Question

A pendulum clock is a clock that uses a swinging weight (a pendulum) as its

time-keeping element. At t=0 s, the pendulum is 5 cm to the right of the
resting position (+5 cm). At t= 2 s, the pendulum is 5 cm to the left of the
resting position (-5 cm).
a) Determine the equation of the sinusoidal function that describes the distance
of the pendulum from its resting position in terms of time.

Answer 1

Answer: x(t) = 5cm*cos(t*pi/2s)

Explanation:

This is a sinusoidal equation, so we can write this as:

x(t) = A*cos(c*t + p) + B

where B is the axis around the movement, as the resting position is x = 0, we have B = 0

so x(t) = A*cos(c*t + p)

A is the amplitude of the oscilation, c is the frequency and p is a phase.

We know that when t = 0s, we have x(2s) = 5cm

if this is the maximum displacement, then knowing that the maximum of the cosine is cos(0) = 1

then we must have that p = 0

x(0s) = A*cos(0) = 5cm

then we have A = 5cm

Now, when t = 2s, we have:

x(2s) = 5cm*cos(2s*c) = -5cm

then 2s*c is the minimum of the cos(x) function, this is:

cos(pi) = -1

then 2s*c = pi

c = pi/2s.

then our function is:

x(t) = 5cm*cos(t*pi/2s)