Question

3. A scientist recorded the movement of a pendulum for 10 s. The scientist began recording when the pendulum was at its resting position. The pendulum then moved right (positive displacement) and left (negative displacement) 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 6 in. Graph the function that represents the pendulum’s displacement as a function of time. (a) Write an equation to represent the displacement of the pendulum as a function of time. (b) Graph the function.

Answer 1

The pendulum starts being timed at equilibrium position, this suggests you use a sine function since sin(0)=0
The time it takes to make one full revolution is what's called the period, represented by a T. You are given T= 4 sec. The distance from equilibrium is the amplitude, A of the function which is given to be 6 in.

General wave function:

f(t) = A*sin( t*2pi/T - b) + c

A = |amplitude| ; distance from the midpoint
T = period ; time it takes to make one full revolution
b = shifts the graph right/left
c = shifts the graph up/down

This graph lies right on the axis so there will be no need to shift the graph


f(t) = 6*sin(t*pi/2)

y-axis will be units of inches and x-axis, seconds

Answer 2

The function for displacement of the pendulum is a sinusoidal function

in which the obtained from the given amplitude and period of motion.

Response:

• The equation to represent the displacement is; $x = 6 \cdot sin \left(\dfrac{\pi}{2} \cdot t \right)$

• Please find attached the graph of the function created with MS Excel

How is displacement function of the pendulum obtained?

The duration the scientist records the movement of the pendulum = 10 s

The position of the pendulum at the start = The resting position

The time it took the pendulum to complete a cycle, T = 4 s

Furthest distance of the pendulum from on either side, A = 6 in.

(a) The motion of the pendulum is a sinusoidal motion with the general

function presented as follows;

x = A·sin(ω·t + ∅)

Where;

$T = \mathbf{\dfrac{2 \cdot \pi}{ \omega}}$

Which gives;

$\omega = \dfrac{2 \cdot \pi}{4} = \mathbf{ \dfrac{\pi}{2}}$

A = 6

At t = 0, x = 0, therefore;

A·sin(ω × 0 + ∅) = 0

Which gives;

∅ = 0

The equation to represent the displacement of the pendulum as a

function of time is therefore;

• $\underline{x = 6 \cdot sin \left(\dfrac{\pi}{2} \cdot t \right)\pi }$

(b) The graph of the function created with MS Excel by using the Formula

and Equation functions is attached'

Learn more about sinusoidal function here:

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