Question

A piston in a gasoline engine is in simple harmonic motion. The engine is running at the rate of 3600 rev/min. Taking the extremes of its position relative to its center point as 65.00cm, find the magnitudes of the
(a) the maximum velocity
(b) the maximum acceleration of the piston.

Answer 1

$x=Asin(\omega t+\varphi),$

$v=A\omega cos(\omega t+\varphi), ~v_{max}=A\omega,$

$a=-A\omega^2 sin(\omega t+\varphi), ~a_{max}=A\omega ^2.$

$A=65 ~cm=0.65 ~m,$

$n=3600~\frac{rev}{min}=60~\frac{rev}{s},$

$\omega=2\pi n=2\pi \cdot 60=120\pi~\frac{rad}{s}.$

a)

$v_{max}=0.65\cdot 120\pi=245~\frac ms,$

b)

$a_{max}=0.65\cdot 120^2\cdot \pi^2=92380 ~\frac{m}{s^2}.$