Write the equation of motion of a simple pendulum and use it to obtain the
condition under which its motion is simple harmonic in nature. Obtain an
expression for its time period under this condition.
Answers
Answer:
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Explanation:
The Equation of Motion
A simple pendulum consists of a ball (point-mass) m hanging from a (massless) string of length L and fixed at a pivot point P. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion. By applying Newton's secont law for rotational systems, the equation of motion for the pendulum may be obtained
τ
=
I
α
⇒
−
m
g
sin
θ
L
=
m
L
2
d
2
θ
d
t
2
and rearranged as
d
2
θ
d
t
2
+
g
L
sin
θ
=
0
If the amplitude of angular displacement is small enough, so the small angle approximation ($\sin\theta\approx\theta$) holds true, then the equation of motion reduces to the equation of simple harmonic motion
d
2
θ
d
t
2
+
g
L
θ
=
0
The simple harmonic solution is
θ
(
t
)
=
θ
o
cos
(
ω
t
)
,
where
θ
o
is the initial angular displacement, and
ω
=
√
g
/
L
the natural frequency of the motion. The period of this sytem (time for one oscillation) is
T
=
2
π
ω
=
2
π
√
L
g
.
Small Angular Displacements Produce Simple Harmonic Motion
The period of a pendulum does not depend on the mass of the ball, but only on the length of the string. Two pendula with different masses but the same length will have the same period. Two pendula with different lengths will different periods; the pendulum with the longer string will have the longer period.
How many complete oscillations do the blue and brown pendula complete in the time for one complete oscillation of the longer (black) pendulum?
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