Reading x, v, and a

Topic 7.3 is about representing simple harmonic motion as a function of time and reading its derivatives. The position of an oscillator is $x(t)=A\cos(\omega t+\phi)$, where $A$ is the amplitude, $\omega$ the angular frequency, and $\phi$ the phase constant set by the initial conditions.

From that one function the velocity and acceleration follow by differentiation: $v(t)=-A\omega\sin(\omega t+\phi)$ and $a(t)=-A\omega^2\cos(\omega t+\phi)$. Each derivative multiplies by a factor of $\omega$ through the chain rule, so the speed peaks at $A\omega$ and the acceleration at $A\omega^2$.

Almost every mistake here breaks one of three habits: dropping the factor of $\omega$ when differentiating, mishandling the phase constant, or losing track of how the position, velocity, and acceleration line up in time.

The position function $x(t)=A\cos(\omega t+\phi)$ traces a cosine in time. Differentiating with respect to time gives the velocity $v(t)=-A\omega\sin(\omega t+\phi)$, and differentiating again gives the acceleration $a(t)=-A\omega^2\cos(\omega t+\phi)$.

Stacked on a common time axis the three curves share the same period but peak at different moments. The velocity peaks a quarter cycle before the position, and the acceleration is the position scaled by $\omega^2$ and flipped in sign, so it is exactly upside down relative to $x$.

One function generates all three: $x=A\cos(\omega t+\phi)$, $v=-A\omega\sin(\omega t+\phi)$, and $a=-A\omega^2\cos(\omega t+\phi)$.

Each time derivative of a cosine or sine in $\omega t$ brings down one factor of $\omega$ from the chain rule. Differentiating $x=A\cos(\omega t+\phi)$ once gives $v=-A\omega\sin(\omega t+\phi)$, with a single $\omega$; differentiating again gives $a=-A\omega^2\cos(\omega t+\phi)$, with $\omega^2$.

The peaks follow the same pattern: the maximum speed is $v_{\max}=A\omega$ and the maximum acceleration is $a_{\max}=A\omega^2$, so $a_{\max}=\omega\,v_{\max}$. The common slip is to differentiate the cosine into a sine but forget the $\omega$, leaving the speed as $A$, or to attach $\omega^2$ where one $\omega$ belongs.

Every derivative multiplies by $\omega$: $v_{\max}=A\omega$ and $a_{\max}=A\omega^2$. Drop that factor and the speed and acceleration come out wrong by powers of $\omega$.

The phase constant $\phi$ shifts the cosine along the time axis, and it is fixed by where the motion starts. At $t=0$ the position is $x(0)=A\cos\phi$ and the velocity is $v(0)=-A\omega\sin\phi$, so the starting position and the direction of motion together fix $\phi$.

Three common starts make the pattern clear: released from rest at $x=+A$ gives $\phi=0$; passing through equilibrium in the positive direction gives $\phi=-\pi/2$; passing through equilibrium in the negative direction gives $\phi=+\pi/2$. Setting $\phi=0$ by default is right only for the first of these, and reading $\phi$ from the position alone misses the sign that the direction of motion fixes.

The phase constant is set by the initial conditions: $x(0)=A\cos\phi$ and $v(0)=-A\omega\sin\phi$ together fix $\phi$. It shifts the motion in time and is not free to drop.

The three quantities reach their extremes at different points in the cycle. The speed is greatest at equilibrium, where the displacement is zero, and zero at the turning points, where the displacement is largest. The acceleration runs the opposite way: largest at the turning points and zero at equilibrium.

In timing terms, $v=A\omega\cos(\omega t+\phi+\pi/2)$ leads the position by a quarter cycle, and $a=-\omega^2 x$ is half a cycle out of phase, exactly opposite. Two facts keep it straight: the block is fastest where the restoring force is zero, and the acceleration is strongest where the block is momentarily at rest.

Velocity leads position by a quarter cycle; acceleration is exactly opposite it. Speed peaks at equilibrium, acceleration at the turning points.

Reading position, velocity, and acceleration off the same motion goes wrong in three recurring places.

• Dropping the factor of ωDifferentiating $x(t)$ without the chain-rule factor of $\omega$. Fix: each derivative multiplies by $\omega$, so $v_{\max}=A\omega$ and $a_{\max}=A\omega^2$.
• Mishandling the phase constantSetting $\phi=0$ by default or reading it from the position alone. Fix: $x(0)=A\cos\phi$ and $v(0)=-A\omega\sin\phi$ together fix $\phi$.
• Confusing the phase relationsMisplacing where $x$, $v$, and $a$ peak or how they line up. Fix: $v$ leads $x$ by a quarter cycle, and $a$ is exactly opposite $x$.

Keep the $\omega$ on every derivative, read the phase constant from both initial conditions, and remember that the velocity leads while the acceleration opposes. The whole topic comes down to those three habits.