At rest the pendulum of a grandfather clock is 8 inches abov

At rest, the pendulum of a grandfather clock is 8 inches above its base. At the highest point of the swing, it is 12 inches above th base. If it takes 3 seconds to complete a full swing starting from the highest point on the left and returning to the highest point on the left, find an equation that models the height of the pendulum above the base of the grandfather clock as a function of time.

Solution

Let the oscillation be
x=a sin w t.
velocity = dx/dt= aw cos wt. It is maximum when t=0.
maximum velocity v = sqrt(2 g h)
h is the height fall= 12-8= 4 in. g= 32 ft/s.
v= sqrt(2*32*4/12)=6.07.

We have
6.07 = a w. w=2 pi/t=2 pi/3. .

Therefore a= 6.07 *3/2/pi=2.9 ft, w=2.09
x= 2.9 sin 2.09 t, x is in feet. This is the model equation.