(455) HL SHM phase angle and more detailed equations [IB Physics HL]

Simple Harmonic Motion

• Definition: Motion that repeats in a regular cycle.

Key Equations

• Acceleration: ( a = -\omega^2 x )

  • Describes relation between acceleration and displacement.

  • Can be solved as a differential equation.

• Displacement: ( x = x_0 \cdot \sin(\omega t + \phi) )

  • ( T ): time (seconds)

  • ( x ): displacement (meters)

  • ( x_0 ): amplitude (maximum displacement, meters)

  • ( \omega ): angular frequency (radians/second)

  • ( \phi ): phase angle (radians)

Graphical Representation

• Displacement vs Time graph is a sine curve.

  • Amplitude: Maximum height of the sine curve.

  • Angular Frequency: Stretches or shrinks the period of the sine curve.

  • Phase Angle: Shifts the sine curve left or right.

Phase Angle ( \phi )

• Measures where in the cycle the oscillation starts (in radians).

  • One complete cycle = ( 2\pi ) radians.

Examples

• ( \phi = 0 ):

  • ( x = x_0 \cdot \sin(\omega t) )

  • Graph starts at the origin.

• ( \phi = \frac{\pi}{4} ):

  • Graph shifted left by ( \frac{\pi}{4} ).

Transformations and Scaling

• Shifting the Graph:

  • Left movement of ( \phi ) leads to a leftward phase shift.

  • Each phase shift corresponds to a scaling of the complete cycle in radians.

Derivatives and Motion Equations

• Velocity:

  • Derivative of position gives velocity: ( v = \frac{d}{dt}(x_0 \sin(\omega t + \phi)) )

  • Velocity equation: ( v = \pm \omega \sqrt{x_0^2 - x^2} )

• Acceleration:

  • Formula derived from displacement: ( a = -\omega^2 x )

  • Can be expressed as ( a = -\omega^2 (x_0 \sin(\omega t + \phi)) ).

Conclusion

• Understanding simple harmonic motion involves familiarity with key concepts, equations, and graphical interpretations.