Mechanics
Simple Harmonic Motion (HL)
5th Year · 6th Year (Leaving Cert)
- ✓By the end of this lesson students will be able to define Simple Harmonic Motion (SHM).
- ✓By the end of this lesson students will be able to state and apply the defining equation of SHM, a = -ω²x.
- ✓By the end of this lesson students will be able to derive and apply the formula for the period of a simple pendulum.
- ✓By the end of this lesson students will be able to derive and apply the formula for the period of a mass-spring system.
- ✓By the end of this lesson students will be able to describe and calculate the energy transformations in SHM.
Key concepts
Simple Harmonic Motion is a type of periodic motion where the restoring force acting on an object is directly proportional to its displacement from the equilibrium position and acts in the opposite direction to the displacement. This results in an acceleration that is directly proportional to the displacement and always directed towards the equilibrium position. The motion is oscillatory and repetitive.
Angular frequency (omega) is a measure of the rate of oscillation in SHM. It is related to the frequency (f) and period (T) of the oscillation. It represents the number of radians swept out per unit time.
The period (T) of a simple pendulum is the time taken for one complete oscillation. For small angles of swing (typically less than 10-15 degrees), the motion approximates SHM. The period depends on the length of the pendulum (l) and the acceleration due to gravity (g), but not on the mass of the bob or the amplitude.
The period (T) of a mass-spring system is the time taken for one complete oscillation of a mass attached to an ideal spring. The motion is SHM. The period depends on the mass (m) attached to the spring and the spring constant (k), which is a measure of the spring's stiffness.
In ideal Simple Harmonic Motion, the total mechanical energy (sum of kinetic energy and potential energy) remains constant. Energy continuously transforms between kinetic energy (maximum at equilibrium, zero at maximum displacement) and potential energy (zero at equilibrium, maximum at maximum displacement). For a spring, potential energy is stored in the stretched or compressed spring.
Key facts to remember
- 1Simple Harmonic Motion (SHM) is defined by the equation a = -ω²x, where 'a' is acceleration, 'ω' is angular frequency, and 'x' is displacement from equilibrium.
- 2The negative sign in a = -ω²x indicates that the acceleration is always directed towards the equilibrium position, opposite to the displacement.
- 3Angular frequency (ω), frequency (f), and period (T) are related by ω = 2πf = 2π/T.
- 4The period of a simple pendulum is T = 2π√(l/g), valid for small angles of swing.
- 5The period of a mass-spring system is T = 2π√(m/k), where 'k' is the spring constant.
- 6In ideal SHM, the total mechanical energy (Kinetic Energy + Potential Energy) is conserved.
- 7Kinetic energy is maximum at the equilibrium position (x=0) and zero at maximum displacement (x=±A).
- 8Potential energy is zero at the equilibrium position (x=0) and maximum at maximum displacement (x=±A).
Worked examples
Example 1
A particle undergoes SHM with an acceleration of -1.8 m s⁻² when its displacement from the equilibrium position is 0.2 m. Calculate the angular frequency and the period of the motion.
Answer
Angular frequency (ω) = 3 rad s⁻¹, Period (T) ≈ 2.09 s.
The negative sign in the SHM equation indicates that acceleration is always directed opposite to displacement.
Example 2
A simple pendulum has a length of 0.8 m. Calculate its period of oscillation. (Take g = 9.8 m s⁻²)
Answer
The period of oscillation (T) ≈ 1.80 s (to 2 decimal places).
This formula is valid for small angles of swing only.
Example 3
A mass of 0.5 kg is attached to a spring and oscillates with a period of 1.2 s. Calculate the spring constant (k) of the spring.
Answer
The spring constant (k) ≈ 13.7 N m⁻¹ (to 1 decimal place).
Ensure units are consistent (kg, s, N m⁻¹).
Example 4
A 0.2 kg mass oscillates on a spring with an amplitude of 0.1 m and an angular frequency of 5 rad s⁻¹. Calculate the total energy of the system and the kinetic energy when the displacement is 0.06 m.
Answer
Total energy (E) = 0.025 J, Kinetic energy (KE) when x = 0.06 m is 0.016 J.
The total energy remains constant throughout the oscillation, assuming no damping.
Common mistakes
- ✗Forgetting the negative sign in the defining equation of SHM (a = -ω²x), which is crucial for indicating the direction of acceleration.
- ✗Confusing angular frequency (ω) with frequency (f) or using incorrect units (e.g., degrees instead of radians for ω in calculations).
- ✗Applying the simple pendulum formula T = 2π√(l/g) for large angles of swing, where the motion is not truly SHM.
- ✗Incorrectly rearranging formulas, especially when solving for variables under a square root or in the denominator.
- ✗Not understanding the energy transformations in SHM, leading to errors in calculating KE or PE at different points in the oscillation.
Exam tips
- ★Always start by writing down the relevant formula before substituting values. This helps in earning partial marks even if there's a calculation error.
- ★Pay close attention to units. Ensure all quantities are in SI units (metres, kilograms, seconds) before performing calculations.
- ★When asked to derive a formula, show all intermediate steps clearly and logically, explaining any assumptions made (e.g., small angles for pendulum).
- ★Practise qualitative descriptions of SHM, such as where velocity and acceleration are maximum or minimum, and how energy transforms.
Ready to practise?
Try a problem on this topic
Snap a photo or type a question — get step-by-step working instantly.