Energy, Connection between SHM and Uniform Circular Motion

 Energy, Connections between SHM and Uniform Circular Motion

  

    What are some similarities we know of both SHM and UCM? They both repeat, they both have predictable motion, forces, velocities, and we know of the equations that describe both. For UCM, the acceleration is constant in magnitude, but changes direction to point toward the center of the circle always. For SHM, the force points toward the center as well, and scales with the distance from equilibium. SHM is modeled by sinusoidal functions and in 1 dimension, while UCM is in 2. Wait, thats it!

Diagram Projecting the Circular Motion of a Rotating Ball

    
    Projecting the "shadow" of the ball in uniform circular motion isolates the motion in 1 dimension, and whats left is the shadow moving in simple harmonic motion! When the force is pointed in the vertical direction, or has components vertically, it isnt contributing to the motion in that dimension, so it isn't reflected by the motion of the shadow. What's left is the horizontal component of a constant force (F), which we can construct as shown below. 




Video Explanation of the Same Topic

    Key video notes: just remember that the shadow is in simple harmonic motion because it isolates one component of the force with an otherwise constant magnitude. We can model this relation with a sin function, and the model used in the video is shown below:
Diagram: Circle to Component Using Cos

    To further prove this, we know the maximum displacement is some value (A), and that if a particle starts at angle 0, its angle at time t is ωt fand that omega is 2pi*f, since the displacement corresponds to cos (horizontal component), we get the following equation for displacement as a function of a time from SHM.

x(t) = Acos(2pi*f*t)

    And it's the exact same equation that we got for displacement in UCM, but derived from the characteristics of SHM in 1 dimension!


    SHM to UCM is an important relation to make when trying to understand SHM, but so are energy transformations in SHM. With the example of a spring, we can model the transformation of energy:

Diagram 1: Spring-Block System in Equilibrium

Diagram 2: Spring-Block System Displaced

    For the sake of efficiency, we will assume you understand the displacement, velocity, acceleration, and forces associated with either diagram and their positions. The energy for the system is constant as there is no work done anywhere, with change in E = Fd or work, so it is transformed from one form (potential) to another (expressed, kinetic). Kinetic energy is a function of velocity and mass, as KE = 1/2mv^2, and potential energy can vary, but in the case of UCM it will be dependent on displacement (with a spring it's 1/2 kx^2). This means when the spring is displaced the most, the system has the most potential energy, and 0 kinetic energy, as it is instantaneously at 0 velocity. The inverse, when it is at its equilibrium point, the velocity is at its max (v(t) graph) thus so is kinetic energy, and it is displaced 0 from the point of equilibrium point, meaning it has 0 (minimum) potential energy.

Graph of KE vs Time overlayed over Usp vs Time

    Notice how for any value on this graph, the sum of both energies is constant. This goes back to the point made previously that energy is not lost to the surroundings by the system as there is no work contributed externally or external work contributed on the system. There is a restriction of either graph, as these graphs only apply for positive energies in either function (kinetic energy is always positive or 0, with a maximum of the total energy in the system, and a minimum of 0, meaning kinetic energy is also restricted as they are bound by a total sum). 
    
    Using these relations we can write some equations:

E_total = U + Ke 
U_max = K_max
1/2kA^2 = 1/2mv^2
v^2 = A^2 * k/m
v_max = A * sqrt(k/m)
2piA/T = A sqrt(k/m)
T = 2pi(sqrt(m/k))
f = 1/2pi(sqrt(k/m))


    Which is just another way to derive maximum velocity in SHM, along with period, frequency, and a bunch of other stuff.


    There's been a lot of stuff in this blog post, but it isn't unconquerable! Happy Studying!

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