Question

A scientist recorded the movement of a pendulum for 12 s. The scientist began recording when the pendulum

was at its resting position. The pendulum then moved right and left several times. The pendulum took 4 s to swing to the right and the left and then return to its
resting position. The pendulum's furthest distance to either side was 8 in. Graph the function that represent
the pendulum's displacement as a function of time. I need the equation

Answer 1

The displacement of the pendulum as a function of time in simple harmonic motion is d(t) = 8 sin(πt / 2) with an amplitude of 8 inches and a period of 4 seconds.

Step-by-step explanation:

The motion of a pendulum is a classic example of simple harmonic motion (SHM). When graphing the displacement of the pendulum as a function of time, we can describe this movement with a sinusoidal function. Given that the furthest distance to either side is 8 inches and it takes 4 seconds for a complete to-and-fro motion, which is the period (T) of the pendulum, the displacement (d) as a function of time (t) can be expressed with the following equation:

d(t) = 8 sin(πt / 2)

This equation indicates that the pendulum starts from its resting position (displacement equals zero) and reaches a maximum displacement of 8 inches. As the pendulum swings, it passes through the resting position again at t = 2 seconds, reaches the opposite maximum displacement at t = 3 seconds, and returns back to the starting position at t = 4 seconds, completing a cycle.

The amplitude of the displacement is 8 inches, the period T is 4 seconds, and we use π to account for the sinusoidal shape of the motion because the pendulum moves back and forth in a periodic manner. The factor of π/2 ensures that the period of the sine function matches the period of the pendulum's motion.