Creating Circular Motion from Sine and Cosine Curves

Creating Circular Motion from Sine and Cosine Curves

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Mr. P: Good morning. Let’s start with a dot moving in the y-direction in simple harmonic motion, just like a vertical mass spring system. Flipping Physics If we set the dot in motion at a constant velocity to the right, Bo what sort of function does that create? Bo: That looks just like a sine curve. Mr. P: Correct Bo. The Y position of a dots moving in simple harmonic motion like this creates a sine curve. Now let’s add a dot moving in the x-direction in simple harmonic motion. This time just like a horizontal mass spring system. If we set the dot in motion at a constant velocity down, Billy what sort of function does that create? Billy: I think it’s still a sine curve, right? Bo: Actually, no the dot starts at its maximum value, so it’s actually a cosine curve. Billy: Right cosine, it’s a cosine curve because it starts at its maximum value. Thanks. Mr. P: Correct, Bobby, the curve created by the dots moving in simple harmonic motion in the x-direction, like this, is a cosine curve. Now I actually want to display both
curves on the same axis, so let’s rotate the cosine curve 90 degrees to put it on the same axis as the sine curve. Bo: Oh, yeah, the sine and cosine curves
have the same shape, it’s just they’re shifted from one another by 90 degrees PI over 2 radians or 1/4 of a revolution along the horizontal axis. Mr. P: Let’s go back to looking at the dots on the left and add lines to indicate where the intersection of these motions occur. Can anybody tell me what the shape of the intersection of those lines creates as a function of time? Bo: It’s a circle
Bobby: Wow, it is a circle Billy: Wait a minute, when you combine the motions described by a sine and cosine curve like that you get a circle? Bo: Yep
Mr. P: That is correct That means if you combine simple harmonic motion in the x-direction with simple monic motion in the y-direction, like we have done here, you get circular motion. This is just like the circular motion of this marker cap moving at a constant angular velocity on this turntable. Billy: I like that
Bobby: Whoa, cool.
Bo: Sure. Mr. P: Thank you very much for learning with me today,
I enjoyed learning with you

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