Derive expression for two natural frequencies
for small
oscillations of the pendulum srain in figlire 3 in the plane of
the paper, assumes rodes massless and rigld Also obtain
expressions for angular amplitude ratio in the two rods.
कागज के plane में चित्र 3 में पेंडुलम के हुग के छोटे दोलनों के लिए दो
प्राकृतिक आवृतियों के लिए व्युत्पन्न अभिव्यक्ति, छड को भारहीन तथा कठोर
मानते हैं। छड़ में कोणीय आयाम अनुषार के लिए अभिव्यक्ति प्राप्त करें।
LIKA
a
min
Figure 3
a
om
bra
Answers
Answer:
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
.
Explanation:
Answer:
plzz post it correctly ...