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Island Physics

Simple Harmonic Motion

Dolphins jumping during a delivery trip across the Mediterrean.  Photo: Mark Roworth
Home >> AP Physics I >> Simple Harmonic Motion >> Mass on Spring
Rotational Mechanics >>

6.1 - Mass on Spring Oscillator

The YouTube clip shows a mass-spring oscillator alongside an object moving in a circle at constant angular speed. Note that the vertical position of both the objects is the same as viewed from the side.  At an advanced level we can use the similarities between SHM and circular motion to accurately model oscillations.

The objectives for the AP course are:
  • energy transfers and amplitude calculations
  • period and frequency calculations.
  • lab work to verify equations
Assignments 2018
A Mass on a Spring Oscillator
The energy that drives the oscillation of a spring comes from stretching the spring in the first place.  We met this back in Unit 5:  Work, Energy and Power as the elastic potential energy.  The object at the end of the spring achieves its maximum speed when all of this potential energy has been transferred to kinetic energy.  This also gives us a relationship between the stiffness of the spring (spring constant, \(k\)), the amplitude, \(x\), and the maximum speed, \(v\), of the mass.
​
From the principle of conservation of energy:
​\[EPE_{lost} = KE_{gained}\]
​\[\frac{1}{2}kx^{2}=\frac{1}{2}mv^{2}\]
\[​v = \sqrt{\frac{k}{m}}x\]

​\[KE_{ends}+EPE_{ends}= KE_{middle} +EPE_{middle}\]
Picture
The energy transfers for a spring oscillator.  Image from hyperphysics.  This page has a simulation that enables you to change parameters and see the effect on the oscillation.


The period, \(T\), of an oscillator is the time taken to complete one oscillation back and forth.  We will be carrying out a detailed lab on this as well as deriving the relationship from first principles.  Essentially the period (and hence the frequency) are governed by a) the mass being oscillated and b) the stiffness of the spring.  The frequency is the number of oscillations per second and is simply the reciprocal of the period.
\[​T = 2\pi \sqrt{\frac{m}{k}}\]
Derivation is based on the circular motion concept - see animation above:
\[v = \frac{2 \pi x}{T}\]
and from the above:
\[v = \sqrt{\frac{k}{m}} x\]
substituting and rearranging yields:
​\[​T = 2\pi \sqrt{\frac{m}{k}}\]


Other Resources

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  • HOME
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    • Solids, Liquids and Gases
    • Waves
    • Astrophysics
    • Electricity
    • Magnetism
    • Nuclear Physics
    • IGCSE revision
  • Physics I
    • Kinematics >
      • Variables and Units
      • Describing Motion
      • Equations of Motion
      • Problem Solving
      • Projectiles
      • Kinematics Animations
    • Dynamics >
      • Forces
      • Static Equilibrium
      • Newton's Laws
      • Friction
      • Advanced Problems
    • Gravitation and Orbits >
      • Circular Motion
      • Vertical Circles
      • Universal Gravitation
      • Orbits
      • Circular Motion Animation
    • Energy >
      • Work
      • Springs
      • PE and KE
      • Conservation of Energy
      • Work-Energy Theorem
    • Linear Momentum >
      • Impulse
      • Conservation of Momentum
      • Types of Collision
      • 2-D Collisions
    • Simple Harmonic Motion >
      • Mass on Spring
      • Pendulums
      • SHM Animations
    • Rotational Mechanics >
      • Torque
      • Rotational Kinematics
      • Rotational Dynamics
      • Angular Momentum
      • Rotational Energy
    • Mechanical Waves >
      • Waves on a String
      • Sound
    • AP-1 Revision
    • AP Physics C (Mechanics)
  • Physics II
    • Fluid Mechanics
    • Thermal Physics
    • Electrostatics
    • Magnetic Fields >
      • EM Induction
    • Interference and Diffraction
    • Optics
    • Modern Physics
    • AP 2 Revision
  • OCEANOGRAPHY
    • The World Ocean >
      • What is Oceanography
      • History
      • Lat and Long
      • Size and Origin
      • Plate Tectonics
    • Seawater >
      • A Salty Sea
      • Measuring Salinity
      • Thermal Properties
      • Density Profiles
      • Drinking Seawater
    • Circulation and Climate >
      • Global Heating
      • Coriolis Effect
      • Surface Currents
      • Vertical Motion
      • Thermohaline Circulation
      • El Nino
      • Carbon Cycle
    • Waves and Tides >
      • Wave Motion
      • Formation of Waves
      • Beaches
      • Tsunamis
      • Tides
    • Observation Systems >
      • Challenges
      • The CTD
      • Moorings
      • Sound Waves
      • Robotics
      • Satellites
    • Weather and Navigation >
      • Weather Systems
      • Weather Forecasting
      • Hurricanes
      • Navigation
      • Life at Sea
    • Oceans and Mankind >
      • Ocean Acidification
      • Pollution
      • Fish Stocks
      • Climate Change
      • Energy Resources
    • Atlantic Explorer Cruise
    • ROVs
  • COLLEGE
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