An object is aftached to a coiled spring. The object is pulled down (negative direction from the rest position) 8 centimeters, and then released. Write an equation for the distance of
the object from its rest position, after seconds if the amplitude is 8 centimeters and the period is 5 seconds.
The equation for the distance 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 Owen Hanson. in the United States

Nov 04,2024

Question

An object is aftached to a coiled spring. The object is pulled down (negative direction from the rest position) 8 centimeters, and then released. Write an equation for the distance of
the object from its rest position, after seconds if the amplitude is 8 centimeters and the period is 5 seconds.
The equation for the distance 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 Owen Hanson. 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 formula for simple harmonic motion:

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

where:
- $$ A $$ is the amplitude,
- $$ T $$ is the period,
- $$ \phi $$ is the phase shift.

Given:
- The amplitude $$ A = 8 $$ cm,
- The period $$ T = 5 $$ seconds,
- The object is pulled down (negative direction) 8 cm, which means it starts at the maximum negative displacement.

Since the object is released from the maximum negative position, the phase shift $$ \phi $$ will be $$ \pi $$ (or 180 degrees), because the cosine function starts at its maximum value, and we need it to start at its minimum value.

Now, substituting the values into the equation:

1. The angular frequency $$ \frac{2\pi}{T} = \frac{2\pi}{5} $$.
2. The phase shift $$ \phi = \pi $$.

Thus, the equation becomes:

$$
d(t) = 8 \cos\left(\frac{2\pi}{5} t + \pi\right)
$$

Using the property of cosine, $$ \cos(x + \pi) = -\cos(x) $$, we can simplify the equation:

$$
d(t) = 8 \left(-\cos\left(\frac{2\pi}{5} t\right)\right) = -8 \cos\left(\frac{2\pi}{5} t\right)
$$

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

$$
d(t) = -8 \cos\left(\frac{2\pi}{5} t\right)
$$
The equation for the distance $$ d $$ of the object from its rest position after $$ t $$ seconds is $$ d(t) = -8 \cos\left(\frac{2\pi}{5} t\right) $$.

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