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A particle moves on the x-axis according to the equation x = x0 sin2ωt. The motion is simple harmonic
  • a)
    with amplitude x0
  • b)
    with amplitude 2x0
  • c)
    with time period 2π/ω
  • d)
    with time period π/ω
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
A particle moves on the x-axis according to the equation x = x0 sin2&o...
(ωt), where x0 is the amplitude of the motion and ω is the angular frequency.

To find the velocity of the particle, we need to take the derivative of the position equation with respect to time:

dx/dt = x0 (2ω) cos(2ωt)

This gives us the velocity of the particle as a function of time. We can also find the maximum velocity by taking the absolute value of the derivative and setting it equal to zero:

|dx/dt| = x0 (2ω) |cos(2ωt)| = 0

cos(2ωt) = 0

2ωt = (2n+1)π/2, where n is an integer

t = (2n+1)π/4ω

Substituting this value of t back into the velocity equation gives us the maximum velocity:

vmax = |dx/dt|max = x0 (2ω) |cos(π/4)| = x0 √2 ω

Therefore, the velocity of the particle is given by:

v(t) = x0 (2ω) cos(2ωt)

and the maximum velocity is:

vmax = x0 √2 ω
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A particle moves on the x-axis according to the equation x = x0 sin2&o...
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A particle moves on the x-axis according to the equation x = x0 sin2ωt. The motion is simple harmonica)with amplitude x0b)with amplitude 2x0c)with time period 2π/ωd)with time periodπ/ωCorrect answer is option 'D'. Can you explain this answer?
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