SOLUTION: A pendulum is connected to a rope 3 m long, which is connected to a ceiling 4 m high. The angle between its widest swing and vertical hanging position is π/3. If the pendulum swin

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Question 1205131: A pendulum is connected to a rope 3 m long, which is connected to a ceiling 4 m high. The angle between its widest swing and vertical hanging position is π/3. If the pendulum swings out to its widest position in 2 seconds, model the horizontal displacement of the pendulum using a sinusoidal function, considering vertical to be x = 0.


Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
a rope 3m+long
a ceiling 4m high

so, from the floor to the pendulum is 1m+=> minimum height
the vertical distance to the ceiling is :
h=3m%2Acos%28alpha%29+
since the highest angle is just alpha=+pi%2F3+and cos%28pi%2F3%29+=+1%2F2, the smallest distance to the ceiling is h=3m%2A%281%2F2%29+=+1.5m+

and
the highest point is h%5B1%5D=1m%2B1.5m=2.5m from the ground

then, cosine function will be:
h%28t%29=D-Acos%28Bt%29
range of the pendulum's height above the ground is
1%3C=+h%28t%29+%3C=+2.5, where
h%280%29=1
h%282%29=2.5
h%284%29=1
amplitude is A=%282.5-1%29%2F2=0.75
midline is D=%281%2B2.5%29%2F2=1.75
the period is T=4+=>B=2pi%2FT=2pi%2F4=pi%2F2
and, function is:
h%28t%29=1.75-0.75%2Acos%28%28pi%2F2%29%2At%29
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