- Parent Category: 02 Newtonian Mechanics
- Category: 09 Oscillations
- Created: Thursday, 20 August 2015 15:00
- Last Updated: Monday, 31 December 2018 16:08
- Published: Monday, 25 April 2016 15:00
- Hits: 10207
220.127.116.11.3 Understanding Phase Difference through Uniform Circular Motion
Simple harmonic motion can in some cases be considered to be the one-dimensional projection of uniform
circular motion. If an object moves with angular speed ω around a circle of radius A centered at the origin of the x−y plane, then its motion along each coordinate is simple harmonic motion with amplitude A and angular frequency ω.
Q1: given that, a circular motion can be described by x = A cos(ω t) and y A sin(ω t) what is the y-component model-equation that can describe the motion of a uniform circular motion?
A1: y = Asin (ωt)
Q2: When the x-component of the circular motion is modelled by x = A cos(ω t) and y A sin(ω t) suggest an model-equation for y.
A2: y = Acos (ωt) for top position or y = - Acos (ωt) for bottom position
Q3: Explain why are the models for both x and y projection of a uniform circular motion, a simple harmonic motion?
A3: both x = A cos(ω t) and y A sin(ω t) each follow the defining relationship for SHM as ordinary differential equations of and respectively.
Q4: In, the diagram above, what is the phase difference between blue and magenta color y direction motion?A4: the phase difference is 90 degrees, or more precisely blue lead red by 90 degrees angle.
http://youtu.be/0IaKcqRw_Ts This video shows how a pendulum's oscillations and the shadow of rotating object are related. This could be used to demonstrate that the projection of a circular motion is actually a simple harmonic motion.
18.104.22.168.3.2 Run Model:
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What is in Phase?