Physics · Ch 10 — Oscillations

Time period, frequency, phase, phase difference and epoch in SHM

10.2.3

Time period, frequency, phase, phase difference and epoch in SHM

The time period $T$ is the time taken to complete one full oscillation. Since one complete revolution on the reference circle corresponds to $\omega t=2\pi$ , setting $t=T$ gives $\omega T=2\pi$ , so $T=2\pi/\omega$ . The displacement can then be written as $y(t)=A\sin\!\left(\dfrac{2\pi}{T}t\right)$ ; replacing $t$ by $t+T$ leaves the sine unchanged since $\sin(\theta+2\pi)=\sin\theta$ , confirming $y(t+T)=y(t)$ -- the formal definition of a periodic function. Frequency $f$ is the number of complete oscillations per second, related to the period by $f=1/T$ , SI unit s $^{-1}$ or hertz (Hz). Angular frequency $\omega$ (cycles per second expressed in radians) is related to $f$ by $\omega=2\pi f$ , SI unit rad s $^{-1}$ . The phase of a vibrating particle at any instant, $\varphi=\omega t+\varphi_0$ , completely specifies its position and direction of motion relative to the mean position; the general displacement equation is $y=A\sin(\omega t+\varphi_0)$ . At $t=0$ , the phase equals $\varphi_0$ , called the epoch (initial phase), and $\varphi_0$ itself is the angle of epoch. For two SHMs of the same $\omega$ but different epochs, $y_1=A\sin(\omega t+\varphi_1)$ and $y_2=A\sin(\omega t+\varphi_2)$ , the phase difference is the constant $\Delta\varphi=\varphi_2-\varphi_1$ . Differentiating the displacement eq …

Figure 10.11The phase of a vibrating particle at two instants of time

What this figure shows. A sinusoidal displacement-versus-time curve is drawn with two marked instants, t = 0 (where the curve sits at height A sin(phi_i), the epoch) and a later instant t_i (where the curve sits at height A sin(phi(t_i))). Arrows indicate the phase angle at each instant, phi_i at t = 0 and phi(t_i) = omega t_i + phi_i later, together with the amplitude envelope +A to -A and the period 2 pi/omega marked along the time axis, illustrating that the phase is simply the running angle argument of the sine function that pins down exactly where in its cy …

Misc Example 10.5Heart beat frequency from time period

Worked out. A nurse reports a patient's average heartbeat time period as 0.8 s, and the question asks for this rate expressed as beats per minute. Frequency is the reciprocal of the time period, so f = 1/T = 1/0.8 = 1.25 per second. Since one minute contains 60 seconds, converting from beats-per-second to beats-per-minute means multiplying by 60: f = 1.25 x 60 = 75 beats per minute, a normal resting heart rate, showing the everyday usefulness of the simple T-to-f conversion for a genui …

Misc Example 10.6Amplitude, angular frequency and phase from a given SHM equation

Worked out. Three different SHM equations are given -- y = 0.3 sin(40 pi t + 1.1), y = 2 cos(pi t), and y = 3 sin(2 pi t - 1.5) -- and the task is to read off amplitude, angular frequency, frequency, time period, and initial phase directly from each, by matching each equation to the standard form y = A sin(omega t + phi0). For the first, A = 0.3 units, omega = 40 pi rad/s, so f = omega/(2 pi) = 20 Hz and T = 1/f = 0.05 s, with initial phase phi0 = 1.1 rad. For the second, A = 2 units, omega = pi rad/s, f = 0.5 Hz, T = 2 s, and phi0 = 0 rad (a pure cosine has zero phase relative to the cosine reference). For the third, A = 3 units, omega = 2 pi rad/s, f = 1 Hz, T = 1 s, and phi0 = 1.5 rad, reading each constant straight off …

Misc Example 10.7Phase differences between displacement, velocity and acceleration

Worked out. Starting from displacement y = A sin(omega t), the task is to show that the phase difference between displacement and velocity is pi/2 (90 degrees), between velocity and acceleration is also pi/2, and between displacement and acceleration is pi (180 degrees). Differentiating once gives velocity v = A omega cos(omega t), which can be rewritten as A omega sin(omega t + pi/2) using the cosine-to-sine identity, so velocity leads displacement by exactly pi/2. Differentiating again gives acceleration a = -A omega^2 sin(omega t), rewritten as A omega^2 sin(omega t + pi) using sin(theta+pi) = -sin(theta), so acceleration leads velocity by a further pi/2, and therefore leads displacement itself by pi/2 + pi/2 = pi, i.e. is exactly out of step with displacement (always pointing the opposite way when the sign co …

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