An object is attached to a coiled spring. The object begins at its rest position at seconds. It is then propelled downward. Write an equation for the distance of the object from its
rest position after seconds, if the amplitude is 2 inches and the period is 3 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 Weston Osborne. 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 downward. Write an equation for the distance of the object from its
rest position after seconds, if the amplitude is 2 inches and the period is 3 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 Weston Osborne. in the United States

Nov 04,2024

UpStudy AI · Accepted Answer

The equation for the distance $$ d $$ of the object from its rest position in simple harmonic motion is given by:

$$ d(t) = A \sin(\frac{2\pi}{T}t) $$

where:
- $$ A $$ is the amplitude of the motion,
- $$ T $$ is the period of the motion,
- $$ t $$ is the time elapsed.

Given:
- Amplitude $$ A = 2 $$ inches,
- Period $$ T = 3 $$ seconds.

Substitute the values of $$ A $$ and $$ T $$ into the equation to find the equation for the distance $$ d $$ of the object from its rest position after $$ t $$ seconds.
Simplify the expression by following steps:
- step0: Solution:
  $$2\sin\left(\frac{2\pi }{3}t\right)$$
The equation for the distance $$ d $$ of the object from its rest position after $$ t $$ seconds is:

$$ d(t) = 2\sin\left(\frac{2\pi}{3}t\right) $$

Therefore, the exact equation for the distance of the object from its rest position after $$ t $$ seconds is $$ 2\sin\left(\frac{2\pi}{3}t\right) $$.
The equation for the distance $$ d $$ of the object from its rest position after $$ t $$ seconds is $$ 2\sin\left(\frac{2\pi}{3}t\right) $$.

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