11 OSCILLATORY MOTION 11.6 Uniform circular motion
±motion simultaneously along both the x- and the y -axes. Note, however, that
these two motions are 90 ◦ (i.e., π/2 radians) out of phase. Moreover, the am-
plitude of the motion equals the radius of the circle. Clearly, there is a close
relationship between simple harmonic motion and circular motion.
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 7 cm, and
that the mass of the piston is 4 kg, find the maximum velocity of the piston when
the steam engine is running at 4000 rev./min. What is the maximum accelera-
tion?
Answer: We are told that the amplitude of the oscillation is a = 0.07 m. Moreover,
when converted to cycles per second (i.e., hertz), the frequency of the oscillation
becomes
f =Hence, the angular frequency is
4000
60= 66.6666 Hz.ω = 2 π f = 418.88 rad./sec.Consulting Tab. 4 , we note that the maximum velocity of an object executing
simple harmonic motion is vmax = a ω. Hence, the maximum velocity is
vmax = a ω = 0.07 × 418.88 = 29.32 m/s.Likewise, according to Tab. 4 , the maximum acceleration is given by
amax = a ω^2 = 0.07 × 418.88 × 418.88 = 1.228 × 104 m/s^2.Worked example 11.2: Block and spring
Question: A block attached to a spring executes simple harmonic motion in a
horizontal plane with an amplitude of 0.25 m. At a point 0.15 m away from the