(i) simple harmonic motion; defining equation a x 2 ~ =-

The idea

Simple harmonic motion (SHM) is oscillation in which the acceleration is directly proportional to the displacement from equilibrium and always directed towards that equilibrium position. The defining equation is a = −ω²x, where a is acceleration in m s⁻², ω is angular frequency in rad s⁻¹ and x is displacement in m. The minus sign shows that acceleration and displacement are always in opposite directions, so the restoring acceleration points back towards equilibrium. For example, if ω = 4.0 rad s⁻¹ and x = 0.10 m, then a = −(4.0)² × 0.10 = −1.6 m s⁻², directed towards equilibrium. The proportionality a ∝ −x distinguishes SHM from other oscillations.

What earns the marks

• SHM is defined by acceleration directly proportional to displacement from equilibrium and directed towards equilibrium.
• The defining equation is a = −ω²x.
• The minus sign indicates that acceleration is always opposite in direction to displacement.
• Acceleration is measured in m s⁻², displacement in m and angular frequency in rad s⁻¹.
• The constant of proportionality between a and x is ω², so a graph of a against x is a straight line through the origin with negative gradient.
• SHM requires a restoring force or acceleration that always acts towards the equilibrium position.

Common mistakes and how to put them right

• Omitting the minus sign in a = −ω²x.

  Put it right: Without it the equation no longer shows that acceleration opposes displacement.

• Writing a = −ωx or a = −ω²x².

  Put it right: The correct relationship is a = −ω²x, with ω squared and x to the first power.

• Confusing acceleration with velocity.

  Put it right: In SHM the acceleration is maximum at the extremes of motion, where velocity is zero.

• Treating ω as frequency in hertz.

  Put it right: ω is angular frequency in rad s⁻¹, related to frequency by ω = 2πf.

Exam tips

• Quote a = −ω²x and explain the meaning of the minus sign when asked to define SHM.
• Check the sign of your calculated acceleration against the direction of displacement.
• Remember that a graph of a against x for SHM is a straight line through the origin with negative gradient.