(a) Sketch a graph of \(x(t)=A \sin \omega t\) (the position of an object in SHM that is at the equilibrium point at \(t=0\) ). (b) By analyzing the slope of the graph of \(x(t),\) sketch a graph of $v_{x}(t) .\( Is \)v_{x}(t)$ a sine or cosine function? (c) By analyzing the slope of the graph of \(v_{x}(t),\) sketch \(a_{x}(t)\) (d) Verify that \(v_{x}(t)\) is \(\frac{1}{4}\) cycle ahead of \(x(t)\) and that \(a_{x}(t)\) is \(\frac{1}{4}\) cycle ahead of \(v_{x}(t) .\) (W) tutorial: sinusoids)

Question: Sketch the graphs for the position, velocity, and acceleration functions for simple harmonic motion. Verify the relationships between their phases. Answer: In simple harmonic motion, we have the position function \(x(t) = A \sin(\omega t)\). The graph of this function is a sine wave with amplitude A and period \(T = \frac{2\pi}{\omega}\). The velocity function is given by \(v_x(t) = A\omega \cos(\omega t)\), and the graph resembles a cosine wave with amplitude A\(\omega\). The acceleration function is \(a_x(t) = -A\omega^2 \sin(\omega t)\) which is equal to \(- \omega^2 x(t)\), having a similar graph as the position function but inverted and scaled by a factor of -\(\omega^2\). The relationships between the phases can be verified as follows: the velocity function \(v_x(t)\) is \(\frac{1}{4}\) cycle ahead of the position function \(x(t)\) because the cosine function reaches its maximum value \(\frac{1}{4}\) cycle ahead of the sine function. The acceleration function \(a_x(t)\) is \(\frac{1}{4}\) cycle ahead of the velocity function \(v_x(t)\) because a \(\sin(\omega t)\) function reaches its minimum (most negative) value at \(\frac{1}{4}\) cycle later than a \(\cos(\omega t)\) function reaching its maximum value.