Topic 7.3 Representing and Analyzing SHM: write the sinusoidal position, velocity and acceleration of an oscillator, relate their amplitudes and phases, and read the motion from graphs and initial conditions.

What this topic is asking

The College Board (Topic 7.3) wants you to write the sinusoidal position, velocity and acceleration of an oscillator, to relate their amplitudes and phases, and to read the motion from graphs and initial conditions. This is the analysis side of SHM: given $x(t) = A\cos(\omega t + \phi)$, differentiate to get $v$ and $a$, and interpret where the speed and acceleration are largest.

The three sinusoidal functions

All three quantities oscillate sinusoidally at the same angular frequency $\omega$, but with different amplitudes and phases. The position has amplitude $A$; the velocity has amplitude $A\omega$; the acceleration has amplitude $A\omega^2$. Each factor of $\omega$ comes from a differentiation. This is the calculus heart of analyzing an oscillator: you are handed one of the three functions and asked to produce the others by differentiating (or integrating).

Maximum speed and acceleration

From the amplitudes of the three functions, the maximum speed and maximum acceleration are

These occur at different points in the cycle. The speed is greatest at the equilibrium position ($x = 0$), where all the energy is kinetic; the acceleration is greatest at the extremes ($x = \pm A$), where the restoring force (and hence $a = -\omega^2 x$) is largest. So when the oscillator races through the center it has zero acceleration, and when it pauses at a turning point it has maximum acceleration. Reading off these maxima from the amplitude and frequency is a standard exam task.

Phase relationships

The three functions are shifted in phase. Because the velocity is the sine (the derivative of the cosine), it leads the position by a quarter cycle ($\pi/2$): the velocity peaks when the position is zero. The acceleration is $-\omega^2 x$, exactly out of phase ($\pi$) with the position: it is most negative when the displacement is most positive. Visualizing these phase relationships, position and acceleration mirror-imaged, velocity a quarter-turn ahead, lets you sketch all three graphs from any one of them.

Reading amplitude and phase from initial conditions

The constants $A$ and $\phi$ are not part of the physics of the oscillator; they encode how it was started. If the oscillator is released from rest at maximum displacement $x_0$, then $A = x_0$ and $\phi = 0$ (pure cosine). If it is launched from equilibrium with speed $v_0$, then $\phi = \pm\pi/2$ (a sine) and $A = v_0/\omega$. In general you evaluate $x(0)$ and $v(0)$ from the expressions and solve the two equations for $A$ and $\phi$. The amplitude can also be found from $A = \sqrt{x_0^2 + (v_0/\omega)^2}$, which combines the initial position and velocity.

Try this

Q1. An oscillator has amplitude $0.15$ m and angular frequency $8.0$ rad/s. Calculate its maximum speed and maximum acceleration. [2 points]

• Cue. $v_{max} = A\omega = 1.2$ m/s; $a_{max} = A\omega^2 = 9.6$ m/s squared.

Q2. State where in its cycle an SHM oscillator has zero velocity and maximum acceleration. [2 points]

• Cue. At the extremes of the motion ($x = \pm A$), where the displacement (and restoring force) is largest.