• Two examples of simple harmonic oscillators are:
  • A simple pendulum
  • A mass-spring system
• Considering equations related to the restoring force leads to expressions for the?time periods?of these scenarios
  • These relationships are useful for investigating simple harmonic motion experimentally

For a pendulum, the restoring force is provided by the component of the bob's weight that is perpendicular to the tension in the pendulum's string. For a mass-spring system, the restoring force is provided by the force of the spring

Time Period of a Simple Pendulum

• A simple pendulum is a type of simple harmonic oscillator
  • The pendulum consists of a string and a bob (a weight, generally spherical) at the end
• An oscillating pendulum can be modelled as simple harmonic motion when the angle of oscillation is small

Forces on a pendulum when it is displaced. Assuming θ < 10°, the small angle approximation can be used to describe the time period of a simple pendulum such as this.

• The restoring force,?F, which returns an oscillating pendulum bob to the equilibrium position is:

F = ?mg?sin?θ

• Where:
  • m?= the mass of the pendulum bob (kg)
  • g?= acceleration due to gravity (m s^?2)
  • θ?= angle between the bob and the vertical (°)

• Using Newton’s Second Law:

F?=?ma?= ?mg?sin?θ

• Both sides can be divided by?m?to give an expression for the acceleration,?a:

a?= ?g?sin?θ

• In this case, the small-angle approximation can be used, this is where?sin?θ???θ
  • This is assuming the angle the pendulum makes with the vertical is less than 10°
• The expression for acceleration then becomes:

a?= ?gθ

• The displacement,?x, is equal to the length of the arc made by the bob,?x?=?Lθ
  • Where?L?= length of the pendulum (m)
• Rearranging this for?θ?and substituting it into the acceleration equation gives:

• This equation shows:
  • For small values of?x, the condition for SHM is satisfied as restoring force,?F?is proportional to ?x
  • For large values of?x, the acceleration of a simple pendulum is not proportional to the displacement

• Using the defining equation of SHM:

Time Period of a Mass–Spring System

• A mass-spring system is another type of simple harmonic oscillator
• The restoring force,?F, which returns a mass-spring system to its equilibrium position is given by:

F?=??kx

• • x?= extension of the spring (m)
  • k?= spring constant (N m^?1)Where:

• Using Newton’s Second Law:

F?=?ma = ?kx

• Rearranging for the acceleration,?a:

• This equation applies for both a?horizontal?or?vertical?mass-spring system

A mass-spring system can be either vertical or horizontal. The time period equation applies to both.

• The equation shows that:
  • The?higher?the spring constant?k, the?stiffer?the spring and hence, the?shorter?the time period of oscillation
  • The time period (and hence, frequency) of a mass-spring system is independent of the force of gravity

• A consequence of this is that oscillations would have the?same time period?on Earth and the Moon

Step 1: List the known quantities

• • Length of the pendulum,?L?= 80 cm = 0.8 m
  • Acceleration due to gravity,?g?= 9.81 m s^?2

Step 2: Write down the relationship between angular frequency,?ω, and period,?T

Step 3: Write down the equation for the time period of a simple pendulum

• • This equation is valid for this scenario since the maximum angle of displacement is less than 10°

Step 4: Equate the two equations and rearrange for?ω

Step 5: Substitute the values to calculate?ω

= 3.50 rad s^?1

Step 6: State the final answer to the correct number of significant figures

ω?= 3.5 rad s^?1

Step 1: Write down the known quantities

• • Mass,?m?= 2.0 kg
  • Spring constant,?k?= 0.9 N m^?1