Cycle, Period, Amplitude, Frequency, Wavelength
Cycle: One complete repeat of the pattern/vibration
Period: The time required to complete one cycle. (unit is s)
Amplitude: The distance from the equilibrium position (resting position) to the maximum displacement when in motion.
Frequency: the reciprocal of period (unit is Hz: 1 Hz = 1 s -1).
Wavelength: the length of one complete wave cycle.
Simple Harmonic Motion
Simple harmonic motion is any periodic motion in which:
- The acceleration of the object is directly proportional to its displacement from its equilibrium position.
- The acceleration is always directed towards the equilibrium position.
2 examples of simple harmonic motion are the spring and the pendulum.
Spring - Horizontal
Hooke's Law
When an elastic object - such as a spring - is stretched, the increased length is called its extension. The extension of an elastic object is directly proportional to the force applied to it:
F = k * x
F is the force in newtons, N
k is the 'spring constant' in newtons per metre, N/m
x is the extension in metres, m
This equation works as long as the elastic limit (the limit of proportionality) is not exceeded. If a spring is stretched too much, for example, it will not return to its original length when the load is removed.
The spring constant k is different for different objects and materials.
Restoring Force & Acceleration
When the mass m is pulled a displacement x, a restoring force of F = - k * x will act on it when released. Since F = m * a, where a is acceleration, we also have:
m * a = - k * x, or a = -(k/m) * x
What this means is that the spring mass system is a typical simple harmonic motion as the acceleration is directly proportional to the negative of the displacement (extension).
Displacement
We can calculate the displacement of the object at any point in it’s oscillation using the equation below.
x = A cos (2Πft)
Where
x is spring extension or displacement in meter, A is amplitude (maximum displacement) in meter, t is the time since the oscillation began in seconds, and f is frequency of the oscillation (f = 1/T where T is period of the oscillation).
When using the equation above, your calculator must be in radians not degrees!
Now let's prove it.
The proof can be done by thinking of simple harmonic motion as a projection of a uniform circular motion.
Picture mass M is performing a uniform circular motion in a vertical plane as shown. Its shadow on the x-axis performs a back-and-forth simple harmonic motion. In the following analysis, ω is angular veolocity (also called angualr frequency, radian frequency, or radial frequency, whose unit is radian/second) of the uniform circular motion, t is time since the oscillatoin began in seconds, T is period in seconds, and f is frequency of the oscillation.
From the diagram:
cosθ = x/A; x = A cosθ
Since ω = θ/t, we have θ = ωt. Therefore:
x = A cosθ = A cos (ωt)
Since ω = θ / t = 2Π / T = 2Π f, we have
x = A cos (2Πft)
Period and Frequency
The period T of the spring is given by the following equation:
Where m is mass, and k is spring constant
The frequency f = 1/T
Potential and Kinetic Energy
Maximum potential energy PEmax = (1/2) kA2, where k is spring constant, A is ampilude (or maximumly stretched displacement)
Maximum kinetic energy KEmax = (1/2) mv2max = (1/2) kA2, where v is maximum velocity (achieved at equilibrium point)
Potential energy at displacement x PE = (1/2) kx2
Kinetic energy at displacement x PK = (1/2) mv2, where v is velocity at displacement x
(1/2) kA2 = (1/2) kx2 + (1/2) mv2
Velocity
From above, you can also derive the equation for velocity at displacement x:
Spring - Vertical
Consider a spring of negligible mass hanging from a stationary support. A block of mass m is attached to its end and allowed to come to rest, stretching the spring a distance d. At this point, the block is in equilibrium, ie, the upward spring force = the downward gravitational force:
kd = mg -> d = mg/k, where k is spring constant, and g is gravity.
If the spring is further stretched to d+y, then the net force on the block will be:
F = k(d+y) - mg = ky
From above, we can see that vertical spring is just anothe SHM, and you can apply the same equations of horizontal spring system, with the only difference being the equilibrium position is not at the spring's natural length.
Pendulum
Free Body Diagram Analysis
There are 2 forces acting on the pendulum: gravity and tension.We are ignoring the influence of air resistence.
The gravity is always of the same magnitude - mass*9.8 N/kg, and its direction is always pointed downward.
The tension force is less predictable. The direction of the tension force is always towards the pivot point. Its magnitude changes as the pendulum moves back and forth.
The force of gravity is resolved into two components: One of the components is directed tangent to the circular arc along which the pendulum moves; this component is labeled Fgrav-tangent. The other component is directed perpendicular to the arc; it is labeled Fgrav-perp.
Fgrav-perp is in the opposite direction of the tension force, and its magnitude is smaller than the tension force, so that there will be a net centripetal force.
Fgrav-tangent is a net force which acts as the restoring force. As the pendulum bob moves to the right of the equilibrium position, this force component is directed opposite its motion back towards the equilibrium position.
Period
Period of pendulum is given by the following equation: