15.6
Simple Harmonic Motion and Uniform Circular Motion
While simple harmonic motion and uniform circular motion may be two separate concepts, they correlate and interlink with each other. Simple harmonic m…
Consider the Moon at a distance A from the center of the Earth, rotating in a circular motion with a constant angular velocity.
Let the Earth's center be the origin of the displacement-time coordinate system. When the Moon moves to a position P, its projection P' on the x-axis makes an angle Ф.
As the Moon moves further around the Earth at any time t, it makes an angle ωt+Ф. Based on the Moon's projection on the x or y-axis, the position of the projection can be denoted by either a cosine or sine function.
The period of the Moon can be given by the circumference of the Earth's orbit over its velocity. Recalling the velocity equation from energy conservation and modifying it, the period of the Moon's projection is determined.
The velocity of the Moon acts tangentially, while the acceleration of the Moon is directed radially inward.
The x-component of the velocity and acceleration of the Moon is equal to the velocity and acceleration of the Moon's projection. Their magnitudes are obtained by recalling the velocity and acceleration equations.
As observed, the time period, position, velocity, and acceleration equations of the Moon's projection are similar to that of a simple harmonic oscillator.
Therefore, the projection of a uniform circular motion along the diameter of a circle on which the circular motion occurs represents simple harmonic motion.
View the full transcript and gain access to JoVE Core videos
Q1: How does the projection of uniform circular motion relate to simple harmonic motion?
The projection of an object undergoing uniform circular motion along the diameter of its circular path produces simple harmonic motion. As the object rotates at constant angular velocity, its projection oscillates back and forth, exhibiting position, velocity, and acceleration equations identical to those of a simple harmonic oscillator. This relationship demonstrates that simple harmonic motion is fundamentally the one-dimensional shadow of circular motion.
Q2: What mathematical functions describe the position of a rotating object's projection?
The position of a projection from uniform circular motion can be expressed using either cosine or sine functions, depending on the initial phase angle. At time t, the angle swept is ωt plus the initial phase angle Ф. This mathematical representation captures how the projection oscillates sinusoidally as the object completes its circular path, with the period determined by the circumference divided by velocity.
Q3: How do velocity and acceleration components differ between circular motion and its projection?
In uniform circular motion, velocity acts tangentially while acceleration points radially inward. The x-component of both velocity and acceleration equals the corresponding values for the projection. The magnitudes of these components can be derived from standard velocity and acceleration equations, revealing that the projection's motion follows the same kinematic relationships as a simple harmonic oscillator.
Q4: What is the period of a Moon orbiting Earth in circular motion?
The period of the Moon's circular orbit equals the circumference of its orbital path divided by its orbital velocity. This period remains constant throughout the motion since the Moon maintains uniform circular motion at constant angular velocity. The same period applies to the Moon's projection, which undergoes simple harmonic motion with identical temporal characteristics.
Q5: Why is observing a projection easier than observing a large-scale simple harmonic oscillator?
Observing the projection of uniform circular motion is often simpler than constructing a precise large-scale simple harmonic oscillator because circular motion naturally produces visible, measurable oscillations. Examples include a ball's shadow on a floor from a rotating turntable or a pen tracing waves on paper beneath a rotating record player. These practical demonstrations make the connection between circular and oscillatory motion immediately apparent.
Q6: What is the relationship between Hooke's law and uniform circular motion?
Hooke's law typically describes systems with simple harmonic motion rather than uniform circular motions at constant angular velocity. While Hooke's law governs the restoring force in oscillatory systems, uniform circular motion involves constant speed and centripetal acceleration. However, the projection of uniform circular motion exhibits forces and accelerations consistent with Hooke's law, linking the two concepts through the projection relationship.
Q7: How can you experimentally demonstrate that circular motion projects into simple harmonic motion?
Attach a pen to a rotating turntable or record player and drag paper beneath it to capture the pen's motion as a wave pattern. This wave represents the projection of the circular motion onto a line, visually demonstrating simple harmonic motion. Alternatively, observe the shadow of a ball on a turntable projected onto a floor, which oscillates back and forth as the ball rotates, providing direct evidence of the projection relationship.