SIMPLE HARMONIC MOTION

SIMPLE HARMONIC MOTION [SHM]

Oscillatory motion is the to and fro motion of particles about a fixed point called the origin of motion. Its equation is written as f(x)=a0+a1x+a2x^2+a3x^3...
Most of the oscillatory motions are governed by forces acting on them and they are called SHM when there is linear dependance of f(x) on the displacement of particle,i.e., only a1 in the f(x) equation is non-zero. The forces are written as F=-kx. Negativity occurs as the force acts opposite to the displacement.
From newton's second law, F=ma. Thus, comparing, k=m, x=a=d2x/dt2.
Thus, m.d2x/dt2=-kx
d2x/dt2=-kx/m=-k'x
k'=k/m. Thus the double derivative of the function gives a similar type of equation ( F(x)=-kx and F"(x)=-k'x ).
This relation is shown in sine and cosine function. The equation is hence written as
x=x'cos(wt + o).
x'=any multiplicative constant, w=angular velocity.
To determine the meaning of constants we differentiate the equation two times.
dx/dt=-wx'sin(wt+o)
d2x/dt2=-w^2x'cos(wt+o)
Thus, recalling the previous eqn., d2x/dt2=-kx/m,
-w^2x'cos(wt+o)=-kx'cos(wt+o)/m
and, if w is a constant, w^2=k/m.
When x=maximum,i.e., x=x', (wt+o) = 0. And at t=o, (wt+o)=o.
Thus,o = initial phase or epoch or phase constant.
Velocity of SHM = v =dx/dt=-x'wsin(wt+o)
v=w(x'^2-x^2)^1/2.
Acceleration of SHM=a=d2x/dt2=-x'w^2cos(wt+o).
Time period of oscillation is given by,
T=2(pie)/w.
The function 'x' thus repeats itself after 'T'.
Since w^2=k/m,
thus,
T=2(pie)(m/k)^1/2
m/k=inertia factor/elasticity factor.
Energy in SHM,
v=w(x'^2-x^2)^1/2.
Thus, Kinetic energy=1/2mv^2=1/2mw^2(x'^2-x^2).
Potential energy can be derived by integrating, dU=F.dx.
PE=1/2mw^2x^2.
Total energy=KE+PE=1/2mw^2x'^2, which is constant for constant m.

Spring cases:
a)for one spring, attached to a support and the other end to a block of mass m over a frictionless surface when displaced by x,
restoring force F=-kx,
ma=-kx
a+kx/m=0
a+w^2x=0
w^2=k/m
thus, T=2(pie)(m/k)^1/2

b)for a series of spring connected end to end,
equivalent spring constant = k', where,
1/k'=1/k1+1/k2+1/k3+...+1/kN.
w^2=k'/m
T=2(pie)/w as usual.

c) a series of springs attached to the block of mass m from the wall,
k'=k1+k2+k3+...+kN.
T calculated as usual.

d) when 2 spring systems of constant k and k' attaches two sides of a block of mass m to adjacent walls, k"=k+k' and T is calculated by the general formula.

e) when a block of mass M hanging from the ground is attached to a spring of mass m, and length l, connecting a wall.
Mass per unit length of spring=m/l
If M displaces by j, an element located at a distance s in the spring is displaced by js/l and its velocity is
v=s/l.(dj/dt)
Hence, KE of spring=1/2mv^2
and by integrating, we get KE of spring=1/6m(dj/dt)^2
KE of block=1/2M(dj/dt)^2
Total KE=1/2(M+m/3)(dj/dt)^2
PE=integral k.jdj=1/2kj^2
Hence total energy of system=PE+KE
=1/2(M+m/3)(dj/dt)^2+1/2kj^2
as we are in the domain of a conservative field,
dE/dt=0
d2j/dt2=-kj/(M+m/3)
we get,
w={k/(M+m/3)}^1/2
and,
T=2(pie)/w.