2. 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 ?
3. 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.
1
2
3
4
4. 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
5. 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 = 0
Velocity = Positive
Acceleration = Negative
Point 2:
Position = π/2
Velocity = Negative
Acceleration = Negative
Point 3:
Position = π
Velocity = Negative
Acceleration = Positive
Point 4:
Position = 3π/2
Velocity = Positive
Acceleration = Positive
0
3π/2
π
π/2
7. Graphical Representation by Quadrant
π/2 π 3π/2 2π
First
Quadrant
Velocity:
Positive
Acceleration
: Negative
Second
Quadrant
Velocity:
Negative
Acceleration
: Negative
Third
Quadrant
Velocity:
Negative
Acceleration
: Positive
Fourth
Quadrant
Velocity:
Positive
Acceleration
: Positive
8. 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
9. 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/mechanic
s/shm/level4/figures/SHM_SHMgraph_4.svg
