Source: Principles of Physics
Problem
A stone of mass $m$ is attached to the bottom of a vertical spring and set vibrating. If the maximum speed of the stone is $v_{max}$ and the period is $T$, find the following.
- The spring constant of the spring.
- The amplitude of the motion.
- The frequency of oscillation.
[Q1] $k = \frac{4\pi^2 m}{T^2}$ [Q2] $A = \frac{v_{max}T}{2\pi}$ [Q3] $f = \frac{1}{T}$
The frequency $f$ is the reciprocal of the period $T$.
$$f = \frac{1}{T}$$The angular frequency $\omega$ is related to the period and frequency.
$$\omega = 2\pi f = \frac{2\pi}{T}$$For a mass-spring system, the angular frequency is also given by $\omega = \sqrt{k/m}$. We can solve for the spring constant $k$.
$$\left(\frac{2\pi}{T}\right)^2 = \frac{k}{m} \implies k = m\left(\frac{2\pi}{T}\right)^2 = \frac{4\pi^2 m}{T^2}$$In simple harmonic motion, the velocity is given by $v(t) = -A\omega \sin(\omega t + \phi)$. The maximum speed $v_{max}$ occurs when the sine term is $\pm 1$, so $v_{max} = A\omega$. We can solve for the amplitude $A$.
$$A = \frac{v_{max}}{\omega} = \frac{v_{max}}{2\pi/T} = \frac{v_{max}T}{2\pi}$$