S
mM ×4 ¤ À W ¥ ² oI í Ä8 ý ÇÊ Ý õ u § w \ X ¢ ú n ÞI í Ä Ò ×y ¢8 ý ¤X N Ë
ý
¡) o - > ∗
â
§¹ ¢ ¤ @ / < Æ § õ < Æ §¹ ¢ ¤ õ , ; 407-753
T ~ ç ¡* å †
ô
Dz D G õ < Æl Õ ü t" é ¶ Ó ü t o < Æõ , @ / 305-701 (2004¸ 2 Z 4 16{ 9 ~ à Î6 £ §)
r
ç ß l 2 ¤ > \ ¦ s 6 x # × æ§ 4 5 Å q ¸ g\ ¦ 8 £ ¤& ñ ½ + É M :\ r ç ß l 2 ¤ > \ _ K ¿ º 7 á x À Ó_ ¹ 1 ϧ 4 ({ 9 & ñ ô
Ç ¹ 1 ϧ 4 õ Å Òl & h ¹ 1 ϧ 4 )s 7 á x s _ s á Ô\ ¦ : x # z Ó ü t ^ \ K t H X < Õ ª ´ òõ H H & h Ü ¼
Ð ß ¼l η
tot { 9 & ñ ô Ç 8 ú x Ä »´ ò ¹ 1 ϧ 4 s K t H כ õ ° ú . s M : z Ó ü t ^ _ 5 Å q ¸ G H Õ ª Ó ü t
^
_ | 9 | ¾ Ó m\ _ > r 9 H & h Ü ¼ Ð G = g − η
tot/m Ü ¼ Ð Å Ò# Q . z ´+ « > X <s ' \ _ ô Ç G @ / m
−1_ Õ ªA á Ô\ @ / # f d ¹ 1 Ôl \ ¦ Ã º' < ÊÜ ¼ Ð+ g = 9.66 ± 0.03 m/s
2( © @ / ¸ = 1.43 %)ü <
η
tot= 0.035 ± 0.002 N` ¦ % 3 % 3 .
PACS numbers: 01.50.Pa, 01.55.+b, 06.30.Gv Keywords: × æ§ 4 5 Å q ¸, r ç ß l 2 ¤ > , ¹ 1 ϧ 4
×
æ§ 4 5 Å q ¸ g H l : r& h Ó ü t o | ¾ Ó{ 9 ÷ rë ß m t ½ ¨ Ó
ü
t o , ¸| ¾ Ó+ þ A < Æ(metrology), " é ¶ Ã Ð \ " f B Ä º × æ כ ¹
. : r 7 Hë H \ " f H × æ§ 4 5 Å q ¸ 8 £ ¤& ñ ` ¦ 0 A # §¹ ¢ ¤ & ³ ©
\
" f ¥ y 6 x ÷ & H r ç ß l 2 ¤ > ~ ½ ÓZ O ` ¦ > h ¦ ô Ç
. Õ ª\ · ú ¡" f þ j' é ß × æ§ 4 > _ & ³S ! õ Ó ü t o §¹ ¢ ¤ & ³ © \
"
f 6 x ÷ & H ª ô Ç ~ ½ ÓZ O [ þ t` ¦ è> h H כ É r u e ` ¦
כ s .
s Ö 0 q H F g < Æ ç ß [ O > Ð s À Ò# Q × æ§ 4 > H f ~ ½ Ó
¾ ÓÜ ¼ Ð & ñ § > = ) a ¼ 1 Ï\ " f Ä »z H ï - Ç ©Ú Ô(corner cube)\ ¦ 6 xô Ç . s ( r & h ) Ç ©Ú Ô_ f ~ ½ Ó ¾ Ó î r1 l x
É
r ç ß [ O ñ РÒ' ½ + É Ã º e . s × æ§ 4 > H & ñ S X
¸ ∆g/g = 2 × 10 −9 þ j ¦_ ] X @ / × æ§ 4 > (absolute gravimeter) Ð" f É r 8 £ ¤& ñ [ þ t_ l ï r s ) a [1]. " é ¶
\
¦ z Ó ü t ^ Ð 6 x½ + É â Ä º\ H Y Us $ F g s " é ¶ \ ¦ d
> §ê ø Í Ù ¼ Ð s Ö 0 q H ç ß [ O > \ ¦ 6 x½ + É Ã º \ O ¦, Y U s
$ Ð Í ty ) a " é ¶ [ þ t_ ì r à º(atomic fountain)\ l í ô
Ç " é ¶ ç ß [ O > Ð s À Ò# Q × æ§ 4 > \ ¦ 6 xô Ç . s © u
\ " f " é ¶ H Ñ ü t s © _ / B Nç ß & h Ü ¼ Ð b # Q " é ¶ © I [
þ
t_ × æ^ o ? © I \ Z ~ # . s © I [ þ t É r y y ô Ç > h_
