When a car hits a certain bump on the road, a shock absorber on the car is compressed a distance of 6 in., then
released. The shock absorber vibrates in damped harmonic motion with a frequency of 2 cycles per second. The damping constant for this particular shock absorber is 2.8.
How long does it take for the amplitude of the vibration to decrease to 0.5 in.?
Why did the shock absorber go to therapy?
Because it was experiencing some "damped" emotions!
Now, to answer your question, we can use the equation for damped harmonic motion:
A(t) = A(0) * e^(-ζωn*t) * cos(ωd*t)
Where:
A(t) = Amplitude at time t
A(0) = Initial amplitude
ζ = Damping constant
ωn = Natural frequency of the system
t = Time
ωd = Damped frequency
We know that A(t) = 0.5 in., A(0) = 6 in., ζ = 2.8, and ωd = 2π * 2 cycles/s = 4π rad/s.
By plugging in these values and rearranging the equation, we can solve for the time t:
0.5 = 6 * e^(-2.8 * ωn * t) * cos(4π * t)
Unfortunately, we don't have the natural frequency (ωn) of the system. Without that information, it's impossible to determine the time it takes for the amplitude to decrease to 0.5 in.
But hey, at least the shock absorber's emotions are well-damped!