1
The pendulum and gravity
1. Measuring
pendulum
g
using
The linear pendulum
d
d
g
! = " and
!=" #
dt
dt
l
Initial conditions:
! (0) " 0 and ! (0) = 0
" g%
! (t ) = ! (0) cos$ t '
# l &
g $ g'
! (t ) = " # (0) sin& t)
l % l (
a
2
A linear pendulum (l = 1 m )
with ! (0) = 30° and ! (0) = 0
3
The period T of the pendulum
" g %
! (0) = ! (T ) = ! (0) cos$ T '
# l &
! g $
g
cos# T & = 1 ' T = 2 (
" l %
l
4
l
T = 2!
g
• Independent of the amplitude
! (0)
(Galileo in a cathedral)
• Longer for longer l
l
l
g = 4! 2 " 39 2
T
T
2
• Can measure g by measuring T
5
39
For l = 1 m : g ! 2
T
T=
(Time for ten cycles)
10
10T (sec)
T (sec)
2
g(m / sec )
The average value for g:
____________ m / sec
2
6
2. Solving differential equations
numerically (on a computer)
Initial data
Differential
equations
Time
evolution
of the system
A linear pendulum (small ! )
7
A linear pendulum
d
! ="
dt
d
g
!=" #
dt
l
#! (0) = 30° = 0.52 rad
$
%" (0) = 0
" g%
! (t ) = ! (0) cos$ t '
# l &
g $ g'
! (t ) = " # (0) sin& t)
l % l (
8
(1) At time t > 0:
! (t ) = 0° = 0 rad
! (t ) = "1.63 rad / sec
! g$
g (
cos# t & = 0 ' t =
" l %
l
2
! l
t=
= 0.50 sec
2 g
9
(2) Our goal: to find ! (t + "t )
and ! (t + "t )
Choose !t to be short so that
! (t + "t ) # ! (t )
and
! (t + "t ) # ! (t )
Choose: !t = 0.01 sec
(Note: this !t is different from
the one chosen in “Class
Notes”)
10
(3) Approximate the rates of
change
d
Change of ! during #t
!"
dt
#t
! (t + #t ) $ ! (t )
=
#t
d
Change of ! during #t
!"
dt
#t
! (t + #t ) $ ! (t )
=
#t
11
(4) Approximate the equations
! (t + "t ) # ! (t ) d
$ ! = % (t )
"t
dt
! (t + "t ) # ! (t ) d
$ !
"t
dt
g
= # % (t )
l
(5)
! (t + "t ) = ! (t ) + "t# (t )
g
! (t + "t ) = ! (t ) # "t $ (t )
l
12
(6)
! (t + "t ) = ! (t ) + "t# (t )
= (0 rad)
+ (0.01 sec)($1.63 rad / sec)
= $0.016 rad
! (t + "t = 0.51 sec)
# g
&
= (0.52 rad) cos$ (0.51 sec)'
% l
(
= )0.013 rad
13
g
! (t + "t ) = ! (t ) # "t $ (t )
l
= (#1.63 rad / sec)
(0.01 sec)(9.8 m / sec )(0)
#
(1 m )
= #1.63 rad / sec
2
! (t + "t = 0.51 sec)
'
g $ g
= #(0.52 rad) sin % (0.51 sec)(
& l
)
l
= #1.63 rad sec
14
(7) Repeat the procedure
! (t + 2"t )
= ! (t + "t ) + "t# (t + "t )
Compare with
! (t + 2"t )
# g
&
= (0.52 rad) cos$ (0.52 sec)'
% l
(
= )0.030 rad
15
! (t + 2"t )
g
= ! (t + "t ) # "t $ (t + "t )
l
Compare with
! (t + 2"t )
'
g $ g
= #(0.52 rad) sin % (0.52 sec)(
& l
)
l
= #1.63 rad / sec
16
(7)’ Repeat the procedure
! (t + 2"t )
= ! (t + "t ) + "t# (t + "t )
= ($0.016 rad)
+ (0.01 sec)($1.63 rad / sec)
= $0.032 rad
! (t + 2"t )
# g
&
= (0.52 rad) cos$ (0.52 sec)'
% l
(
= )0.030 rad
17
! (t + 2"t )
g
= ! (t + "t ) # "t $ (t + "t )
l
= (#1.63 rad / sec)
9.8 m / sec )
(
# (0.01 sec)
2
(1 m)
% ( #0.016 rad)
= (#1.63 rad / sec)
# (0.0016 rad / sec)
= #1.63 rad / sec
18
! (t + 2"t )
'
g $ g
= #(0.52 rad) sin % (0.52 sec)(
& l
)
l
= #1.63 rad / sec