Simple Harmonic Systems

Test yourself

SHM Experiment (Mass-Spring System)

A mass suspended on the end of a spring is also an excellent example of SHM.

Equipment

Suspend a mass, m, from a spring as shown in the diagram.

Method

The formula for the period, T, is $T=2\pi \sqrt{\frac{m}{k}}$ where k is the spring constant of the spring.
Find the length of time for 10 complete oscillations, using the pointer against the vertical scale.

Method 2

Repeat this and find the average time for 10 oscillations. Divide that time by 10 to find the period, T.
The mass should be varied and the period remeasured for each value of m.
Goggles/safety specs should be worn because there is a spring under tension.

Getting quality data

The fiducial mark should be horizontal. Some form of pointer could be attached to the suspended mass.
You can use a small section of string between the clamp and the mass-hanger to reduce the effect of any side to side motion given to the mass in releasing it.

Obtaining quality data 2

You should check the mass, m, with a balance.
The extension should not be so large that the elastic limit of the spring is reached. The spring should not be compressed so much that it no longer provides a restoring force.
Keep the pointer close to but not touching the scale.
Observe the pointer from eye level. Do this to reduce parallax errors.

Analysis

You should plot a graph of T² against m to verify the equation.
You would expect to plot a straight line through the origin with a gradient $=\frac{4\pi^2}{k}$ .
The straight line might miss the origin because the effective mass of the spring has not been taken into account.
Plotted or given error bars would help to provide an estimate of the uncertainty in g by finding the worst acceptable line of fit.

SHM Experiment (Simple Pendulum System)

The simple pendulum is a classic experiment to demonstrate SHM.

Equipment

Suspend a pendulum bob by a thread from a clamp.
The thread should be squashed between two small blocks of wood or a bung/cork split in two. This makes the suspension point easy to define.

Method

Time for 10 oscillations. Repeat and get an average.
Divide this average time by 10 to find the period.
Repeat for different lengths of pendulum thread.

Obtaining quality data

Use a ruler to measure the length of the string from the suspension point to the centre of the bob.
- A micrometer or calipers might be useful to measure the diameter of the bob.
Pull the bob to one side (position A) so that the thread makes a small angle with the vertical (use a protractor to measure this angle to keep it consistent and small. The maximum angle should be 10 degrees).

Obtaining quality data 2

Start and then later stop the timer when the bob passes in front of a fiducial mark placed behind the thread (i.e. position B).
- This can be a card with a vertical line drawn on, or the clamp stand.
- This mark needs to be at the equilibrium position. This is where the bob is travelling fastest, so means there is the smallest error in starting and stopping the timer.

Analysis

The equation for this is supposed to be $T = 2\pi \sqrt{\frac{l}{g}}$ . So you should plot a graph of T² against l to verify the equation.
You would expect to see a straight line through the origin with gradient $= \frac{4\pi^2}{g}$ .
You can find a value of g from the gradient ( $g = \frac{4\pi^2}{gradient}$ ).
Plotted or given error bars would help to provide an estimate of the uncertainty in g by finding the worst acceptable line of fit.

