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Question
A needle of a sewing machine moves along a path of amplitude 4 cm with a frequency of 5 Hz. Find its acceleration `(1/30)` s after it has crossed the mean position.
Solution
Data: A = 4 cm, n = 5 Hz, t = `(1/30)` s, a = - ω2 x, |a| = ω2x
as x = A sin ω t
∴ |a| = ω2 A sin ω t
`= (2 π " n")^2 "A" sin (2 π " n") xx 1/30`
`= 4 pi^2 "n"^2 " A" sin (2 pi 5) 1/30`
`= 4pi^2 xx 25 xx 4 sin (10 pi)/30`
`= 100 pi^2 xx 4 sin pi/3`
`= 100pi^2 xx cancel(4)^2 xx sqrt3/cancel(2)_1`
`= 100 pi^2 xx 2 xx 1.732`
`= 100 xx pi^2 xx 3.464`
`= 346.4 xx 9.87`
= 3420 cm/s2
= 3420 × 10-2 m/s2
|a| = 34.2 m/s2
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