ª % i < Æ& h 0 A © ½ Ó\ _ K l Õ ü t ÷ &# Qt ¦, × æ \ ½ + Ë5 g t
" f Ð ç ß [ O ` ¦ { 9 Ü ¼ . " é ¶ ì r à º\ " f Y Us $ Ð Í t y
) a [ j¸ o u(caesium) " é ¶ \ ¦ 6 xô Ç ë ß " é ¶ ç ß [ O > \
∗
E-mail: [email protected]
†
E-mail: [email protected]
_
ô Ç × æ§ 4 5 Å q ¸ 8 £ ¤& ñ _ & ñ S X ¸ H & h ì r r ç ß \ 1.3
í é ß { 9 8 £ ¤& ñ Å Òl Ê ê\ H 2 × 10 −8 , 1 ì r Ê ê\ H 3 × 10 −9 , 2 { 9 Ê ê\ H 1 × 10 −10 s [1]. þ j H \ H " é ¶ ì
r à º ¿ º > h Ð " é ¶ ç ß [ O > ¿ º > h\ ¦ ë ß [ þ t # Q " é ¶ â ¸8 £ ¤
&
ñ l (atomic gradiometer) Ð 6 x ¦ × æ§ 4 " é ¶ s ÷ & H | 9
|
¾ Ó(source mass)_ 0 Au \ ¦ or v " f z ´+ « >` ¦ ì ø Í4 ¤
#
¸ ú ñü < > : x ¸ (systematic error)\ ¦ × ¦ # × æ§ 4
5 Å
q ¸ 8 £ ¤& ñ _ & ñ S X ¸\ ¦ 10 −11 Ü ¼ Ð ¾ Ó © r v H z ´+ « >~ ½ ÓZ O s
] jî ß ÷ &% 3 [2]. ¢ ¸ô Ç C 60 Û ¦ QE $ (fullerene) ì r [ þ t` ¦ s
6 xô Ç ì r ç ß [ O > × æ§ 4 5 Å q ¸ 8 £ ¤& ñ \ & h 6 x ÷ &% 3 [3]. t F K t C 60 H ª & h ' 1 l x s S X ) a © 4 ¤¸ ú
¦ Á º î r Ó ü t ^ s . × æ§ 4 5 Å q ¸ ° ú כ É r [ j (grating) Mach-Zehnder ç ß [ O > \ " f \ P c (thermal beam)_ 0 A © s
1 l x` ¦ 8 £ ¤& ñ < ÊÜ ¼ Ð+ Ä » ¸ | ¨ c à º e . Û ¦ QE $ ì r 10 8
>
hü < ç ß s 500 nm \ ¦ 6 x½ + É â Ä º\ 8 £ ¤& ñ _ & ñ S X
¸ H 1 × 10 −4 s .
s
] j Ó ü t o §¹ ¢ ¤ & ³ © \ " f × æ§ 4 5 Å q ¸\ ¦ 8 £ ¤& ñ H ª ô
Ç ~ ½ ÓZ O \ @ / # · ú Ð .
A. Ó ü t ^ \ ¦ f ~ ½ Ó ¾ ÓÜ ¼ Ð Ä »z r ~ ´ â Ä º:
(a) ü < " 3 » ¡ § r > (stopwatch)\ ¦ 6 x # z o sü <
z
r ç ß t\ ¦ y y 8 £ ¤& ñ ¦ s = gt 2 /2\ ¦ s 6 x.
(b) Ä »z # Qé s' (adapter)\ n t _ O s Q '
`
s Û ¼\ ¦ # z r ç ß ` ¦ 8 £ ¤& ñ [4].
(c) í Ð> s à Ô\ ¦ s 6 xô Ç z r ç ß 8 £ ¤& ñ .
-324-
(d) z Ó ü t ^ Ð" f x Ö ¿ -K $ Û ¼(picket-fence)\ ¦ 6 x ¦ í
Ð> s à Ô\ ¦ s 6 x # r ç ß 8 £ ¤& ñ [5–7].
(e) l Ä » ¸\ ¦ s 6 x: f ~ ½ Ó ¾ ÓÜ ¼ Ð { 9 § > = Ð b # Q4 R e
H ¿ º ï{ 9 5 Å q Ü ¼ Ð % ò ½ ¨ $ 3 ` ¦ z r v ¦ ï{ 9 \ µ 1 ÏÒ q t
H Ä » ¸ À Ó\ ¦ PC \ [ O u ) a ¸z ´ ÐÛ ¼ ïá Ô\ ¦ s 6 x
#
8 £ ¤& ñ < ÊÜ ¼ Ð+ ¿ º ï{ 9 ç ß _ z r ç ß ` ¦ 8 £ ¤& ñ [8].
