Answer to Question #173356 in Classical Mechanics for Gergana

Question

A ball weighing 20g, attached to a spring, performs a harmonic oscillation with a frequency of 10 Hz and an amplitude of A = 10cm.
1. Find the maximum force with which the spring acts on the ball.
2. Find the speed of the ball as it passes through the equilibrium position.
3. Find the mechanical energy of the spring pendulum

Answer 1

1) Let's first find the spring constant of the spring:

$$f=\dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}},$$ $$k=4\pi^2f^2m=4\pi^2\cdot(10\ Hz)^2\cdot0.02\ kg=79\ \dfrac{N}{m}.$$

Then, we can find the maximum force with which the spring acts on the ball:

$$F_{max}=kA=79\ \dfrac{N}{m}\cdot0.1\ m=7.9\ N.$$

2) The speed of the ball as it passes through the equilibrium position can be found as follows:

$$v_{max}=A\omega=2\pi fA=2\pi\cdot10\ Hz\cdot0.1\ m=6.28\ \dfrac{m}{s}.$$

3) We can find the mechanical energy of the spring pendulum as follows:

$$E=\dfrac{1}{2}kA^2=\dfrac{1}{2}\cdot79\ \dfrac{N}{m}\cdot(0.1\ m)^2=0.395\ J.$$