Question

1. The graph shows the distance y, in centimeters, a pendulum moves to the left (negative displacement) and right (positive displacement), for a given number of seconds x. (a) From its resting position, how long does it take the pendulum to swing one direction, then the other, and then return to its resting position? (b) What is the pendulum’s maximum displacement? (c) What percent of a full cycle is completed within 1 s? (d) How is the pendulum moving during the time period from x = 5 to x = 5.625? (e) What is the equation of the graph?

Answer 1

The graph is shown in the picture attached.

A) In order to calculate how long it takes the pendulum to swing one direction, then the other, and then return to its resting position, you need to look at the graph and see where the function returns to zero after going once up and once down. This time is called period (T), which is defined as the time taken to perform one complete oscillation.
In our case, T = 5 s (see the second picture attached for reference).

B) In order to find the pendulum's maximum displacement, you need to look at the graph and see how high and low on the y-axis the function goes. This displacement is called amplitude (A), which is defined as the maximum displacement from the resting position in either direction.
In our case, A = 20 cm (see the second picture attached for reference).

C) Since the period of the pendulum is 5 seconds and it swings constantly, the fraction of a cycle completed in 1 second is calculated by simply dividing
p = 1 / 5 = 0.20
Therefore, in 1 second the pendulum completes 20% of its cycle.

D) During the time period from x = 5 to x = 5.625, we can see from the graph that the pendulum has just started its second oscillation, therefore it starts from the resting position and it moves towards the right (positive values of y)decreasing its velocity.

E) The equation of the graph is a sinusoidal function, therefore it will have an equation like
$f(t) = A \cdot sin( \frac{2 \pi}{T} \cdot t - b) + c$

where:
A = amplitude
T = period
b and c = shifting constants

In order to find b and c, we recall that the graph starts at (0,0) and therefore it is not shifted either horizontally or vertically, and therefore we can infer that b = c = 0.

Substituting the values, we find:
$f(t) = 20 \cdot sin( \frac{2 \pi}{5} \cdot t)$