(f) Û ¼à Ô Ð ÐÛ ¼ ïá Ô\ ¦ s 6 x # ` ¦ O É % ò ¦ { 9 & ñ ô Ç
$ 3
F g r ç ß ç ß Ó ü t ^ _ 0 Au \ ¦ 8 £ ¤& ñ # 5 Å q ¸ü < × æ
§
4 5 Å q ¸\ ¦ ½ ¨ô Ç .
(g) n t _ O q n ¸ ì r$ 3 [9]: \ V\ ¦ [ þ t # Q, VideoPoint á Ô Ð Õ
ªÏ þ [10] s 6 x.
(h) (7 á x s _ s á Ô) r ç ß l 2 ¤ > _ s 6 x [11].
(i) Behr Ä »z © u ü < Û ¼ ß ¼ s Q s 6 x [12].
(j) G ' p" f(î r1 l xG ' p" f ¢ ¸ H 5 Å q ¸ G ' p" f)ü < ' ` s Û ¼ s 6 x [6,7].
B. Ó ü t ^ \ ¦ â y θ c ± Ü ¼ Ð ? / 9 Ðè q â Ä º: A_
~
½ ÓZ O [ þ t × æ \ " f (b), (d), (e), (i)\ ¦ ] jü @ô Ç Qt ~ ½ ÓZ O [ þ t
É
r ¸¿ º s 6 x 0 p x . c ± ` ¦ \ # Qà ÔÏ þ Ü ¼ Ð ë ß [ þ t ¦ í
Ð> s à Ô\ ¦ 6 x [5], Ð: x_ c ± ` ¦ 6 x H @ /
3 ' ² ú 2 ; à Ô\ ¦ 6 x # ¹ 1 Ï` ¦ × ¦ s ¦ G ' p" fü < '
` s Û ¼\ ¦ 6 x # [6, 7] à Ô_ 5 Å q ¸ a\ ¦ 8 £ ¤& ñ ô Ç
. Õ ª Q a = g sin θ ' a > \ ¦ s 6 x # × æ§ 4 5 Å q ¸\ ¦
½
¨½ + É Ã º e .
C. í ^ _ íÓ ü t î r1 l x_ â Ä º: A_ ~ ½ ÓZ O [ þ t × æ \ " f (f) Û ¼à Ô Ð ÐÛ ¼ ïá Ô\ ¦ s 6 xô Ç O É % ò ì r$ 3 õ (g) n t _ O
q n ¸ ì r$ 3 [9]` ¦ s 6 x½ + É Ã º e .
D. Atwood l > _ s 6 x: z r ç ß 8 £ ¤& ñ ` ¦ 0 A # " 3
»
¡
§ r > [13], (7 á x s _ s á Ô) r ç ß l 2 ¤ > [14], Û ¼ ß ¼ s
Q [15] ¢ ¸ H í Ð> s à Ôü < ' ` s Û ¼ [6,7]\ ¦ 6 x½ + É Ã
º e .
E. _ s 6 x:
(a) é ß : 1 l x Å Òl \ ¦ 8 £ ¤& ñ < ÊÜ ¼ Ð+ × æ§ 4 5 Å q ¸\ ¦ ½ ¨
H ~ ½ ÓZ O [5]õ , Ó ü t ^ _ 5 Å q ¸\ ¦ Û ¼ ß ¼ s Q` ¦ s 6 x
#
8 £ ¤& ñ ¦ % i < Æ& h \ -t Ð > r Z O g Ë :Ü ¼ РÒ' × æ§ 4
5 Å
q ¸\ ¦ ½ ¨ H ~ ½ ÓZ O [16]s e . (b) Borda [17].
(c) Kater [18]ü < > h| ¾ Ó ) a Kater [19].
0 A\ " f ¶ ú ( R : r Ó ü t o §¹ ¢ ¤ ~ ½ ÓZ O [ þ t × æ \ " f r ç ß l 2 ¤ > ~ ½ Ó Z O
É r, q 2 ¤ ¸A ) a ~ ½ ÓZ O s l H t ë ß , î r1 l x s f ] X & h Ü ¼
Ð ì r$ 3 ÷ &l M :ë H \ §¹ ¢ ¤& h u Z } Ü ¼ 9 f ¸ ; ¤ V ,
> 6 x ÷ & ¦ e . t F K Â Ò' : r 7 Hë H \ " f H Ó ü t ^ \ ¦ f
~ ½ Ó ¾ ÓÜ ¼ Ð Ä »z r v ¦ r ç ß l 2 ¤ > \ ¦ s 6 x # × æ
§
4 5 Å q ¸\ ¦ ½ ¨ H ~ ½ ÓZ O \ í& h ` ¦ ´ ú Æ Òl Ð ô Ç .
