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Question
A particle performing linear S.H.M. of period 2π seconds about the mean position O is observed to have a speed of `"b" sqrt3` m/s, when at a distance b (metre) from O. If the particle is moving away from O at that instant, find the time required by the particle, to travel a further distance b.
Solution
Data:
T = 2πs, v = `"b" sqrt3` m/s at x = b
∴ ω = `(2pi)/"T" = (2pi)/(2pi)` = 1 rad/s
v = `ω sqrt("A"^2 - "x"^2)`
∴ At x = b,
`"b" sqrt3 = (1) sqrt("A"^2 - "b"^2)`
∴ 3b2 = A2 − b2
∴ A = 2b
∴ Assuming the particle starts from the mean position, its displacement is given by x = A sin ωt = 2b sin t
If the particle is at x = b at t = t1,
b = 2b sint1
∴ t1 = sin−1 `1/2` = `pi/6`s
Also, with period T = 2πs, on travelling a further distance b the particle will reach the positive extremity at time t2 = `π/2`s.
∴ The time taken to travel a further distance b from x = b is
t2 − t1 = `pi/2 - pi/6`
= `(3pi - pi)/6`
= `(2pi)/6`
= `pi/3`s
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