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Simple Harmonic Motion and Uniform Circular Motion
Simple harmonic motion is produced by the projection of uniform circular motion onto one of the axes in the xy plane.
Learning Objectives

Describe properties of the velocity vector in uniform circular motion

Express the acceleration in uniform circular motion in a form of equation

Describe relationship between the simple harmonic motion and uniform circular motion
Key Points
 Uniform circular motion describes the movement of an object traveling a circular path with constant speed. The onedimensional projection of this motion can be described as simple harmonic motion.
 In uniform circular motion, the velocity vector v is always tangent to the circular path and constant in magnitude. The acceleration is constant in magnitude and points to the center of the circular path, perpendicular to the velocity vector at every instant.
 If an object moves with angular velocity ω around a circle of radius r centered at the origin of the xy plane, then its motion along each coordinate is simple harmonic motion with amplitude r and angular frequency ω.
Terms

centripetal acceleration
Acceleration that makes a body follow a curved path: it is always perpendicular to the velocity of a body and directed towards the center of curvature of the path.

uniform circular motion
Movement around a circular path with constant speed.
Full Text
Uniform Circular Motion
Uniform circular motion describes the motion of a body traversing a circular path at constant speed. The distance of the body from the center of the circle remains constant at all times. Though the body's speed is constant, its velocity is not constant: velocity (a vector quantity) depends on both the body's speed and its direction of travel. Since the body is constantly changing direction as it travels around the circle, the velocity is changing also. This varying velocity indicates the presence of an acceleration called the centripetal acceleration. Centripetal acceleration is of constant magnitude and directed at all times towards the center of the circle. This acceleration is, in turn, produced by a centripetal force—a force in constant magnitude, and directed towards the center.
Velocity
The above figure illustrates velocity and acceleration vectors for uniform motion at four different points in the orbit. Since velocity v is tangent to the circular path, no two velocities point in the same direction. Although the object has a constant speed, its direction is always changing. This change in velocity is due to an acceleration, a, whose magnitude is (like that of the velocity) held constant, but whose direction also is always changing. The acceleration points radially inwards (centripetally) and is perpendicular to the velocity. This acceleration is known as centripetal acceleration.
Uniform Circular Motion (at Four Different Point in the Orbit)
Velocity v and acceleration a in uniform circular motion at angular rate ω; the speed is constant, but the velocity is always tangent to the orbit; the acceleration has constant magnitude, but always points toward the center of rotation
Displacement around a circular path is often given in terms of an angle θ. This angle is the angle between a straight line drawn from the center of the circle to the objects starting position on the edge and a straight line drawn from the objects ending position on the edge to center of the circle. See for a visual representation of the angle where the point p started on the xaxis and moved to its present position. The angle θ describes how far it moved.
Projection of Uniform Circular Motion
A point P moving on a circular path with a constant angular velocity ω is undergoing uniform circular motion. Its projection on the xaxis undergoes simple harmonic motion. Also shown is the velocity of this point around the circle, v−max, and its projection, which is v. Note that these velocities form a similar triangle to the displacement triangle.
For a path around a circle of radius r, when an angle θ (measured in radians) is swept out, the distance traveled on the edge of the circle is s = rθ. You can prove this yourself by remembering that the circumference of a circle is 2*pi*r, so if the object traveled around the whole circle (one circumference) it will have gone through an angle of 2pi radians and traveled a distance of 2pi*r. Therefore, the speed of travel around the orbit is:
where the angular rate of rotation is ω. (Note that ω = v/r. ) Thus, v is a constant, and the velocity vector v also rotates with constant magnitude v, at the same angular rate ω.
Acceleration
The acceleration in uniform circular motion is always directed inward and is given by:
This acceleration acts to change the direction of v, but not the speed.
Simple Harmonic Motion from Uniform Circular Motion
There is an easy way to produce simple harmonic motion by using uniform circular motion. The figure below demonstrates one way of using this method. A ball is attached to a uniformly rotating vertical turntable, and its shadow is projected onto the floor as shown. The shadow undergoes simple harmonic motion .
