Shown
is a simple pendulum which has a mass
m
m
that is displaced by an angle
θ
θ
.
There is tension
(
T
⃗
T
⃗
)
in the string which acts from the mass to the anchor point. The weight of the
mass is
m
g
⃗
m
g
⃗
and the tension in the string is
T
=
m
g
cos
θ
T
=
m
g
cos
θ
.
There is a tangential restoring force
=
−
m
g
sin
θ
=
−
m
g
sin
θ
.
If we approximate that
θ
θ
is small (we have to make this approximation or else we can not solve the
problem analytically) then
sin
θ
≈
θ
sin
θ
≈
θ
and
x
=
l
θ
x
=
l
θ
.
(note that
sin
θ
sin
θ
is only approximately equal to
x
l
x
l
because
x
x
is the distance along the
x
x
axis) so that we can write:
F
=
m
a
=
m
x
¨
=
−
m
g
sin
θ
≈
−
m
g
θ
≈
−
m
g
x
l
F
=
m
a
=
m
x
¨
=
−
m
g
sin
θ
≈
−
m
g
θ
≈
−
m
g
x
l
or
x
¨
+
g
l
x
=
0
x
¨
+
g
l
x
=
0
(Note
that We should immediately recongnize that this is the equation for simple
harmonic motion (SHM) with
ω
=
g
l
.
ω
=
g
l
.
We could take another approach and use angular momentum to
solve the problem. Recall that:
L
=
I
ω
=
I
θ
˙
L
=
I
ω
=
I
θ
˙
I
=
m
l
2
.
I
=
m
l
2
.
Also
recall that the torque is the time derivative of the angular momentum so that:
τ
⃗
=
r
⃗
×
F
⃗
=
ⅆ
L
⃗
ⅆ
t
−
l
m
g
θ
=
I
θ
¨
τ
⃗
=
r
⃗
×
F
⃗
=
ⅆ
L
⃗
ⅆ
t
−
l
m
g
θ
=
I
θ
¨
θ
¨
+
g
l
θ
=
0
θ
¨
+
g
l
θ
=
0
Again
we would recognize that this is simple harmonic motion with
ω
=
g
l
.
ω
=
g
l
.
"This book covers second year Physics at Rice University."