Comparing Simple Harmonic Motion and Circular Motion - Physics LibreTexts
We can see the connection between the motion of a particle executing uniform circular motion and a particle executing simple harmonic motion, say while. Uniform Circular Motion is Motion in a circle at a constant velocity. There is usually some force that acts upon the object, causing it to accelerate towards the . We can better understand and visualize many aspects of simple harmonic motion by studying its relationship to uniform circular motion. The following animation.
You should go to the projections pull-down menu and select "constrained" to see the things displayed on equal x,y axes. We need to describe the motion by finding a way to predict the position and velocity as functions of time.
How do we do that? Use the equation of motion! Maple can take from here. Ask it to find x t for the equation of motion we specified.
But it is the most general solution. Luckily, we can get Maple to a little more work that's interesting for physics, because we can give it information that is physically relevant.
Let's call that point d.
relation between Circular motion and Simple Harmonic Motion (SHM)
Maple allows us to put that information directly into dsolve. Still, the answer looks funny, doesn't it? Both the spring constant and the mass are magnitudes, and therefore positive. So it looks like the solution involves exponentials of imaginary numbers! Can that be physical? Look at a plot to find out.
harmonic oscillator - Relation between shm and circular motion - Physics Stack Exchange
It even reminds us of some functions we've seen before. Let's try mimicking this solution with a cosine function. 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: 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.JEE Physics - Relation between SHM and uniform circular motion
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. 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 x- axis and moved to its present position. Its projection on the x-axis undergoes simple harmonic motion.