1Space, Time & Motion

1.1Motion

1.1.1Motion Basics

1.1.2Graphs of Motion

1.1.3Uniform Acceleration Equations

1.1.4Projectile Motion

1.1.5Bouncing Ball Example

1.2Forces

1.2.1Friction

1.2.2Terminal Speed

1.2.3Newton's Laws

1.2.4Exam-Style Question - Forces

1.2.5End of Topic Test - Newton's Third Law

1.3Momentum & Impulse

1.3.1Momentum

1.3.2Momentum 2

1.3.3End of Topic Test - Newton's Laws & Momentum

1.4Work, Energy & Power

1.4.1Work & Energy

1.4.2Conservation of Energy

1.4.3Power & Efficiency

1.4.4IB Multiple Choice- Work, Energy & Power

2The Particulate Nature of Matter

2.1Thermal Concepts

2.1.1Molecular Kinetic Theory Model

2.1.2Molecular Kinetic Theory Model 2

2.1.3Thermal Energy Transfer

2.1.4Thermal Energy Transfer Experiments

2.1.5Thermal Equilibrium

2.1.6Thermal Conductivity

2.1.7Heat Energy Transfer

2.2Modelling a Gas

2.2.1Ideal Gases

2.2.2Ideal Gases 2

2.2.3End of Topic Test - Thermal Energy & Ideal Gases

2.2.4Exam-Style Question - Ideal Gases

3Wave Behaviour

3.1Oscillations

3.1.1Progressive Waves

3.1.2Simple Harmonic Motion

3.2Travelling Waves

3.2.1Wave Speed & Phase Difference

3.2.2Longitudinal & Transverse Waves

3.2.3Electromagnetic Waves

3.2.4Electromagnetic Waves 2

3.2.5Sound Waves

3.3Wave Characteristics

3.3.1Polarisation

3.4Wave Behaviour

3.4.1Reflection & Absorption

3.4.2Reflection of Light

3.4.3Refraction

3.4.4Refractive index

3.5Standing Waves

3.5.1Standing Waves

3.5.2Standing Waves 2

3.5.3End of Topic Test - Standing Waves

3.5.4IB Multiple Choice - Waves

3.6Simple Harmonic Motion

3.6.1Simple Harmonic Motion

3.6.2Simple Harmonic Systems

3.6.3Energy in Simple Harmonic Motion

3.6.4End of Topic Test - Simple Harmonic Motion

3.6.5End of Topic Test - Simple Harmonic Motion

3.7Single Slit Diffraction

3.7.1Diffraction

3.8Interference

3.8.1Interference

3.8.2Interference 2

3.8.3Diffraction Gratings

3.8.4End of Topic Test - Diffraction

3.9Doppler Effect

3.9.1Doppler Effect

4Fields

4.1Circular Motion

4.1.1Circular Motion

4.1.2Centripetal Force

4.1.3IB Multiple Choice - Circular Motion

4.2Newton's Law of Gravitation

4.2.1Newton's Law

4.2.2Orbits of Planets & Satellites

4.2.3Escape Velocity & Synchronous Orbits

4.2.4End of Topic Test - Gravitational Fields

4.3Fields

4.3.1Fields

4.3.2Gravitational Field Strength

4.3.3Gravitational Potential

4.3.4Electric Field Strength

4.3.5Electrostatic & Gravitational Forces

4.3.6Electric Potential

4.3.7IB Multiple Choice - Gravitational Fields

4.4Fields at Work

4.4.1Radial Field Strength

4.4.2Orbits of Planets & Satellites

4.4.3Escape Velocity & Synchronous Orbits

4.4.4End of Topic Test - Gravitational Fields

4.4.5Uniform Electric Fields

4.4.6IB Multiple Choice - Electric Field & Potential

4.5Electric Fields

4.5.1Electric Charge

4.5.2Conservation of Charge

4.5.3Coulomb's Law

4.5.4Current

4.5.5Potential Difference

4.5.6IB Multiple Choice - Electric Charge

4.6Magnetic Effect of Electric Currents

4.6.1Magnetism

4.6.2Magnetic Field

4.6.3Magnetic Force

4.6.4Moving Charges in a Magnetic Field

4.6.5IB Multiple Choice - Magnetism

4.7Heating Effect of Currents

4.7.1Circuits

4.7.2Resistance

4.7.3Power and Conservation

4.7.4IB Multiple Choice - DC Circuits

4.8Electromagnetic Induction

4.8.1Electromagnetic Induction

4.8.2Electromagnetic Induction 2

4.8.3Magnetic Flux & Flux Linkage

4.8.4End of Topic Test - Electromagnetic Induction

4.9Power Generation & Transmission

4.9.1Alternating Currents

4.9.2Operation of a Transformer

4.10Capacitance

4.10.1Capacitance

4.10.2Parallel Plate Capacitor

4.10.3Energy Stored by a Capacitor

4.10.4Capacitor Discharge

4.10.5Capacitor Charge

4.10.6Magnetic Flux Density

4.10.7End of Topic Test - Capacitance & Flux Density

5Nuclear & Quantum Physics

5.1Discrete Energy & Radioactivity

5.1.1Atomic Spectra

5.1.2Atomic Structure

5.1.3Photons

5.1.4Discrete Energy Levels

5.1.5Alpha, Beta & Gamma Radiation

5.1.6Scientific Practices - Unstable Nuclei

5.1.7Radioactive Decay

5.1.8IB Multiple Choice - Atomic Physics

5.2Nuclear Reactions

5.2.1Mass & Energy

5.2.2Nuclear Fission

5.3The Interaction of Matter with Radiation

5.3.1Particles, Antiparticles & Photons

5.3.2Particle Interactions

5.3.3Classification of Particles

5.3.4Wave-Particle Duality

5.3.5Wave Functions & Probability

5.3.6IB Multiple Choice - Quantum Physics

5.4Nuclear Physics

5.4.1Nuclear Instability & Radioactive Decay

5.4.2Nuclear Radius

5.4.3IB Multiple Choice - Nuclear Physics

6Measurements

6.1Measurements & Errors

6.1.1Fundamental Units

6.1.2SI Prefixes, Standard Form & Converting Units

6.1.3End of Topic Test - Units & Prefixes

6.2Uncertainties & Errors

6.2.1Limitation of Physical Measurements

6.2.2Uncertainty

6.2.3Estimation

6.2.4End of Topic Test - Measurements & Errors

6.3Vectors & Scalars

6.3.1Scalars & Vectors

6.3.2Vector Problems

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