Question

In this problem, we study the motion of a pendulum that slows down! This is called damped harmonic motion. You start by pulling the pendulum to an angle of 1 radian and dropping it. The length of the pendulum is l = 10
meters. After 1 period, the pendulum swings to a maximum of 0.8 radians. The pendulum’s motion is modeled by y = A cos(t).The amplitude is a decreasing function modeled by an exponential:
A = C e^(r)(t) Using the given information, find the values of C and r ! go!

Answer 1

Ok by amplitude I am assuming the height the pendulum reaches in the vertical direction.
angle is zero means amplitude is zero
Ok initial values. t=0, A = C
From the fact the angle is 57.3 degrees...i prefer degrees.
the height from the top of pendulum is 10*cos(57.3) = 5.4
10 - 5.4 = 4.6
thus A = C = 4.6
Now 1 period implies t = 2pi
the angle is now 45.84 degrees
the height from the top of pendulum is 10*cos(45.84) = 6.97
10 - 6.97 = 3.03
A = 4.6*e^(2pi*r) = 3.03
Solve for r
e^(2pi*r) = .659
2pi*r = -.417
r = -.066
Now we have equation for A
\[A = 4.6e^{-.066t}\]