Worked example 11.1: Piston in steam engine

Question: A piston in a stream engine executes simple harmonic motion. Given that the maximum displacement of the piston from its centre-line is $\pm 7 {\rm cm}$, and that the mass of the piston is $4 {\rm kg}$, find the maximum velocity of the piston when the steam engine is running at 4000rev./min. What is the maximum acceleration?

Answer: We are told that the amplitude of the oscillation is $a=0.07 {\rm m}$. Moreover, when converted to cycles per second (i.e., hertz), the frequency of the oscillation becomes

$$f = \frac{4000}{60} = 66.6666 {\rm Hz}.$$

Hence, the angular frequency is

$$\omega = 2 \pi f = 418.88 {\rm rad./sec}.$$

Consulting Tab. 4, we note that the maximum velocity of an object executing simple harmonic motion is $v_{\rm max} = a \omega$. Hence, the maximum velocity is

$$v_{\rm max} = a \omega = 0.07\times 418.88 = 29.32 {\rm m/s}.$$

Likewise, according to Tab. 4, the maximum acceleration is given by

$$a_{\rm max} = a \omega^2 = 0.07 \times 418.88\times 418.88 = 1.228\times 10^4 {\rm m/s^2}.$$