I read somewhere that there is an analogy between the motion of a mass on a spring and a mass going in circles in a certain velocity. How can they be analogous? I mean aren't they totally different?
In many books, the representative diagram of a uniform circular motion is used to represent an oscillatory system.
the spring attached to a mass m and if it is extended by a displacement say x then the spring force acts opposite to the displacement and pulls the body back.
however, this force generates a kinetic energy of the body and if it comes back to its original length the spring force is zero but the velocity pushes the spring compressing it by a length x and there its kinetic energy becomes zero and spring gets a potential energy to push back the mass and an oscillatory motion is set up.
IF the spring stretches to a maximum value x =A then this becomes the amplitude of oscillation.
If no dissipative forces are present and the spring is massless/ideal the motion is a simple harmonic one.
The Force F = -k.x where k is the spring constant
which can be written as
mass.acceleration =- k.x
or a = -(k/m).x = -w^2.x
where x is instantaneous displacement of the mass m from mean position.
The acceleration a =0 at x=0 and is maximum at x= +A or -A (A being the amplitude)
the velocity is max. at x=0 and zero at x= +A or -A
this motion can be represented on a diagram -
draw a circle of radius A the amplitude and move a radius vector from x=A to x= 0
then from x=0 to x=-A
drop a perpendicular from the tip to the x-axis, say at an angle theta from the x-axis.
The projection will be A.Cos (theta)
as the tip moves from x-axis to y-axis the projection will move from x=A to x=0 and again to x=-A.
Imagine a mass m moving at the tip of the projection
then one can see that the mass is oscillating between +A and -A as the radius vector travel the full circle.
Therefore the length of the base, i.e position of mass m can be written as
x = A Cos(wt) if the instantaneous position of the radius vector can be written as
theta = w.t
where w can be represented as the angular velocity of the radius vector and in a time T it completes one full rotation of 2.pi
w = (2.pi) /T
and each quadrant represents a time span of T/4
and the half circle represents T/2.
Thus the mass m which is placed at the base of the perpendicular oscillates between A and -A during a complete time period of T which gets related to w.
As the force is proportional to displacement in the opposite direction- when the mass reaches A or -A the force is max. i.e. the acceleration is max. but the velocity is momentarily zero and the velocity is maximum when it passes through x=0.
the above picture is a way to represent harmonic motion on a circular diagram. It is actually composed of two SHM's of identical time period but perpendicular to each other with a phase difference pi/2.
As one can draw a perpendicular on the y-axis and again the base of the perpendicular will oscillate along the y-axis with
y = A Sin (wt) and a mass can be imagined to oscillate along y with similar amplitude and w.