C.1.3 Energy transformations in oscillations (HL only) Notes

Energy Changes Over a Cycle in SHM

In simple harmonic motion (SHM), energy continuously transforms between kinetic energy and potential energy.

Kinetic and Potential Energy Variations

1. At Maximum Displacement ($x = \pm x_0$):
   1. Velocity is zero, so kinetic energy ($E_K$) is zero.
   2. Potential energy ($E_P$) is at its maximum: $$E_P = \frac{1}{2} k x_0^2$$
2. At Equilibrium Position ($x = 0$):
   1. Velocity is maximum, so kinetic energy is at its maximum: $$E_K = \frac{1}{2} m v_{\text{max}}^2$$
   2. Potential energy is zero.
3. At Intermediate Positions ($0 < x < x_0$):
   1. The system has both kinetic and potential energy.
   2. Total energy ($E_T$) is the sum of both: $$E_T = E_K + E_P = \frac{1}{2} k x_0^2$$

Phase Representation of SHM

The phase angle ($\phi$) is a crucial concept in SHM, providing insight into the timing of oscillations.

Understanding Phase Angle ($\phi$)

1. Phase angle ($\phi$) determines the starting point of the oscillation.
2. It is measured in radians and shifts the displacement-time graph horizontally.

Phase Difference

1. Phase difference ($\Delta \phi$) describes how out of sync two oscillations are.
2. It is calculated using the formula: $$
   \Delta \phi = \frac{\Delta t}{T} \times 2\pi
   $$ where $\Delta t$ is the time difference between corresponding points on the oscillations and $T$ is the period.

Motion Equations in SHM

The motion of an object in SHM can be described using equations for displacement, velocity, and acceleration.

Displacement Equation

The displacement $x$ of an object in SHM is given by:

$$
x = x_0 \sin(\omega t + \phi)
$$

1. $x_0$: Amplitude (maximum displacement).
2. $\omega$: Angular frequency ($\omega = 2\pi f = \frac{2\pi}{T}$).
3. $t$: Time.
4. $\phi$: Phase angle.

Velocity Equation

The velocity $v$ in SHM is derived from the displacement equation:

$$
v = \pm \omega \sqrt{x_0^2 - x^2}
$$

Derivation of Velocity and Acceleration in SHM using Differentiation

As mentioned earlier, the motion of an object undergoing simple harmonic motion (SHM) can be described by the displacement equation:

$$x = x_0 \sin(\omega t + \phi)$$

1. To derive the velocity equation, we differentiate $x$ with respect to time: $$v = \frac{dx}{dt} = \frac{d}{dt} \left[ x_0 \sin(\omega t + \phi) \right]$$
2. Using the derivative of sine, $\frac{d}{dt} \sin(\theta) = \cos(\theta)$, and applying the chain rule: $$v = x_0 \omega \cos(\omega t + \phi)$$
3. This shows that velocity is 90° out of phase with displacement, meaning it is maximum when displacement is zero and zero when displacement is at its maximum.
4. To derive the acceleration equation, we differentiate velocity with respect to time: $$a = \frac{dv}{dt} = \frac{d}{dt} \left[ x_0 \omega \cos(\omega t + \phi) \right]$$
5. Since the derivative of cosine is $⁡-\sin$, we get: $$a = - x_0 \omega^2 \sin(\omega t + \phi)$$
6. Using $x = x_0 \sin(\omega t + \phi)$, we substitute: $$a = -\omega^2 x$$

This confirms that acceleration in SHM is directly proportional to displacement but acts in the opposite direction, meaning the motion always accelerates toward the equilibrium position.

Energy Relations in SHM

The total energy in SHM is constant and can be expressed in terms of kinetic and potential energy.

Total Energy

The total energy $E_T$ is given by:

$$
E_T = \frac{1}{2} m \omega^2 x_0^2
$$

This energy is conserved and remains constant throughout the motion.

Potential Energy

The potential energy $E_P$ at displacement $x$ is:

$$
E_P = \frac{1}{2} m \omega^2 x^2
$$

Kinetic Energy

The kinetic energy $E_K$ is the difference between the total energy and the potential energy:

$$
E_K = E_T - E_P = \frac{1}{2} m \omega^2 (x_0^2 - x^2)
$$