Applications of Trigonometric Functions

Question

An object is attached to a coiled spring. The object begins at its rest position at t=0 seconds. It is then propelled downward. Write an equation for the distance of the object from its rest position after t seconds, if the amplitude is 6 inches and the period is 4.5 seconds.

The equation for the distance d of the object from it's rest position is:

Answer 1

d = a sin bt

is the equation for the dependence of distance from rest position (x), and time (t)

where a = the amplitude in inches, and b = 2π/T where T is the period in seconds

We are given that the amplitude is 6 inches and the period is 4.5 seconds

This means that a = 6 and b = 2π/4.5 in the equation above. So we have

d = 6 sin (2πt/4.5)

In the equation the value of d will fluctuate between a

1. minimum of –6 in and
2. maximum of 6 in

where positive or negative indicates which side of the rest position (above or below) the object is on.

If the question requires distance as a positive number then we just take the absolute value of d and the equation is changed to

d = 6 |sin (2πt/4.5)|

Answer

Depending what the question is asking, the answer is either

d = 6 sin (2πt/4.5)

or

d = 6 |sin (2πt/4.5)|