The displacement of a harmonic oscillator is given by y = 8.4 sin(14t), where the units of y are meters and t is measured in seconds. What is its maximum velocity?
y(t) = 8.4 sin(14t) m
v(t)=dy/dx = 8.4*14 cos(14t) m/s
max. v(t) occurs when cos(14t)=1, or
max v(t)=8.4*14=117.6 m/s
Thank you!
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To find the maximum velocity of a harmonic oscillator, we need to find the derivative of the displacement function and evaluate it at the point where the displacement is at its maximum.
The displacement function is given by y = 8.4 sin(14t).
Now, let's find the derivative of y with respect to time (t):
dy/dt = 8.4 * 14 * cos(14t)
To find the maximum velocity, we need to find when the derivative is equal to zero.
Setting dy/dt = 0:
8.4 * 14 * cos(14t) = 0
Now, solve for t:
cos(14t) = 0
To find the values where cosine is equal to zero, we can use the inverse cosine function (also known as arccosine or cos^(-1)):
14t = arccos(0)
cos^(-1)(0) = π/2
Therefore, 14t = π/2
Solving for t:
t = π/2 * (1/14)
t ≈ 0.1124 seconds
Now, we have found the time (t) at which the maximum velocity occurs. To find the maximum velocity, substitute this value back into the derivative equation:
dy/dt = 8.4 * 14 * cos(14 * 0.1124)
dy/dt ≈ 8.4 * 14 * cos(1.5746)
dy/dt ≈ -147.3437 meters per second
Therefore, the maximum velocity of the harmonic oscillator is approximately -147.3437 meters per second.