Period of Simple Harmonic Oscillators (Edexcel IAL Physics)

Period of a simple pendulum

Spec point — 145

• A simple pendulum is:

  • an object moving from side to side
  • attached to a string which is fixed to a point above

• The period of a simple pendulum can be calculated using the equation:

$T=2π\sqrt{\frac{l}{g}}$

• Where:

  • $l$ = the length of the pendulum
  • $g$ = the gravitational field strength on the planet on which the pendulum is set up

Worked Example A child is sitting on a swing that is 200 cm long. What is the period of oscillation?

Answer:

Step 1: Convert length to meters

200 cm = 2 m

Step 2: Substitute the correct values

T = 2π$\sqrt{\frac{l}{g}}$= 2π$\sqrt{\frac{2}{9.81}}$=2.84 s

Step 3: Confirm the answer

The time period of 1 oscillation of the swing is 2.84 s

Period of a Mass-Spring System

Spec point — 145

• A mass-spring system is:

  • an object moving up and down, or side to side
  • attached to the end of a spring

• The equation for the restoring force in SHM is the same as the equation for Hooke's law

$F=-kx$

• The time period T can be calculated using the equation:

$T=2π\sqrt{\frac{m}{k}}$

• Where:

  • $m$ = the mass of the object on the end of the spring
  • $k$ = the spring constant of the spring

Observing the Motion of a Mass-Spring System

• An experimental and graphical method can be used to observe the motion of a simple mass-spring system

  • Tie a pencil together with the mass and set the mass in free oscillations by displacing it downwards slightly

• The oscillations will move the pencil up and down

  • On a piece of graph paper, allow the pencil to trace the path of the oscillations by pulling the paper sideways as the mass-spring system oscillates up and down

• The oscillations will produce a curved, periodic graph

  • This will decrease in amplitude as the mass-spring system slows down

The motion of an oscillator can be observed through a simple mass-spring system