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Connections between SHM and UCM

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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 centripeta...
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Forces, Motion, and Energy

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 How are force, acceleration, velocity, and displacement related? Often, we use F=ma as our starting point when deriving equations, but when discussing simple harmonic motion we can describe the restoring force as F=-kx.  Setting these two equal to each other gives ma=-kx, or a∝-x. This shows how acceleration is proportional to displacement and will always point towards equilibrium.  Velocity, however, is not proportional to displacement since an object doesn't need to be displaced in order to have velocity. For example, in the last post about springs, the spring passed through the equilibrium point where displacement was zero with a high velocity due to the restoring force. Refer to this diagram for a visual representation: Pendulums While we have been using springs for examples up until now, pendulums also exhibit SHM.  In a pendulum, the force of gravity is the restoring force.  This is because the equilibrium position of the pendulum is resting in the...
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Linear Restoring Forces

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Thinking back a few units, we have actually seen restoring forces before. When a spring is compressed or stretched, i t is no longer in equilibrium. The force that wants it to go back to equilibrium is called the restor ing force.  Looking at the diagram above, we can see that the restoring force changes direction dynamically based on the direction of the displacement of the object. If the spring is compressed towards the left, the restoring force is opposite towards the right and vice versa. The formal definition of linear restoring forces are forces that point an object back toward its equilibrium position and are proportional to the displacement. This is due to linear restoring forces being based on Hooke's law (F=-kx), another familiar topic. A restoring force is not linear if the graph of force vs displacement is curved since it does not translate to perfect simple harmonic motion. Graphs should usually look similar to this: One of the most common examples for linear restoring...
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Amplitude, Period, and Frequency

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Simple harmonic motion repeats itself or is periodic. This means that it can be described using amplitude, period, and frequency. Amplitude Definition: The amplitude is the distance from the midline, or the maximum displacement from a state of equilibrium. Represented with A and measured in meters Can be used to describe the range of motion Found by averaging the extremes; the (Max + Min) / 2 Period Definition: The time period it takes for an object to complete one revolution. Represented with T and measured in seconds Can be easily found by comparing similar adjacent extrema, like one maximum to the next. Devices like clocks have very predictable periods Frequency Definition: The number of revolutions per second Represented in Hertz Depends on the period; shorter period means high frequency Since clocks have predictable periods, they consequentially have predictable frequencies f = 1/T, so frequency and period are inverses One example of a pendulum:
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