<![CDATA[Connection between Circular Motion and Simple Harmonic Motion]]> false false false false false false false true true false ]]> ./ReferenceCirclev2/Screen Shot 2018-05-21 at 5.25.49 PM (2).png ./01authorandrewduffy.png;./01authorlookang50x50.png; DESCRIPTION_EDITOR Reference Circle true false _default_ Reference Circle false

The Connection between Circular Motion and Simple Harmonic Motion

In this simulation, you can examine the connection between uniform circular motion and simple harmonic motion. This connection is one way to justify the basic equation of motion for an object experiencing simple harmonic motion. x = A cos (ωt) This equation also corresponds to one dimension of uniform circular motion. In this simulation, we even go beyond the equation above a little, and explore the phase angle, φ. In that case, the equation of motion becomes x = A cos (ωt + φ) There are analogous equations for the velocity and the acceleration as a function of time. For more info: http://weelookang.blogspot.sg/2012/07/ejs-open-source-phase-difference-java.html ]]> DESCRIPTION_EDITOR Activities true false _default_ Activities false

Activities

  1. When the simulation begins, the two blocks have the same parameters, so they move together. Just focus on the connection between the uniform circular motion (the motion of the ball in a circular path at constant speed) and the simple harmonic motion. Describe the relationship between these two motions.
  2. Now, give the two blocks different amplitudes, while keeping everything else the same. How do the motions of the two blocks on springs compare? Can you make sense of this by comparing the harmonic motion to the corresponding circular motion?
  3. Set the amplitudes to the same values, but now change either the mass or the spring constant of one of the blocks so that the blocks have different angular frequencies. How do the motions of the two blocks on springs compare? Can you make sense of this by comparing the harmonic motions to the corresponding circular motions?
  4. Set the amplitudes, masses, and spring constants of the two blocks so that they match, and now just change the phase of one block with respect to the other. How do the motions of the two blocks on springs compare? Can you make sense of this by comparing the harmonic motions to the corresponding circular motions?
  5. What does it mean for two objects to be 180 degrees out of phase?
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This connection is one way to justify the basic equation of motion for an object experiencing simple harmonic motion.
x = A cos (ωt)
Applied to mass 1,
x₁ = A₁ cos (ω₁t)
This equation also corresponds to one dimension of uniform circular motion.
In this simulation, we even go beyond the equation above a little, and explore the phase angle, φ.
In that case, the equation of motion becomes x₂ = A₂ cos (ω₂t + φ).
There are analogous equations for the velocity and the acceleration as a function of time.
In Simple harmonic motion, the restoring spring force is
F = -kx = -kAcos (ωt)
Using Newtons Second Law, F = ma
-kAcos (ωt) = m(-Aω² cos (ωt))
Therefore, k/m=ω²
This means the spring constant, k divided by the mass, m of the spring mass system is equal to the angular frequency squared, ω² of the simple harmonic motion.
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