simple harmonic Motions
An object is said to be in simple harmonic motion if the
- It moves in a straight line.
- A variable force acts on it.
- The magnitude of force is proportional to the displacement
of the mass.
- The force is always opposite in direction to the
- The motion is repetitive and a round trip, back and forth,
is always made in equal time periods.
There are two basic examples of simple harmonic motion:
springs and pendulums.
A spring that is oscillates an attached mass back and forth on
a frictionless surface is an example of simple harmonic motion. The magnitude
and direction of the force in terms of its displacement is:
F = -kx
k is a constant for the particular spring and x
is the displacement. The right side of the equation is negative because the
force always acts opposite the displacement (positions 2 and 4 on the
illustration at right). No force is acting when the mass returns to its original
position (positions 1 and 3), but the mass keeps moving because it was moving
before. In addition, at the original position, the mass is moving at its highest
speed. Once it passes the original position, the force acts to slow it down and
move it in the opposite direction. The entire cycle repeats again and again in a
certain period of time.
If you look at the equation for the period, you'll notice that
it is independent of the maximum displacement. No matter how far you stretch or
compress the spring, the period will remain the same.
The motion of a pendulum can be considered simple harmonic
motion even though the bob hanging at the end of the string moves in a curve.
This is because if the string is relatively long compared to the initial
displacement, the curve made by the bob is close enough to a straight line.
The pendulum works almost like the spring. The force is always pointing opposite
the displacement. The bob is moving the fastest when it passes its lowest point
(position 2 on the illustration at right). The pendulum continuously moves back
and forth. Also, the equation for the period of a pendulum is similar to that of
L is the length of the string and g
is the gravitation acceleration (your good ol' 9.8 m/s2). Note that
the period does not depend at all on the mass or the initial displacement.
Check out the Pendulum
in our games and fun stuff section to experiment with how a pendulum works.