Oscillations Simple Harmonic Motion

Simple Harmonic Motion Simple harmonic motion is the repetitive motion of an oscillating object (simple harmonic oscillator). Example: A spring mass system. Like circular motion, simple harmonic motion has a regular period and frequency. In fact, simple harmonic motion and circular motion are linked both mathematically and graphically.

Equilibrium The spring mass system shown below is at equilibrium ( x = 0 ). Equilibrium for a spring is its natural rest position. This spring can either be stretched or compressed, displacing the mass from its current equilibrium position. To stretch the spring a force, F , must be applied in the +x direction. This new unbalanced force will displace the mass from equilibrium. +F

A force is applied As the mass is displaced the spring generates an increasing force of it own. +F

A force is applied +F −FS As the mass is displaced the spring generates an increasing force of it own. +F −FS

A force is applied −FS +F As the mass is displaced the spring generates an increasing force of it own. +F −FS

A force is applied −FS +F As the mass is displaced the spring generates an increasing force of it own. +F −FS

A force is applied +F −FS As the mass is displaced the spring generates an increasing force of it own. This force is known as a restoring force, as it acts to restore the spring to its original equilibrium position. +F −FS

Amplitude When the applied force, F , and the restoring force, FS , are equal and opposite the oscillator is stationary. The oscillator has reached maximum displacement, xmax . Maximum displacement is known as the amplitude, A . This is equal to the radius, R , of a circle matching the springs displacement. +A = xmax +F −FS

The Oscillator is Released When the applied force, F, is removed, the restoring force moves the mass left. x +A −FS +F

The Oscillator is Released When the applied force, F, is removed, the restoring force moves the mass left. x +A −FS

The Oscillator is Released When the applied force, F, is removed, the restoring force moves the mass left. x −FS +A

The Oscillator is Released When the applied force, F, is removed, the restoring force moves the mass left. x −FS +A

The Oscillator is Released When the applied force, F, is removed, the restoring force moves the mass left. x −FS +A

The Oscillator is Released When the applied force, F, is removed, the restoring force moves the mass left. The radius of the circle is equal to the maximum displacement (amplitude) R = A = xmax The radius, R = A , and displacement, x , are two of the sides of a right triangle. While the mass moved to the left the corresponding point on the circle moved with an angular velocity, ω , through an angle, Δθ . R = A = xmax Δθ ω x +A −FS

Conclusion ω R = A = xmax Δθ x −FS An oscillator’s displacement equals the x-component of the radius of another object that is in uniform circular motion with the same amplitude (R = A = xmax). Cosine solves for x . In uniform circular motion Therefore, +A −FS x R = A = xmax Δθ ω

Conclusion ω R = A = xmax Δθ x −FS Period and frequency are related to angular velocity. Which rearrange to These can be substituted into the previous equation +A −FS x R = A = xmax Δθ ω

Putting all together FS x(t) The motion described by this equation looks like… x(t) −A +A FS

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… ω Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… ω Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ x(t) x(t) The motion described by this equation looks like… Δθ x(t) −A +A x(t)

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ x(t) x(t) The motion described by this equation looks like… Δθ x(t) −A +A x(t)

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ x(t) FS x(t) The motion described by this equation looks like… Δθ x(t) −A +A FS x(t)

Putting all together ω Δθ FS x(t) The motion described by this equation looks like… Δθ x(t) −A +A FS

Graphing Oscillations π 2π x(t) −A +A FS

Putting all together +A −A π 2π ω Δθ x(t) −A +A FS x(t)

Putting all together +A −A π 2π ω Δθ x(t) −A +A x(t)

Putting all together +A −A π 2π ω Δθ x(t) −A +A FS x(t)

Putting all together +A −A π 2π ω Δθ x(t) −A +A FS x(t)

Putting all together +A −A π 2π ω Δθ x(t) −A +A FS x(t)

Putting all together +A −A π 2π ω Δθ x(t) −A +A x(t)

Putting all together +A −A π 2π ω Δθ x(t) −A +A FS x(t)

Putting all together +A −A π 2π ω Δθ x(t) −A +A FS

Analyzing Oscillation Graphs: Amplitude Amplitude is the largest displacement and it can be easily read on the graph.

Analyzing Oscillation Graphs: Period Period is the time of one cycle (1 revolution). One cycle is one complete wave form. This wave form extends from 3 seconds to 15 s. The time of this wave form is 15 − 3 = 12 s

Analyzing Oscillation Graphs: Frequency Frequency is the inverse of period.

Analyzing Oscillation Graphs: Displacement Displacement can be solved using the displacement equation and substituting the values determined in the previous slides. Substitute the given time to determine the matching displacement.

Analyzing Oscillation Graphs: Maximum Speed Maximum speed can be found using the equation for speed in uniform circular motion. Set the radius equal to amplitude.

Analyzing Oscillation Graphs: Qualitatively When is the object moving to the right at maximum speed? This occurs at equilibrium (x = 0) and when object is moving toward a greater displacement value: 9 and 21 seconds When is the object moving to the left at maximum speed? This occurs at equilibrium (x = 0) and when object is moving toward a lesser displacement value: 3 and 15 seconds When is the object instantaneously at rest? This occurs at maximum displacement: 0, 6, 12, 18, and 24 seconds.

Circular Motion R ω y(t) Δθ x(t) Oscillation is only concerned with the component of the circular motion equation that is parallel to the oscillation. If a spring is oscillating horizontally If a spring is oscillating vertically x(t) y(t) Δθ ω R Circular motion involves both equations simultaneously. Amplitude is equal radius R , and Δθ = ω t .