Notes to a video lecture on http://www.unizor.com
Mechanics, as a subject, deals with movements of different objects. Among these movements there are those that we can call "repetitive". Examples of these repetitive movements, occurring during certain time segment, are rotation of a carousel, swinging of a pendulum, vibration of a musical tuning fork, etc. Here we are talking about certain time segment during which these movements are repetitive, because after some time these movements are changing, if left to themselves.
These repetitive movements might be of a kind when the repetitions are to a high degree exactly similar to each other (like in case of a pendulum) or some of the characteristics of the motion change in time (like in case of a tuning fork).
Repetitive movements that can be divided into equal time segments, during which the movements to a high precision repeat exactly each other, are called periodic.
The time segments of such a repetitive movement are called periods.
If a position P of an object making periodic movement with a period T is defined by a set of Cartesian coordinates P=(x,y,z) as a vector function of time P(t), the periodicity means that for any time moment t
P(t) = P(t+T)
which is exactly the mathematical definition of a periodic function.
For example, a period of rotational movement of a carousel equals to a time it takes to make one circle. The period of a movement of a pendulum is the time it moves from left most position all the way to the right most and back to the left.
A case of a vibrating tuning fork is a bit more complex because gradually the vibrations, after being initiated, diminish with time. The period of vibration might be the same during this process, but the amplitude (deviation from a middle point) would diminish with time.
A special type of periodic movement is oscillation. It's characterized by a periodic movement of an object that repeats the same trajectory of movement in alternating directions, back and forth. For example, a pendulum, an object on a spring, a tuning fork, a buoy on a surface of water under ideal weather conditions etc.
In all those systems we can observe a specific middle point position from which an object can deviate in both directions. If put initially at this position, an object would remain there, unless some external force acts on it. This is a point of a stable equilibrium. Then, after some external force is applied, it will move along its trajectory back and forth, each time passing this equilibrium point.
From this point an object can move along a trajectory to some extreme position, then back through an equilibrium point to another extreme position, then back again, repeating a movement along the same trajectory in alternating directions.
Oscillation is only possible if some external forces act on a moving object towards stable equilibrium point. Otherwise, it would never return to an equilibrium. These forces must depend on the position, not acting at the equilibrium point, acting in one direction in case an object deviated from an equilibrium to one side along its trajectory and acting in the opposite direction in case an object deviated to the other side along a trajectory.
A very important type of oscillations are so-called harmonic oscillations.
An example of this type of a movement is an object on an initially stretched (or squeezed) spring with the only force acting on an object during its movement to be the spring's elasticity.
According to the Hooke's Law, the force of elasticity of a spring is proportional to its stretch or squeeze length and directed towards a neutral point of no stretch nor squeeze.
If a string is positioned along the X-axis on a Cartesian system of coordinates with one end fixed to some point with negative coordinate on this axis, while its neutral point at x=0, the position of an object attached to this spring and oscillating can be described as a function of time x(t) that satisfies two laws:
(1) the Second Newton's Law connecting the force of elasticity F(t) to the mass m and acceleration (second derivative of position)
F(t) = m·x"(t) = m·d²x(t)/dx²
(2) the Hooke's Law connecting the force of elasticity F with a displacement of a free end of a spring from its neutral position
F(t) = −k·x(t)
(where k is a coefficient of elasticity that is a characteristic of a spring).
From these two equations we can exclude the force F(t) and get a simple differential equation that defines the position of an object at the free end of a spring x(t).
m·x"(t) = m·d²x(t)/dx² = −k·x(t)
x"(t) = −(k/m)·x(t)
Obviously, trigonometric functions sin(t) and cos(t) are good candidates for a solution to this equation since their second derivative looks like the original function with some coefficients
sin"(t) = -sin(t)
cos"(t) = -cos(t)
General solution to the above linear differential equation is
x(t) = C1·cos(ωt) + C2·sin(ωt)
where ω depends on coefficients of the differential equation and constants C1 and C2 depend on initial conditions (initial displacement of the object off the neutral position on a spring and its initial speed).
x'(t) = −C1·ω·sin(ωt) +
x"(t) = −C1·ω²·cos(ωt) −
x"(t) = −(k/m)·x(t)
we conclude that
−(k/m)·x(t) = −C1·ω²·cos(ωt) −
+ C2·sin(ωt)] =
from which immediately follows
ω = √k/m
Assume, initially we stretch a spring by a distance a from the neutral position (that is, x(0)=a) and let it go without any push (that is, x'(0)=0).
From these initial conditions we can derive the values of constants C1 and C2
a = x(0) =
= C1·cos(0) + C2·sin(0) = C1
0 = x'(0) =
= −C1·ω·sin(0) + C2·ω·cos(0) =
from which immediately follows
C1 = a
C2 = 0
and the solution for our differential equation with given initial conditions is
x(t) = a·cos(√k/m·t)
The oscillations described by the above function x(t) in its general form x(t)=a·cos(ω·t) are called simple harmonic oscillations.
Parameter a characterizes the amplitude of harmonic oscillations, while parameter ω represents the angular speed of oscillations.
Function cos(t) is periodical with a period T=2π.
Function cos(ωt) is also periodical with a period T=2π/ω.
= cos(ω(t+2π/ω)) =
= cos(ωt+2π) =
Therefore, the simple harmonic oscillations in our case have a period (the shortest time the object returns to its original position)
T = 2π/ω = 2π·√m/k
If one full cycle the oscillation process makes in time T, we can find how many cycles it makes in a unit of time (1 sec) using a simple proportion
1 cycle - T sec
f cycles - 1 sec
Hence, f = 1/T
Therefore, the object on a spring we deal with makes
f = 1/T = (1/2π)·√k/m
oscillations per second.