A particle performing linear S.H.M.
of period 2π seconds about the mean
position O is observed to have a speed
of b 3 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. [Ans: π/3
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The time required by the particle, to travel a further distance b is π/3ω seconds.
Given:
Time period = 2π/ω s = T
Speed = bω√3 m/s = v
To find:
The time required to travel the distance 'b'.
Explanation:
The velocity of the particle is given by the formula:
v = ω√(A² - x²)
Where,
x = Distance = b
Now,
bω√3 = ω√(A² - b²)
3b²ω² = ω²(A² - b²)
3b² = A² - b²
4b² = A²
∴ A = 2b
The time taken by the particle to travel from mean position is given as:
x = A sinωt
b = 2b sinωt
sinωt = 1/2 ⇒ ωt = sin⁻¹ 1/2
ωt = π/6
∴ t = π/6ω s
The time taken by the particle to reach mean position is given as:
t = T/4 - π/6ω
t = π/2ω - π/6ω
∴ t = 2π/6ω = π/3ω s
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