Great work, you explained it well :) but is it really necessary to supstitute sin("theta")="theta" ? you could easy supstitute sin("theta")=y/l, and get a=g*sin("theta")=g*y/l
That is the defining characteristic of a simple harmonic motion. A simple pendulum is an example of a simple harmonic motion. The acceleration of the system is directly proportional to its displacement and acceleration always acts in the opposite direction of displacement, towards its equilibrium displacement. By comparing the displacement of the pendulum/system relative to its equilibrium to a circular motion, displacement x can be expressed as x_0 cos(omega * theta), where x_0 is the amplitude of displacement, theta is the angle of the displacement from the equilibrium in a complementary circular motion and omega is the angular velocity, which can be derived and simplified to be 2pi/period. By differentiating the equation twice, to obtain acceleration, it can be observed that a = - (omega)^2 * x, the defining equation of simple harmonic motion.
I liked this video very much and it has helped in studying. It would have been nice if you had explained that "w" meant angular frequency. Thanks very much for making this video.
It necessary because when equating a = w^2(y), y here is the value along the curve, aka s. The proper equation should have been: a = F/m = g sin(theta) ~= g*theta = w^2*s We know from the definition of an angle, theta = s/r, in this case theta = s/L, therefore s = theta * L. Plugging in theta * L for s, we get: g*theta = w^2 * theta * L g/L = w^2. From here, we get the proper period value. The person in this video made the mistake of writing y/L = sin(theta), while in reality y/L = theta, since y here actually is s. Because of this mistake the small angle approximation seems unnecessary.
