IB Physics Topic C.1.3 Energy Transformations in Oscillations Revision Notes (HL)

IB Physics Topic C.1.3 Energy Transformations in Oscillations Revision Notes (HL) | RevisionDojo

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Energy Changes Over a Cycle in SHM

In simple harmonic motion (SHM), energy continuously transforms between kinetic energy and potential energy.
This section quantifies those energy changes; the qualitative picture is introduced in C.1.1.

Hint

The 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 ($x = \pm x_0$):
1. Velocity is zero, so kinetic energy ($E_K$) is zero.
2. Potential energy ($E_P$) is at its maximum, taking equilibrium as the zero reference: $E_P = \frac{1}{2} k x_0^2 = \frac{1}{2} m \omega^2 x_0^2$
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 ($E_P$) is zero at the equilibrium position, the chosen zero reference.
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: $$E_T = E_K + E_P = \frac{1}{2} k x_0^2 = \frac{1}{2} m \omega^2 x_0^2$$

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

Example

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: $$E_K = E_T - E_P = 25 - 9 = 16 \, \text{J}$$

Tip

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.

Example

If $\phi = 0$, the motion starts at the equilibrium position.
If $\phi = \frac{\pi}{2}$, the motion starts at maximum displacement.

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IB Physics C.1.3 Energy Transformations in Oscillations Notes

Energy Changes Over a Cycle in SHM

Kinetic and Potential Energy Variations

Phase Representation of SHM

Understanding Phase Angle ($\phi$)

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Energy Changes Over a Cycle in SHM

Kinetic and Potential Energy Variations

Phase Representation of SHM

Understanding Phase Angle ($\phi$)

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Topic A - Space, time and motion5 subtopics

Topic B - The particulate nature of matter5 subtopics

Topic C - Wave behaviour5 subtopics

Topic D - Fields4 subtopics

Topic E - Nuclear and quantum physics5 subtopics

IB Physics C.1.3 Energy Transformations in Oscillations Notes

Topic A - Space, time and motion5 subtopics

Topic B - The particulate nature of matter5 subtopics

Topic C - Wave behaviour5 subtopics

Topic D - Fields4 subtopics

Topic E - Nuclear and quantum physics5 subtopics

Energy Changes Over a Cycle in SHM

Kinetic and Potential Energy Variations

Phase Representation of SHM

Understanding Phase Angle ($\phi$)

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Energy Changes Over a Cycle in SHM

Kinetic and Potential Energy Variations

Phase Representation of SHM

Understanding Phase Angle ($\phi$)

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