Question

2) A mass-on-a-spring oscillates with an amplitude of 45.0 cm. It has a maximum acceleration of 2.84 m's at the end points of the motion. What is the period of oscillation? What is the spring constant if the mass is 2.00 kg?

Answer 1

For an oscillation, acceleration is

$\\a = - \omega^2 x$

where $\omega$ is the angular frequency and is the displacement

T

and

$\\\omega = \frac{2 \pi}{T}$......where t is the period

Given, maximum displacement is amplitude(at the ends of motion)

Given, maximum acceleration is

we have

$a_{max} = \omega^2 A$

$\\\omega^2 = \frac{a_{max}}{A} \\\\\omega^2 = \frac{2.84}{0.45} \\\\\omega^2 = 6.31 \\\\\omega = \sqrt{6.31} \\\\\omega = 2.512 \ rad/s$

that is,

$\\\frac{2 \pi}{T} = 2.512 \\\\T = \frac{2 \pi}{2.512} \\\\T = 2.501 \ s$ ......period of oscillation

Given, mass is $\\m = 2 \ kg$

Let    be the spring constant,

We have,

$\\\omega = \sqrt{\frac{k}{m}} \\\\\omega^2 = \frac{k}{m} \\\\k = m \omega^2 \\\\k = 2 * 2.512 \\\\k = 5.024 \ N/m$ .......spring constant