Oscillations

Oscillations occur in many areas of physics with simple harmonic motion (shm) a special case of oscillation that appears in various natural phenomena.

• Oscillations involve the interchange between kinetic and potential of energy.
• In Simple harmonic motion (SHM) the acceleration is proportional to the displacement from equilibrium but in the opposite direction:
  $a = −\omega^2x$ where $\omega$ is the angular frequency. $\omega$ has units of rad per second.
• This acceleration is caused by a restoring force that is always directed towards the mean position and is also proportional to the displacement from that mean position.
  
• Displacement is a vector quantity which measures the position of a point relative to its equilibrium position.
• The maximum displacement is called the amplitude.
• SHM consists of periodic oscillations with a period $\rm (T)$ that is independent of the amplitude: $\rm T = 2\pi/\omega$
  
  In this example $\rm T = 20~s$ and the amplitude $\rm = 2.0~cm$.
• The frequency $(f)$ of an oscillation is the number of cycles per second. $f= 1/\rm T$
  As $\mathrm T = 1/f =2\pi/\omega$. This gives $\omega = 2\pi f$
• The displacement can be described by a sin or cos curve depending on the initial displacement.
  For example, with an initial displacement of $0$, an amplitude $0 =1$ and $\omega=0.20$:
  

  $x = x_0\sin \omega t$

  For this curve $\rm T = 2\pi/\omega$ $= 2\pi/0.2$ $= 10 \pi$ $\rm = 31.4~s$
• The velocity can be determined from the slope of the displacement-time graph. In this example it is a maximum at $t = 0$ and so follows a cosine curve.
  

  $v = v_0 \cos\omega t$ Note: $v_0 = 0.20$ $v = \omega x_0\cos\omega t$

  It should be noted that $v_0 = 0.20 = \omega x_0 = 0.2 \times 1.0$
• The acceleration can be determined from slope of the velocity-time graph.
• In this example it is a minimum at $t = 0$ and so follows a -sin curve.
  

  $a = -a_0\sin\omega t$ Note: $a_0 = 0.04$ $a = -\omega^2 x_0 \sin\omega t$

  It should be noted that $a_0 = 0.04 = 0.20^2 \times 1.0 = \omega^2 x_0$

• In SHM, kinetic energy gets transformed to potential energy and vice versa.
• The kinetic energy of a mass m that undergoes SHM is given by:
  $\mathrm{E_K}=1 / 2 m v^{2} \quad \mathrm{E}_{\mathrm{K}}=1 / 2 m \omega^{2} x_{0}^{2} \cos ^{2}(\omega t)$. (When $x=0$ at time $t=0$)
  For example, with an initial displacement of $0$, an amplitude $x_0 = 1$ and $\omega = 0.20$ as before, and $a$ mass $m = 10~\rm kg$
  
  Note the kinetic energy is always positive with the time to undergo a full cycle $\rm = 15.7 = T/2$
• The maximum kinetic energy is therefore $\mathrm{E_k^{max}} = ½m\omega^2x_0^2$. This is also the total energy of the system.
• In this case $\mathrm{E_k^{max}} = ½ \times 10 \times 0.2^2 \times 1 = 0.2~\rm J$
  
  The potential energy is $\mathrm{E_\mathrm{P}} = 1 / 2 m \omega^{2} x_{0}^{2}\left(1 - \cos ^{2}(\omega t)\right)$ $=1 / 2 m \omega^{2} x_{0}^{2} \sin ^{2}(\omega t)$ $=m \omega^{2} x^{2}$
  
• The velocity in SHM is given by $v = \pm\omega\sqrt{(x_0^2 − x^2)}$.
  It should be noted that when $x = \pm x_0 v = 0$ and when $x = 0$, $v = \pm\omega x_0$.

• Simple pendulum: A mass hanging on a string oscillates with SHM since the force increases as the mass moves away from the equilibrium position.
  
  The restoring force $\rm = T \sin \theta$ where $\rm T$ is the tension in the string.
• Mass on a spring: The tension in the spring increases as the mass is pulled down so the acceleration is proportional to the displacement. The force is in the opposite direction to the extension.
  
• Two bodies oscillating with the same frequency are in phase if they travel through the maximum at the same time. Phase difference is measured by phase angle.
  

  The bodies are completely out of phase if the phase angle $= \pi$.
  The bodies have a phase difference $= \pi/2$. The red curve follows a cosine curve and the blue curve a sin curve.