Question

A mass oscillating up and down on the bottom of a spring (assuming perfect elasticity and no friction or air resistance) can be modeled as harmonic motion. If the weight is displaced a maximum of 6 cm, find the modeling equation if it takes 4 seconds to complete one cycle. Be to show, illustrate and explain your work.

Answer 1

The motion of the mass as it moves on the bottom of the spring is a

repetitive motion.

• $\mathrm{The \ motion \ of \ the \ mass \ attached \ to \ the \ spring \ is \ }\displaystyle d = 6 \cdot sin\left(\frac{\pi}{2} \cdot t \right)$

Reasons:

The general form of the equation of the simple harmonic motion of the

mass is d = a·sin(ω·t)

Where;

d = The distance of the mass from the rest position

a = The maximum displacement of the mass from the equilibrium position = 6 cm

ω = The frequency of rotation

t = The time of motion

ω = The frequency of rotation

$\displaystyle \omega = \mathbf{\frac{2 \cdot \pi}{T}}$

Where;

T = The time to complete one cycle (the period of oscillation) = 4 seconds

$\displaystyle \omega = \frac{2 \cdot \pi}{4} = \frac{\pi}{2}$

Combining the above values gives the modelling equation as follows;

$\displaystyle d = 6 \cdot sin\left(\frac{\pi}{2} \cdot t \right)$

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