An object is attached to a coiled spring. The object begins at its rest position at seconds. It is then propelled dowrnward. Write an equation for the distance of the objed from its
rest position after seconds, if the amplitude is 8 inches and the period is 2.5 seconds.
The equation for the distance d of the object from its rest position is .
(Type an exact answer, using as needed. Use integers or fractions for any numbers in the equation.)

Ask by Ford Page. in the United States

Nov 04,2024

Question

An object is attached to a coiled spring. The object begins at its rest position at seconds. It is then propelled dowrnward. Write an equation for the distance of the objed from its
rest position after seconds, if the amplitude is 8 inches and the period is 2.5 seconds.
The equation for the distance d of the object from its rest position is .
(Type an exact answer, using as needed. Use integers or fractions for any numbers in the equation.)

Ask by Ford Page. in the United States

Nov 04,2024

UpStudy AI · Accepted Answer

To write the equation for the distance $$ d $$ of the object from its rest position after $$ t $$ seconds, we can use the cosine function, which is appropriate for modeling the motion of an object attached to a spring.

Given:
- Amplitude $$ A = 8 $$ inches
- Period $$ T = 2.5 $$ seconds

The general form of the cosine function for this scenario is:

$$
d(t) = A \cos\left(\frac{2\pi}{T} t + \phi\right)
$$

where:
- $$ A $$ is the amplitude,
- $$ T $$ is the period,
- $$ \phi $$ is the phase shift (which we will determine based on the initial conditions).

Since the object starts at its rest position and is propelled downward, we can assume that at $$ t = 0 $$, the object is at its maximum displacement (which corresponds to the amplitude). Therefore, we can set the phase shift $$ \phi = 0 $$.

Now, we need to calculate $$ \frac{2\pi}{T} $$:

$$
\frac{2\pi}{T} = \frac{2\pi}{2.5} = \frac{2\pi}{\frac{5}{2}} = \frac{2\pi \cdot 2}{5} = \frac{4\pi}{5}
$$

Now we can substitute the values into the equation:

$$
d(t) = 8 \cos\left(\frac{4\pi}{5} t\right)
$$

Thus, the equation for the distance $$ d $$ of the object from its rest position after $$ t $$ seconds is:

$$
d = 8 \cos\left(\frac{4\pi}{5} t\right)
$$
The equation for the distance $$ d $$ of the object from its rest position after $$ t $$ seconds is $$ d = 8 \cos\left(\frac{4\pi}{5} t\right) $$.

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