Connections between SHM and UCM

Connecting SHM and UCM

When we talk about uniform circular motion, we are describing an object moving at a constant speed in a circle. Simple harmonic motion is actually just the projection of circular motion onto one axis. If a light shone on an object moving in UCM, it's shadow would be a representation of SHM. Because of this, every quantity in SHM can be directly connected to a quantity in circular motion. For example:

  • The time it takes for one full revolution in UCM is the same as the time for one full oscillation in SHM
  • The radius of the circle becomes the amplitude of the motion
  • The maximum velocity in SHM is the tangential speed of the object in UCM
  • The centripetal acceleration in UCM (a = ω²r) becomes the acceleration in SHM (a = −ω²x) (negative since it goes in the direction of the restoring force).

In a previous post, we said that acceleration in SHM is proportional to displacement and opposite in direction. This relationship is proven here, since the inward centripetal acceleration of circular motion turns into the restoring force that always pulls the object back toward equilibrium in simple harmonic motion.

Graphing SHM

When an object rotates at a constant speed, its vertical or horizontal displacement follows a sinusoidal pattern, meaning it constantly reaches the same maximums and minimums using the same path. Assuming that the system begins oscillating at the maximum positive point of displacement, the amplitude of the equation is the maximum displacement, A. Using this knowledge, we can create three equations relating position, velocity, and acceleration to time.



1. x(t) = Acos(ωt)

The position vs time equation is a positive cosine function because we assume the initial position to be equivalent to A, so the graph starts in the positive region at the maximum displacement.

2. v(t) = -Aωsin(ωt)

The velocity vs time equation is a negative function because velocity is equivalent to the slope of a displacement vs time graph, and x(t) begins with a negative slope as the graph goes down from the maximum to the minimum. It is a sine function because at the maximum displacement, the object is changing directions and momentarily has a velocity of 0, so the graph also starts at 0.   

3. a(t) = -Aω²cos(ωt)

The acceleration vs time equation is a negative function because acceleration is equivalent to the slope of a velocity vs time graph, and v(t) also initially continues downward from the midline to the minimum with a negative slope. It is a cosine function because at the maximum displacement, the restoring force is at it's greatest value, so given F=ma the acceleration is also at it's maximum value and opposite to the positive direction of displacement.

How do SHM principles apply in the real world?

One real-world example of simple harmonic motion are earthquake-resistant buildings. Engineers design these buildings so that when the building begins to oscillate, the internal mass moves in a controlled way that reduces the overall impact of the shaking. When an earthquake occurs, the ground moves back and forth, causing the building to sway around its equilibrium position. This swaying motion can be modeled as simple harmonic motion because the building moves back and forth in a repeating pattern, and the structural supports provide the restoring forces that pull it back towards the center. This motion is also sinusoidal and can be described using the position vs time equation x(t) = Acos(ωt).  This shows how the same principles that connect uniform circular motion and simple harmonic motion can be used to protect real structures during earthquakes.

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