09.1 - SMH

Essential idea: The solution of the harmonic oscillator can be framed around the variation of kinetic and potential energy in the system.

Nature of science:

Insights: The equation for simple harmonic motion (SHM) can be solved analytically and numerically. Physicists use such solutions to help them to visualize the behaviour of the oscillator. The use of the equations is very powerful as any oscillation can be described in terms of a combination of harmonic oscillators. Numerical modelling of oscillators is important in the design of electrical circuits. (1.11)

General Resources:

Understandings:

The defining equation of SHM
Energy changes

Applications and skills:

Solving problems involving acceleration, velocity and displacement during simple harmonic motion, both graphically and algebraically
Describing the interchange of kinetic and potential energy during simple harmonic motion
Solving problems involving energy transfer during simple harmonic motion, both graphically and algebraically

Guidance

Contexts for this sub-topic include the simple pendulum and a mass-spring system

Data booklet reference:

Utilization:

Fourier analysis allows us to describe all periodic oscillations in terms of simple harmonic oscillators. The mathematics of simple harmonic motion is crucial to any areas of science and technology where oscillations occur
The interchange of energies in oscillation is important in electrical phenomena
Quadratic functions (see Mathematics HL sub-topic 2.6; Mathematics SL sub-topic 2.4; Mathematical studies SL sub-topic 6.3)
Trigonometric functions (see Mathematics SL sub-topic 3.4)

Aims:

Aim 4: students can use this topic to develop their ability to synthesize complex and diverse scientific information
Aim 6: experiments could include (but are not limited to): investigation of simple or torsional pendulums; measuring the vibrations of a tuning fork; further extensions of the experiments conducted in sub-topic 4.1. By using the force law, a student can, with iteration, determine the behaviour of an object under simple harmonic motion. The iterative approach (numerical solution), with given initial conditions, applies basic uniform acceleration equations in successive small time increments. At each increment, final values become the following initial conditions.
Aim 7: the observation of simple harmonic motion and the variables affected can be easily followed in computer simulations

MOS and Pendulum Description of Motion.

From your experimentation, you should be able to:

Describe the relationships amongst the four graphs to the right.
Produce free body diagrams for 3 key positions while oscillating.
Describe the energy changes as the mass completes one oscillation.
Describe how energy can dissipate as it continues to oscillate.
Describe how the equation:

can be used to define SHM.

6. Create and describe the shape of a velocity v. position graph of the oscillating MOS or pendulum.

For addition information please see this page produced by Tim Brzezinski: Graphing Sine and Cosine Functions

The SHM Equations:

Keys to Success:

Where are you starting from? Stated in the problem.
All Calculations are in rads. Check your MODE.
Be careful with negative displacement (x = -0.3m) values in the instantaneous velocity and kinetic energy equations. Know your order of operations!!!

The total energy in the simple harmonic motion of a particle is:

Directly proportional to its mass
Directly proportional to the square of the frequency of oscillations and
Directly proportional to the square of the amplitude of oscillation.

The SHM Equations

The SHM Equations Document

Position v. Time

From Equilibrium:

The Blue Wave is a sine wave. Modeled with the equation:

From Max Displacement:

The RED Wave is a cosine wave. Modeled with the equation:

See Circle Exploration in GC / Desmos

Kinetic Energy:

From Equilibrium and Max Displacement:

Both waves are COSINE Squared cos(ωt)2 relationships. Although the cos(ωt)2 does not appear in the equation:

Notice that the frequency of the wave has doubled as the kinetic energy is at a maximum each time the mass passes thru the equilibrium point (x=0m) and is zero (E_k= 0 J) when the displacement is at a maximum (x = xo).

Calculating with Pendulum and MOS

Using the equation below, determine the k-value [Nm-1] of the spring you used during the practical.

2. Obtain a 'mystery mass', from the bench.

3. Using the oscillations on your spring, determine its mass.

4. Calculate your percent error.

Sample SHM Practice Problems

A long flat steel spring is clamped at its lower end and a 2 kg mass is fastened to the top end. When the top is displaced to one side then released, the period is measured as 1.40 seconds.
1. Find the spring constant k.
2. Find the restoring force when the end is displaced 20 cm.

Compute the acceleration due to gravity at a place where a simple pendulum 150 cm long makes 100 cycles in 246 seconds. Identify the location.

A body of mass 100 g undergoes simple harmonic motion with amplitude of 20 mm. The maximum force which acts upon it is 0.05 N. Calculate:
1. Its maximum acceleration.

Its period of oscillation.

A simple pendulum has a period of 4.2 s. When it is shortened by 1.0 m the period is only 3.7 s.
1. Without assuming a value for g, calculate the original length of the pendulum.
2. Calculate the acceleration due to gravity g suggested by the data.

Equations in Use:

SHM Problems - 9-1

Create a copy of the above document HERE.

Spreadsheet with values from Blue/Red Graph

Instantaneous Velocity v. Position

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