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Question
A particle performs linear S.H.M. of period 4 seconds and amplitude 4 cm. Find the time taken by it to travel a distance of 1 cm from the positive extreme position.
Solution
Given:
T = 4 s, A = 4 cm = 0.04 m,
x = 1 cm from extreme position = 4 − 1 = 3 cm = 0.03 m
To find: Time taken (t)
Formula: x = A sin (ωt + Φ)
Calculation:
Particle starts from a positive extreme position.
∴ Φ = `pi/2`
From formula,
x = `"A" sin((2pi"t")/"T" + Φ)` ........`(∵ ω = (2pi)/"T")`
∴ 3 = 4 sin`((2pi"t")/4 + pi/2)`
∴ cos`((2pi)/4)"t" = 3/4` ......`[∵ sin(pi/2 + θ) = cosθ]`
∴ `(pi/2)"t" = cos^-1(3/4)`
∴ t = `2/pi xx 41.1^circ xx pi/180 = 41.4/90 = 0.46 "s"`
Time taken by it to travel a distance of 1 cm from the positive extreme position is 0.46 s.
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