A simple pendulum 22 inches in length is oscillating in harmonic motion. The bob at the end of the pendulum swings through an arc of 20 inches (from the far left to the far right, or one-half cycle) in about 0.4 seconds. What is the equation model for this harmonic motion?

Question

Quizplus · Accepted Answer

The equation for simple harmonic motion of a pendulum is given by: T = 2π * sqrt(L/g), where T is the period, L is the length of the pendulum and g is the acceleration due to gravity.

Using the given information, we can calculate the period of the pendulum as follows:

T = 2 * 0.4 = 0.8 seconds

Next, we can use the formula for the period to solve for g:

g = (4 * π^2 * L) / T^2

Substituting the given values, we get:

g = (4 * 3.1416^2 * 22) / 0.8^2

g = 96.19 inches/second^2

Now that we know g, we can write the equation for the harmonic motion of the pendulum as:

y = A * cos(2π * t / T)

where A is the amplitude and t is the time. To find A, we use the fact that the bob swings through an arc of 20 inches. Since the amplitude is half of the total arc length, we get:

A = 20 / 2 = 10 inches

Thus, the equation for the harmonic motion of the pendulum is:

y = 10 * cos(2π * t / 0.8)

which simplifies to:

Answer: C
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Note: The equation for the harmonic motion of a pendulum can also be written in terms of the period as y = A * cos(2π * t / (2L/g)). This would give the same answer as above.

A Simple Pendulum 22 Inches in Length Is Oscillating in Harmonic