Final answer:
Correct option; B. 5π/6 units.
The displacement equation for a particle in simple harmonic motion is x(t) = X cos (ωt + φ). To find the velocity when the displacement is 3 units, substitute x = 3 into the displacement equation and solve for t. Then substitute the value of t into the velocity equation to find the velocity.
Step-by-step explanation:
The displacement of a particle in simple harmonic motion is given by the equation x(t) = X cos (ωt + φ), where X is the amplitude, ω is the angular frequency, t is the time, and φ is the phase angle.
In the given equation, x = 5sin(4t−π/6). From this equation, we can identify that the amplitude (X) is 5, the angular frequency (ω) is 4, and the phase angle (φ) is -π/6.
To find the velocity when the displacement is 3 units, we can differentiate the equation for displacement with respect to time.
The velocity function is given by v(t) = -ωX sin (ωt + φ).
Substituting the given values, we have v(t) = -4(5)sin(4t−π/6).
Now, to find the velocity when the displacement is 3 units, we substitute x = 3 into the displacement equation and solve for t.
3 = 5sin(4t−π/6)
Solving this equation, we find t ≈ 0.5426
Finally, substituting this value of t into the velocity equation, we can find the velocity:
v(0.5426) ≈ -4(5)sin(4(0.5426)−π/6)
v(0.5426) ≈ -5π/6 units