![was the motion you observed in part c simple harmonic was the motion you observed in part c simple harmonic](https://i.ytimg.com/vi/xNkMZyCklqE/maxresdefault.jpg)
(These characters are often identical in some fonts.) f = The symbol for frequency is a long f but a lowercase italic f will also do. Mathematically, it's the number of events ( n) per time ( t). ⎡įrequency is the rate at which a periodic event occurs. The SI unit of period is the second, since the number of events is unitless. The symbol for period is a capital italic T although some professions prefer capital italic P. Mathematically, it's the time ( t) per number of events ( n). (A system where the time between repeated events is not constant is said to be aperiodic.) The time between repeating events in a periodic system is called a period. How does one thing relate to another? Since the short answer is "abstractly" the reasonable thing to do is to avoid ω altogether and use a coefficient grounded in physical reality.Ī periodic system is one in which the time between repeated events is constant. Angular frequency is great for systems that rotate (spin) or revolve (travel around a circle), but our system is oscillating (moving back and forth). It has no physical meaning - in this context. The SI unit of angular frequency is the radian per second, which reduces to an inverse second since the radian is dimensionless. That thing is called angular frequency, which in this case is the rate of change of the phase angle ( φ) with time ( t). The way around this is to add a coefficient that changes our input variable (time) into something a trig function can handle (radians). In a sense, a radian is a unit of nothing. Using SI units would give us meters over meters, which dimensional analysis reduces to nothing. From the mathematical definition, an angle ( φ) is the ratio of arc length ( s) to radius ( r). The only unit you can really put into a trig function is the radian.
![was the motion you observed in part c simple harmonic was the motion you observed in part c simple harmonic](https://i.ytimg.com/vi/Mj7Pr42rliI/maxresdefault.jpg)
All the trig functions are ratios, which makes them dimensionless (the more precise mathematical term) or unitless (the term I prefer). The solution to our differential equation is an algebraic equation - position as a function of time ( x ( t)) - that is also a trigonometric equation.
![was the motion you observed in part c simple harmonic was the motion you observed in part c simple harmonic](https://memberdata.s3.amazonaws.com/pt/ptx2013/photos/ptx2013_photo_gal__photo_1760568713.png)
We also need coefficients to handle the units. From a physical standpoint, we need a phase term to accommodate all the possible starting positions - at the equilibrium moving one way ( φ = 0), at the equilibrium moving the other way ( φ = π), all the way over to one side ( φ = π 2), all the way over to the other side ( φ = 3π 2), and everything in between ( φ = whatever). I think I'll go with the sine function and add an arbitrary phase shift or phase angle or phase ( φ, "phi") so that our analysis covers sine ( φ = 0), cosine ( φ = π 2), and everything in between ( φ = whatever). Trigonometric functions and derivatives function Nothing else is affected, so we could pick sine with a phase shift or cosine with a phase shift as well. When a trig function is phase shifted, it's derivative is also phase shifted. We have two possible functions that satisfy this requirement - sine and cosine - two functions that are essentially the same since each is just a phase shifted version of the other. The solution for this equation is a function whose second derivative is itself with a minus sign. On the right side we have the second derivative of that function. On the left side we have a function with a minus sign in front of it (and some coefficients). This is a second order, linear differential equation. Replace acceleration with the second derivative of displacement. There is only one force - the restoring force of the spring (which is negative since it acts opposite the displacement of the mass from equilibrium). (A restoring force acts in the direction opposite the displacement from the equilibrium position.) If the spring obeys Hooke's law (force is proportional to extension) then the device is called a simple harmonic oscillator (often abbreviated sho) and the way it moves is called simple harmonic motion (often abbreviated shm).īegin the analysis with Newton's second law of motion. The system will oscillate side to side (or back and forth) under the restoring force of the spring. Pull or push the mass parallel to the axis of the spring and stand back. Start the system off in an equilibrium state - nothing moving and the spring at its relaxed length. Fix one end to an unmovable object and the other to a movable object. Start with a spring resting on a horizontal, frictionless (for now) surface.