Simple harmonic motion. Every object possesses the natural quality in which it oscillates back and forth. Harmonic oscillation is the fundamental theory in which all of physics is based upon. So as physicists, it's essential to have an understanding of this phenomenon. Particularly, we will be learning about resonance. What is simple harmonic motion? Harmonic oscillation is a motion in which it repeats itself periodically in the particular manner. The time dependence means that at given any interval, the displacement from the starting position can be determined according to a sinusoidal graph. This can be determined through the following relation. From Newton's second law, we can determine the type of forces required to give oscillation to a particle. If you look at the terms, acceleration is proportional to some constant k, which gives the restoring force. To solve for displacement, we solve the second order differential equation analytically. We will define the term omaga zero, and this describes the natural frequency of a particular system. In the perfect vacuum, the oscillation motion will continue forever. However, this is not true in our case. The motion will gradually stop due to the interference. So to counteract this, we introduce a damping term, which makes the damping force become. Where lambda is the damping coefficient. So the final equation of motion can be defined as this. In our case, the damping term is the error friction. Let's solve the differential equation. Consider the solution of the form by substituting this into the equation. We can finally express alpha in the form of. For simplicity, we define together with omaga zero. We obtain the value of alpha as the variations between gamma and omaga zero. We result in three different scenarios of damping. However, all three scenarios result in decline in motion caused by the damping force. So we must introduce a driving force to continue the oscillation. Now let's look at the response of the oscillating system when external force is applied periodically. Then the equation of motion becomes like this. Let's consider a particular solution of the form, which has exactly the same period omaga one as the applied force. However, it may include some face difference, which balances out the offset at initial time t equals zero. Simplify in the expression we get. Separate the real and imaginary parts, and compare the coefficients we now have. Rearranging the equations, the amplitude of displacement of oscillation can be shown as. To get face angle, we divide the real and imaginary parts. For a special case of post-amp oscillation, when driving frequency is approaching the natural frequency, resonance will occur at resonance frequency, the amplitude will reach its maximum. As seen in this demonstration, I set the driving frequency resonance occurs. The equation for amplitude is, by taking the first derivative of the amplitude, we can determine the value for driving frequency. So that the amplitude will be at its maximum. Let's go back to the demonstration. As we can see, different objects actually need different driving frequency to reach its resonance.