Lesson 3. Simple Harmic Motion in 1 and 2 Dimensions

Stephen Dickman, Jacob Kelter
Physics
Estimated time for students: two 45 minute periods.
AP Physics 1
v3

Overview

Simple Harmonic Motion(SHM) is represented here using a mass that is attached to a spring. The other end of the spring is fixed to the middle of a horizontal surface. In this model there is no friction. This SHM simulation was designed to help students better conceive the relationship between position, velocity and acceleration.

The simulation allows the student to vary the mass of the oscillating mass and the strength constant of the spring. Thus, the student will be able to make conjectures on subsequent motion as the inertia and net force have changed. These variances will help the student gain a deeper understanding of Newton's Laws.

Finally, the SHM simulation can be done in 2 dimensions creating elliptical motion. Newton wrote this is the Principia and the motion is similar to planetary orbits.

Standards

Computational Thinking in STEM 2.0
  • Computational Modeling and Simulation Practices
    • [CT-MODEL-1] Using computational models to understand a complex phenomenon
    • [CT-MODEL-2] Using computational models to hypothesize and test predictions
    • [CT-MODEL-3] Using a computational tool to understand a system's compenents and dynamics
Next Generation Science Standards
  • Physical Science
    • [HS-PS2-1] Analyze data to support the claim that Newton’s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration.
    • [MS-PS4-2] Develop and use a model to describe that waves are reflected, absorbed, or transmitted through various materials.

Credits

The coding was done by Jacob Kelter. Help uploading and editing the lesson was given by Kelvin Lao.

Activities

  • 1. Simple Harmonic Motion Model
  • 2. Thinking Computationally
  • 3. Simple Harmonic Motion in 2 Dimensions!

Student Directions and Resources


Learning Objectives:

  • Gain a better understanding of motion by analyzing position-time, velocity-time and acceleration-time graphs of a mass undergoing SHM.
  • Learn that a mass connected to a spring that has one end fixed on a horizontal surface has an elliptical trajectory.
  • Get more practice on measuring frequency and period.
  • Get a better understanding on how critical initial conditions are for motion of a system.

     

1. Simple Harmonic Motion Model


Directions:

This is a simulation of a mass and a spring oscillating on a frictionless table in 1 dimension. Observe the code and observe the motion of the turtle. (You will probably need to increase the speed slider for it to move quickly enough)


Question 1.1

Observe the motion of the turtle. What do you notice?

Motion Graphs:

Point out the slope of the x-coordinate vs time graphs and see if it matches the velocity graph below.

Point out the slope of the x-velocity vs time graphs and see if it matches the acceleration graph below.

 



Question 1.2

Calculate the period for one full cycle.



Question 1.3

Calculate the frequency for one full cycle.



Question 1.4

When velocity is zero what is the position? When the position graph has a negative slope, what is the sign of the velocity? The acceleration?



Question 1.5

Increase the mass of the turtle and make a prediction. What should happen? What did happen? Why is this so?



Question 1.6

Increase the spring constant and make a prediction. What should happen? What did happen? Why is this so?



Question 1.7

Draw the force and velocity vectors on the turtle when it is:

  1. left of the equilibrium point moving left
  2. left of the equilibrium point moving right
  3. at furthest point left
  4. at the equilibrium point
  5. right of the equilibrium point moving right
  6. right of the equilibrium point moving left
  7. at the furthest point right
Note: Draw your sketch in the sketchpad below


2. Thinking Computationally


Play with the code as much as you like. Please observe the motion change of the turtle as you change the spring constant and mass of the turtle.


Question 2.1

For the following questions, you can look into the NetLogo code tab under the simulation. What does the “SETUP” button initiate?



Question 2.2

What does the “GO” button initiate?



Question 2.3

Identify which part of the code represents the spring force and paste it below. 



Question 2.4

Why is the force of gravity, mg, not in play here?

3. Simple Harmonic Motion in 2 Dimensions!


Now the turtle can move in 2 dimensions. If Vy=0, will it move in 2D motion? Explore the model and answer the questions below. (Again you will need to increase the slider speed)


Question 3.1

Now, set the turtle in 2 dimensional motion. Play with the model as much as you like. Please observe the motion change of the turtle as you change the spring constant and mass of the turtle. Describe what you see.



Question 3.2

Draw the 2D trajectory and include the velocity and force vectors on the turtle at the far end of ellipse and two other points of your choosing. 

Note: Draw your sketch in the sketchpad below


Question 3.3

What other bodies in nature has a trajectory of this shape?



Question 3.4

If you increase the mass, is the trajectory larger or smaller? Why is this so?



Question 3.5

If you vary the spring constant while keeping the mass fixed, is the trajectory larger or smaller? Why is this so?