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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?
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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
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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