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Consider a mass on a spring moving horizontally. The only force on the mass is the spring itself which we can model using Hooke's Law. Then by Newton's Second Law that F=ma, this gives the differential equation mx''+kx=0. This is a second order, constant coefficient, homogeneous differential equation which we can study! We use our techniques from ODEs to solve this system and then do a bit of trigonometry to convert the answer to the standard form. This type of oscillation is often called either undamped mechanical vibrations or more generally simple harmonic motion as this occurs in a myriad of places in physics beyond just this scenario.
0:00 Mass on a Spring
1:18 Newton's 2nd Law & Hooke's Law
2:51 Solving the ODE
4:28 Rewriting into standard Form
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