Energy Changes Over a Cycle in SHM
In simple harmonic motion (SHM), energy continuously transforms between kinetic energy and potential energy.
HintThe total energy of the system remains constant, assuming no energy is lost to friction or other resistive forces.
Kinetic and Potential Energy Variations
- At Maximum Displacement ($x = \pm x_0$):
- Velocity is zero, so kinetic energy ($E_K$) is zero.
- Potential energy ($E_P$) is at its maximum: $$E_P = \frac{1}{2} k x_0^2$$
- At Equilibrium Position ($x = 0$):
- Velocity is maximum, so kinetic energy is at its maximum: $$E_K = \frac{1}{2} m v_{\text{max}}^2$$
- Potential energy is zero.
- At Intermediate Positions ($0 < x < x_0$):
- The system has both kinetic and potential energy.
- Total energy ($E_T$) is the sum of both: $$E_T = E_K + E_P = \frac{1}{2} k x_0^2$$
| Position | Velocity | Kinetic energy ($E_k$) | Potential energy ($E_p$) |
|---|---|---|---|
| Max displacement | 0 | 0 | Max ($\frac{1}{2}kA^2$ or $\frac{1}{2}m\omega^2 A^2$) |
| Equilibrium | Max | Max | 0 |
| Intermediate | Medium | Intermediate | Intermediate |
Consider a mass-spring system with amplitude $x_0 = 0.5 \, \text{m}$ and spring constant $k = 200 \, \text{N m}^{-1}$.
- Total energy: $$E_T = \frac{1}{2} k x_0^2 $$ $$= \frac{1}{2} \times 200 \times (0.5)^2 = 25 \, \text{J}$$
- At $x = 0.3 \, \text{m}$:
- Potential energy: $$E_P = \frac{1}{2} k x^2 = \frac{1}{2} \times 200 \times (0.3)^2 = 9 \, \text{J}$$
- Kinetic energy: $$E_K = E_T - E_P = 25 - 9 = 16 \, \text{J}$$
- Always keep in mind graphs of energy transformations.
- Potential energy forms a parabola, while kinetic energy is an inverted parabola.
- The total energy is a horizontal line, representing conservation of energy.
Phase Representation of SHM
The phase angle ($\phi$) is a crucial concept in SHM, providing insight into the timing of oscillations.
Understanding Phase Angle ($\phi$)
- Phase angle ($\phi$) determines the starting point of the oscillation.
- It is measured in radians and shifts the displacement-time graph horizontally.
- If $\phi = 0$, the motion starts at the equilibrium position.
- If $\phi = \frac{\pi}{2}$, the motion starts at maximum displacement.
- Think of the phase angle as the starting position in a race.
- If two runners start at different points on a circular track, their positions are out of phase, even if they run at the same speed.
Phase Difference
- Phase difference ($\Delta \phi$) describes how out of sync two oscillations are.
- 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.
- If two oscillations have a time difference of $0.25 \, \text{s}$ and a period of $1.0 \, \text{s}$, the phase difference is: $$
\Delta \phi = \frac{0.25}{1.0} \times 2\pi = \frac{\pi}{2}
$$ - This means the oscillations are 90° out of phase.
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:
$$
x = x_0 \sin(\omega t + \phi)
$$
- $x_0$: Amplitude (maximum displacement).
