Question

a horizontal position of the pendulum of a grandfather clock can be modelled by h(t)=Acos(2(pi)t/T), where A is amplitude of the pendulum in meters, t is time in seconds, and T is the period of the pendulum, in seconds. If the pendulum has a velocity of v(t)=sin Pi(t) find:
A) an equation for h(t)
B) a formula for acceleration of the pendulum at time t.
C) the maximum horizontal displacement
D) the time this displacement occurs.
E) the length of the pendulum, rounded to the nearest meter.

Answer 1

v = dh/dt = sin pi t

then h = - (1/pi)cos pi t
but h = A cos (2 pi t/T)
so
A = -1/pi
pi = 2 pi /T so T = 2
so in the ean
h =-(1/pi) cos pit
= -(1/pi)cos(2 pi t/2)

B)
h =-(1/pi) cos pit
v = dh/dt = sin pi t
a = dv/dt = pi cos pi t

C)1/pi

D) cos 0 = 1 so at t = 0 , 1 , 2 etc

E) T = 2 pi sqrt(L/g)
here T = 2
2 = 2 pi sqrt (L/g)
sqrt (L/g) = 1/pi
L/g = 1/pi^2)
L = g/pi^2