Simple Harmonic Motion

Visualizing the Relationship between Position, Velocity, and Acceleration using Uniform Circular Motion

Uniform Circular Motion and How it Relates to Simple Harmonic Motion

Question: What is a quick and easy way to visualize how the signs of velocity and acceleration change in a graph of Simple Harmonic Motion as we make our way around a circle in Uniform Circular Motion ?

ω= Angular Rate

V= Velocity

a= Acceleration

Visualizing Uniform Circular Motion at Four Different Points

Speed is constant at all points in the circle. Velocity is always tangent to the orbit in the direction of motion.Acceleration is always pointing towards the center of the orbit.

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Let’s analyze the signs of both the velocity and the acceleration at each of the four points around the circle.

Using the standard system of radians, we can label the position at each of the four points as: 0, π/2, π, and 3π/2.

Using this information, we can then plot the curves of position, velocity, and acceleration on the same graph to analyze relationships.

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x

y

Placing this circle into a coordinate system using radians

Let’s analyze the signs of both the velocity and the acceleration at each of the four points around the circle.

Using the standard system of radians, we can label the position at each of the four points as: 0, π/2, π, and 3π/2.

Using this information, we can then plot the curves of position, velocity, and acceleration on the same graph to analyze relationships.

1

2

3

4

x

y

Placing this circle into a coordinate system using radians

Point 1: Position = 0Velocity = PositiveAcceleration = Negative

Point 2:Position = π/2Velocity = NegativeAcceleration = Negative

Point 3:Position = πVelocity = NegativeAcceleration = Positive

Point 4: Position = 3π/2Velocity = PositiveAcceleration = Positive

0

3π/2

π

π/2

Basic Graphical Representation

Graphical Representation by Quadrant

π/2 π 3π/2 2π

First Quadrant

Velocity: PositiveAcceleration: Negative

Second Quadrant

Velocity: NegativeAcceleration: Negative

Third Quadrant

Velocity: NegativeAcceleration: Positive

Fourth Quadrant

Velocity: PositiveAcceleration: Positive

Key take-home messages from the relationship between the circle and the

graph

•When displacement is at a maximum (positive or negative), velocity is zero

•When displacement is at an equilibrium position (x=0), velocity is maximized

•When displacement is positive, acceleration is negative and vice versa

References

Boundless. “Simple Harmonic Motion and Uniform Circular Motion.” Boundless Physics. Boundless, 29 Dec. 2014. Retrieved 24 Jan. 2015 from: https://www.boundless.com/physics/textbooks/boundless-physics-textbook/waves-and-vibrations-15/periodic-motion-123/simple-harmonic-motion-and-uniform-circular-motion-430-6028/

Hawkes, (2014). Physics for Scientists and Engineers: An Interactive Approach. Custom ed. Toronto: Nelson Education Ltd.

University of Cambridge. “Maximum Speed in Simple Harmonic Motion.” Isaac Physics. Retrieved 24 Jan. 2015 from:https://isaacphysics.org/api/images/content/questions/physics/mechanics/shm/level4/figures/SHM_SHMgraph_4.svg