(455) Simple harmonic motion SHM basics [IB Physics SL/HL]

Introduction to Simple Harmonic Motion (SHM)

• SHM often confuses students despite its name.

• SHM is not just periodic motion; periodic motion repeats (e.g., clock, heartbeats).

• SHM has a specific definition relating acceleration and displacement.

Definition of SHM

• Acceleration (a) is proportional to displacement (x) and has a negative sign:

  • a ∝ -x

  • Units for acceleration: m/s²

  • Units for displacement: meters

Example of SHM: Pendulum

• A pendulum swinging back and forth exhibits SHM.

• Key aspects:

  • Acceleration is proportional to displacement from equilibrium.

  • Graph of acceleration vs. displacement is linear with a negative slope.

Key Graphs

• Graph of acceleration vs. displacement:

  • Linear graph through the origin with a negative slope.

  • Serves as a visual definition of SHM.

Relation of Force and Acceleration

• According to Newton's Second Law, force is related to acceleration:

  • Displacing the pendulum creates force opposite to the direction of displacement.

  • Greater displacement leads to greater acceleration and force,

  • Positive displacement = positive acceleration.

  • Negative displacement = negative acceleration.

Other Examples of SHM: Mass on a Spring

• System consists of mass on a frictionless surface with springs.

• When stretched and released, the mass exhibits SHM:

  • Equilibrium position is defined as x.

  • Maximum displacement is when potential energy is highest.

Energy in SHM

• Potential energy (PE) is stored when displaced:

  • Maximum potential energy at maximum displacement (x₀).

  • Kinetic energy (KE) is maximum at equilibrium (v is maximum here).

• Formulas:

  • KE = 1/2 mv²; maximum KE at x = 0.

Graphs of Energy vs. Displacement

• Potential Energy (PE) graph:

  • Zero at equilibrium and maximum at maximum displacement.

  • Smooth curve transitioning between these points.

• Kinetic Energy (KE) graph:

  • Maximum at equilibrium (v max = max KE) and zero at maximum displacement.

• Total Energy (TE):

  • Sum of PE and KE remains constant over time.

Graphs of Energy vs. Time

• For energy with respect to time, starting point influences each curve:

  • If starting at maximum displacement:

    • PE is maximum at t=0 and decreases to zero over time.

    • KE starts at zero and reaches maximum.

• Both PE and KE oscillate over time, maintaining total energy.