Additional Comments by J. Varriano:
The motion is controlled in the Control Window. If the motion is too fast to follow, pause it and advance frame by frame using the arrow keys.
The point of this applet is to demonstrate that one-dimensional simple harmonic motion, such as the motion of an object attached to a spring, is directly analogous to circular motion.
Consider the object traveling in a circle. The blue arrow is the object's velocity vector. The red arrow is the constant centripetal force applied to the object. Note that the object's speed is constant (the blue arrow has a constant length) but that its direction is always changing so that its velocity vector is always tangent to the circle and it keeps moving in a circle. This change in direction is the acceleration provided by the force.
Now consider the simple harmonic motion of the object below the circle. Think of an object attached to a spring. (The object to the right of the circle is also undergoing simple harmonic motion. But, for now, just concentrate on the one below the circle.) Again, the blue arrow represents the object's velocity vector and the red arrow is the force exerted on the object by the spring. Note that the force is NOT constant. This gives rise to a change in speed. Note that the object moves fastest at the equilibrium position where the force is zero.
So what's the connection between this simple harmonic motion and the circular motion?
Look closely at the two motions. Note that the object undergoing simple harmonic motion is always direcly underneath the object traveling in a circle. One oscillation for the object on the spring corresponds directly to one orbit by the object traveling in a circle. In fancy terms, the motion of the object on the spring is found by taking the projection of the circular motion. Look at the two force vectors. Note that the force exerted by the spring is the horizontal component of the centripetal force. And the velocity of the object on the spring is the horizontal component of the velocity vector of the object traveling in a circle! That is why mathematically we can describe the motion of the object on the spring by using a cos(wt). The angle wt is the angle swept out by the object traveling in a circle (measured with respect to the positive x-axis)!
Now look at the oscillator to the right of the circle. Its motion is also described by the projection of the circular motion, except that you use the vertical components of the circular motion force and velocity vectors.
| Author:Sadahisa Kamikawa |