graph of v against x is an ellipse.
Worked Example
A system oscillates with SHM of amplitudeThe maximum displacement of a point on a wave from its equilibrium (rest) position. Measured in metres (m). 0.050 m and frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz). 4.0 Hz. Calculate the speed when the displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). is 0.030 m.
Show Solution
1
Find $\omega$
$\omega = 2\pi f = 2\pi \times 4.0 = 8\pi \text{ rad s}^{-1} = 25.1 \text{ rad s}^{-1}$.
2
Use $v = \omega\sqrt{A^2 - x^2}$.
3
$v = 25.1 \times \sqrt{0.050^2 - 0.030^2}$.
4
$v = 25.1 \times \sqrt{0.0025 - 0.0009}$.
5
$v = 25.1 \times \sqrt{0.0016} = 25.1 \times 0.040 = 1.0 \text{ m s}^{-1}$.
Answer
$v = 1.0 \text{ m s}^{-1}$