11.1 Simple Harmonic Motion

Simple Harmonic Motion

Key Concepts

• Oscillatory Motion: Movement of an object back and forth between two opposing points.

• Periodic Motion: Any motion that repeats in a regular cycle.

• Equilibrium: The position where the net force on an object is zero.

Hooke’s Law

• F_{elastic} = -kx

• Where:

  • F_{elastic} is the spring force (restoring force).

  • k is the spring constant (stiffer springs have higher k values).

  • x is the displacement from the equilibrium position.

• Hooke's Law states that the force needed to extend or compress a spring by some distance is proportional to that distance

Mass-Spring System

• The direction of the spring force is always opposite to the displacement from equilibrium.

• When the spring is unstretched, the spring force and mass's acceleration are zero.

• At maximum compression or extension, the spring force and mass's acceleration reach a maximum.

• Mechanical energy is conserved in an ideal mass-spring system.

Simple Pendulum

• A simple pendulum consists of a mass (bob) attached to a fixed string.

• The restoring force is the component of the bob's weight tangent to its motion. F_{g,x} = F_{g}\sin\theta

• For small angles (<15°), the pendulum's motion approximates simple harmonic motion because F_{g,x} is proportional to the displacement.

Energy in Simple Harmonic Motion

• In a mass-spring system, potential energy is elastic.

• In a simple pendulum, potential energy is gravitational and is defined as zero at the lowest point of the swing.

• Mechanical energy is conserved in an ideal (frictionless) pendulum system.

Conditions for Simple Harmonic Motion

• Oscillation about an equilibrium position.

• Restoring force proportional to displacement.