Theory
point mass suspended
may be described ideally as a
A simple pendulum
by a massless string from some point about which it is allowed to swing
back and forth in a place.in radians is X L
/ , the arc length divided by the length of the
pendulum or the radius of the circle in which the mass moves.When a simple pendulum is displaced from its equilibrium position, there
will be a restoring force that moves the pendulum back towards its
equilibrium position
If the restoring force F is opposite and directly proportional to the
displacement x from the equilibrium position, so that it satisfies the
relationship
F=-KX (1)
then the motion of the pendulum will be simple harmonic motion and its
period can be calculated using the equation for the period of simple
harmonic motion
T = 2??m
k (2)
It can be shown that if the amplitude of the motion is kept small,
Equation (2) will be satisfied and the motion of a simple pendulum will
be simple harmonic motion, and Equation (2) can be used
The restoring force depends on the gravitational force and the displacement
of the mass from the equilibrium position.Similarly, the period is the inverse of the frequency, T = l/f
The maximum distance that the mass
defined as is
(A)
he amplitude T -
is displaced from its equilibrium position.time that it takes to make one complete defined the
is
period( T ) -
oscillation is
of the oscillations is the number of oscillations that )
f
( The frequency-
occur per unit time and is the inverse of the period, f = 1/T.The gravitational force can be resolved into two components,
one along the radial direction, away from the point of suspension, and one
along the arc in the direction that the mass moves.The component of the
gravitational force along the arc provides the restoring force F and is given
by
F = -mgsin?is the angle the pendulum is displaced
the minus sign indicates that the force is opposite to the displacement.Consider Figure 1 where a mass
m is suspended by a string of length l and is displaced from its equilibrium
position by an angle ?The restoring
force is then given by
F = -mg
X
L (5)
and is directly proportional to the displacement x and is in the form of
Equation (1) where K = -
mg
L .If a pendulum is set in motion so that is swings back and forth, its motion
will be periodic.measured
in radians so that Equation (3) can be written as
F = -mg?Substituting this value of k into Equation
(2), the period of a simple pendulum can be found by
T = 2??L
g (6).(3)
where g is the acceleration of gravity
?For
small amplitudes where ?