Question

An object hangs from a spring in a stable (equilibrium) position. The spring is pulled 3 feet downward and the object begins to oscillate, making one complete oscillation every 4 seconds. What is an equation of the motion of this object

Answer 1

The equation of the motion of this object is  $\mathbf{y = 3 cos \Big [ \dfrac{ \pi}{2}(t -2) \Big ] }$

What is oscillation?

An oscillation in a waveform is a periodic set of vibrations an object in the medium produces.

From  the parameters given:

• The spring is pulled 3 feet down ward from the origin(equilibrium position) = -3
• Time to complete one oscillation = 4 seconds

The angular velocity can be computed by using the formula:

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

$\mathbf{\omega = \dfrac{2 \pi}{4.0}}$

The amplitude A from the oscillation is:

$\mathbf{\Big|A \Big| = \Big|\dfrac{3 - (-3)}{2} \Big| = 3}$

It implies that the vertical shift (D) is:

$\mathbf{\dfrac{minimum + maximum}{2} = \dfrac{3+(-3)}{2} = 0}$

However, by using the formula of the general form of a sinusoidal equation, we have:

$\mathbf{y = A cos \Big [ \dfrac{2 \pi}{4}(t -2) \Big ] + 0}$

$\mathbf{y = 3 cos \Big [ \dfrac{ \pi}{2}(t -2) \Big ] }$

Learn more about oscillation in waveform here:

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