Question

Consider the simplified single-piston engine in Figure CQ15.13. Assuming the wheel rotates with constant angular speed, explain why the piston rod oscillates in simple harmonic motion.

Short answer

The yoke and piston move with simple harmonic motion.

Step-by-step solution

Step 1: The general equation of position

The general equation of position is described by

$x=A\cos \left(\theta t+\varphi \right)$

A is the amplitude of the wave

$\theta$is the angular amplitude

t is the time

$\varphi$is the phase angle

Reasoning

The angle of the crank pin is $\theta =\omega t$. Its x coordinate is $x=A\cos \theta =A\cos \omega t$, where Ais the distance from the center of the wheel to the crank pin. This is of the form $x=A\cos \left(\theta t+\varphi \right)$, so the yoke and piston move with simple harmonic motion.