thank you so SO much, I've been searching for HOURS for an explanation and this was by far the clearest and concise one. I seriously seriously appreciate this!
I'm so glad it helped! This is a tricky subject, and it REALLY helps me to think about a pendulum swinging back and forth or a mass-spring moving left and right. If you have any questions about the content, don't hesitate to post them here!
0:05 seconds in, he said "... like... THESE are motion graphs"... I was all in. "but its also true, for a PENDULUM!" 😅 and I have no idea why that was so exciting to me, but there I was alone in the kitchen shouting 'YAY PEN-DU-LUMS!!!!' 😂🤣😂 your presence and energy are RADIANT my friend. YOU are gonna save the world. " with Simple Harmonic Motion. 😊🙏💞
It requires differential equations. SHM is defined as a system where the acceleration is proportional and opposite to the position x from equilibrium: a = -kx, where k is a constant. Acceleration a is the second time derivative of position x. So we have d²x/dt² = -kx. Intuitively, you may know from calculus class that the function we're looking for is sin(x) or cos(x). No other function has a second derivative that is proportional to the original function. For example, the first derivative of cos(x) is (d/dt)( cos[t] ) = -sin[t]. Taking the derivative of this gives the second derivative, (d/dt)(-sin[t]) = -cos[t]. I'm pretending that x = cos[t], though, which isn't quite right, because I've ignored the constant k. If you work it out, this constant ends up being ω², such that a = -ω²x and x = x₀cos(ωt). That's not a full derivation, because I'm skipping some steps, but it's hopefully enough to get you started so that you can search this question and find some good videos!
Thank you so much, it helped me a lot. I have a question, when a block is at equilibrium position acceleration is zero but velocity is a maximum value but how is this possible when there is velocity and no acceleration?
The velocity is a result of its previous acceleration. The block accelerates from the edge all the way till it reaches the middle. That acceleration speeds it up, and briefly (while at the middle) the block is just gliding/coasting at a high speed due to its inertia. It's like being in a car on the highway, speeding up, and then coasting for a brief moment before hitting the brakes. The inertia of the block carries it through the center position, and as soon as it passes to the other side of the center/equilibrium position, it begins slowing down.
