In this case, the motion of the particle (or body)
involved is to and fro along a straight line, besides
the other necessary conditions.
Let a particle `P` perform oscillatory motion between
two fixed points `A` and `B`. Let `O` be the mid point of
`A` and `B`. (fig.) If the particle `P` oscillates about point
`O` in such a way that its acceleration `f` , at any
position when its displacement from O is `x` , can be
mathematically expressed as `f` `x` and is directed
towards `O`, then its motion will be `SHM`. Here, the
point `O` is known as mean or stable or
equilibrium or neutral position, and in case of "simple"
harmonic motion , the maxim tun displacement, of
the particle on either side of the mean position is
the same i.e. `O A = OB` .
Now, taking the motion of the particle to be along Xaxis,
SHM can be mathematically expressed as
`vecf = (-omega^2)vecx...........................(1)`
Here, the negative sign stands for the fact that the
direction of acceleration is opposite to that of
displacement, and `omega^2` is a positive constant.
Again, if m be the mass of the oscillating particle (or
body), then, multiplying both sides of eqn. (1) by m,
we have
`mvecf = (-momega^2)vecx` or `vecF = (-k)vecx......................(2)`
where `mvecf = vecF` is the force acting on the particle and
k = `m omega^2` a positive constant.
Equations (1) or (2) can equally be used to show
that a given motion is SHM.
From equation (2) it is evident that `X = 0 ; F = 0`
which shows that the particle experiences no force,
when at mean position, it will not oscillate.
However, if it is disturbed even slightly by external
force, then new forces (restoring forces) should be
set up in the system which should tend to bring the
particle to its mean position. Depending upon the
nature of restoring forces, we have several types of
SHM. The restoring forces can be electrical,
gravitational, magnetic, elastic etc.
In this case, the motion of the particle (or body)
involved is to and fro along a straight line, besides
the other necessary conditions.
Let a particle `P` perform oscillatory motion between
two fixed points `A` and `B`. Let `O` be the mid point of
`A` and `B`. (fig.) If the particle `P` oscillates about point
`O` in such a way that its acceleration `f` , at any
position when its displacement from O is `x` , can be
mathematically expressed as `f` `x` and is directed
towards `O`, then its motion will be `SHM`. Here, the
point `O` is known as mean or stable or
equilibrium or neutral position, and in case of "simple"
harmonic motion , the maxim tun displacement, of
the particle on either side of the mean position is
the same i.e. `O A = OB` .
Now, taking the motion of the particle to be along Xaxis,
SHM can be mathematically expressed as
`vecf = (-omega^2)vecx...........................(1)`
Here, the negative sign stands for the fact that the
direction of acceleration is opposite to that of
displacement, and `omega^2` is a positive constant.
Again, if m be the mass of the oscillating particle (or
body), then, multiplying both sides of eqn. (1) by m,
we have
`mvecf = (-momega^2)vecx` or `vecF = (-k)vecx......................(2)`
where `mvecf = vecF` is the force acting on the particle and
k = `m omega^2` a positive constant.
Equations (1) or (2) can equally be used to show
that a given motion is SHM.
From equation (2) it is evident that `X = 0 ; F = 0`
which shows that the particle experiences no force,
when at mean position, it will not oscillate.
However, if it is disturbed even slightly by external
force, then new forces (restoring forces) should be
set up in the system which should tend to bring the
particle to its mean position. Depending upon the
nature of restoring forces, we have several types of
SHM. The restoring forces can be electrical,
gravitational, magnetic, elastic etc.