Shadow of a Ball Undergoing Simple Harmonic Motion
The shadow of a ball rotating at constant angular velocity ω on a turntable goes back and forth in precise simple harmonic motion.
The next figure shows the basic relationship between uniform circular motion and simple harmonic motion. The point P travels around the circle at constant angular velocity ω. The point P is analogous to the ball on a turntable in the figure above. The projection of the position of P onto a fixed axis undergoes simple harmonic motion and is analogous to the shadow of the object. At a point in time assumed in the figure, the projection has position x and moves to the left with velocity v. The velocity of the point P around the circle equals v_{max}. The projection of v_{max} on the xaxis is the velocity v of the simple harmonic motion along the xaxis .
To see that the projection undergoes simple harmonic motion, note that its position x is given by:
where θ=ωt, ω is the constant angular velocity, and X is the radius of the circular path. Thus,
The angular velocity ω is in radians per unit time; in this case 2π radians is the time for one revolution T. That is, ω=2π/T. Substituting this expression for ω, we see that the position x is given by:
Note: This equation should look familiar from our earlier discussion of simple harmonic motion.
Key Term Reference
 acceleration
 Appears in these related concepts: Centripetial Acceleration, Position, Displacement, Velocity, and Acceleration as Vectors, and The Second Law: Force and Acceleration
 amplitude
 Appears in these related concepts: Interference and Diffraction, Period of a Mass on a Spring, and Sound
 angular
 Appears in these related concepts: The Physical Pendulum, Stability, Balance, and Center of Mass, and Constant Angular Acceleration
 angular frequency
 Appears in these related concepts: Wavelength, Freqency in Relation to Speed, Driven Oscillations and Resonance, and Period and Frequency
 angular velocity
 Appears in these related concepts: Conservation of Energy in Rotational Motion, Angular Velocity, Omega, and Kepler's Second Law
 axis
 Appears in these related concepts: Rotational Collisions, Torque, and Area Between Curves
 centripetal
 Appears in these related concepts: Kinematics of UCM, Banked and Unbacked Highway Curves, and Overview of NonUniform Circular Motion
 circular motion
 Appears in these related concepts: Water Waves, Sinusoidal Nature of Simple Harmonic Motion, and XRay Diffraction
 circumference
 Appears in these related concepts: Eratosthenes' Experiment, Using Interference to Read CDs and DVDs, and Angular Position, Theta
 coordinates
 Appears in these related concepts: Force at an Angle to Displacement, FourDimensional SpaceTime, and Springs
 displacement
 Appears in these related concepts: Reference Frames and Displacement, Interference, and Introduction to Human Language
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 force
 Appears in these related concepts: Work Done by a Variable Force, Work, and Force of Muscle Contraction
 frequency
 Appears in these related concepts: The de Broglie Wavelength, Antennae, and Beats
 magnitude
 Appears in these related concepts: Adding and Subtracting Vectors Using Components, Multiplying Vectors by a Scalar, and Components of a Vector
 motion
 Appears in these related concepts: Newton and His Laws, Mechanical Work and Electrical Energy, and Friction: Static
 origin
 Appears in these related concepts: Adding and Subtracting Vectors Graphically, Overview of Different Muscle Functions, and ThreeDimensional Coordinate Systems
 perpendicular
 Appears in these related concepts: The Cross Product, Circular Motion, and Normal Forces
 plane
 Appears in these related concepts: Scalars vs. Vectors, Shape and Volume, and Shape
 position
 Appears in these related concepts: Motion with Constant Acceleration, Motion Diagrams, and Graphical Interpretation
 radians
 Appears in these related concepts: Position, Velocity, and Acceleration as a Function of Time, Centripetal Force, and Rotational Angle and Angular Velocity
 rotation
 Appears in these related concepts: Synovial Joint Movements, Lever Systems, and Rotational Kinetic Energy: Work, Energy, and Power
 simple harmonic motion
 Appears in these related concepts: Harmonic Wave Functions, Energy in a Simple Harmonic Oscillator, and Simple Harmonic Motion
 tangent
 Appears in these related concepts: Tangent and Velocity Problems, Newton's Method, and Graphs of Exponential Functions, Base e
 uniform motion
 Appears in these related concepts: GallileanNewtonian Relativity and The First Law: Inertia
 vector
 Appears in these related concepts: Vectors in the Plane, Angular vs. Linear Quantities, and Superposition of Electric Potential
 velocity
 Appears in these related concepts: Rolling Without Slipping, RootMeanSquare Speed, and Force
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Source: Boundless. “Simple Harmonic Motion and Uniform Circular Motion.” Boundless Physics. Boundless, 21 Jul. 2015. Retrieved 22 Jul. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/wavesandvibrations15/periodicmotion123/simpleharmonicmotionanduniformcircularmotion4306028/