Ä »z î r1 l x É r { 9 & ñ ô Ç × æ§ 4 © \ " f { 9 & ñ ô Ç 5 Å q ¸
\
¦ . / B N l $ ½ Ó` ¦ Á ºr , í5 Å q ¸ v 0 Ó ü t ^ _
Fig. 1. The graph of v − v 0 as a function of t. The solid (straight) line is the graph of Eq. (3) in the pres- ence of the frictional forces exerted by the tape timer.
Here we used the values of the parameters: m = 0.03 kg, τ = 0.003 s, T = 1/60 s, η = 0.02 N, η 0 = 1/6 N, g = 9.80 m/s 2 , and η tot = η + η eff = 0.05 N. The solid line turns out to be very close to the graph of v − v 0 = (g − η tot /m)t. The graph in the small rect- angle is the magnified view of the solid line for a short time comparable to T . The dotted line is the graph for free fall without frictional forces.
5 Å
q ¸ H r ç ß _ < ÊÃ º Ð" f v(t) = v 0 +gt Ð Å Ò# Q . s < Ê Ã
º_ Õ ªA á Ô_ l Ö ¦ l g H × æ§ 4 5 Å q ¸s 9 Ä »z
H Ó ü t ^ _ | 9 | ¾ Ó m\ Á º ' a . Ä »z î r1 l x_ 5 Å q ¸
\
¦ 8 £ ¤& ñ l 0 A # r ç ß l 2 ¤ > \ ¦ 6 x H z ´+ « >` ¦ ¦ 9 K
Ð . r ç ß l 2 ¤ > \ z 0 > 7 á x s _ s á Ô_ ô Ç A á ¤ = å Q \ Ó
ü
t ^ \ ¦ # z r v r ç ß l 2 ¤ > H Å Òl T Ð 7 á x s
_ s á Ô\ ¦ M :a Ë >Ü ¼ Ð+ _ s á Ô\ Å Òl & h ¹ 1 ϧ 4 ` ¦ ô Ç
. ¢ ¸ô Ç, 7 á x s _ s á Ô H r ç ß l 2 ¤ > \ ¦ : x õ ½ + É M :\ r ç
ß l 2 ¤ > ü <_ ] X 8 ú ¤` ¦ : x # { 9 & ñ ô Ç ¹ 1 ϧ 4 ` ¦ ~ à ΠH .
s
o # z Ó ü t ^ \ K t H 8 ú x j Ë µ É r × æ§ 4 , r ç ß l 2 ¤
>
\ _ ô Ç Å Òl & h ¹ 1 ϧ 4 õ { 9 & ñ ô Ç ¹ 1 ϧ 4 Ü ¼ Ð ½ ¨$ í ) a
. @ /> h r ç ß l 2 ¤ > z ´+ « >\ " f H ¹ 1 ϧ 4 ` ¦ ¦ 9 t · ú § l
M :ë H \ © { © ô Ç 8 £ ¤& ñ ¸ \ ¦ Ä »µ 1 Ïô Ç . # l \ " f Ä ºo
H × æ§ 4 5 Å q ¸ 8 £ ¤& ñ r ` ¦ É r ° ú כ` ¦ % 3 l 0 A # ¹ 1 ϧ 4 _ ´ òõ \ ¦ à н ¨ ¦ ô Ç ; Ä ºo H s Qô Ç [ j t j Ë µ
`
¦ ¸¿ º ~ à ΠH Ó ü t ^ _ î r1 l x` ¦ l Õ ü t½ + É כ s .
/ B
N l $ ½ Ó É r Á ºr ¦ [20], z Ó ü t ^ \ 6 x H × æ§ 4 õ
r ç ß l 2 ¤ > \ ¦ : x õ H 7 á x s _ s á Ô\ ¦ : x # Õ ª Ó ü t
^
\ 6 x H { 9 & ñ ô Ç ¹ 1 ϧ 4 õ Å Òl & h ¹ 1 ϧ 4 ` ¦ ¸¿ º
¦ 9 , | 9 | ¾ Ós m z Ó ü t ^ _ î r1 l x~ ½ Ó& ñ d É r
m dv
dt = mg − η − η 0
∞
X
n=1
θ(t − t n )θ(t n + τ − t) (1)
Ð Å Ò# Q . # l \ " f θ(t) H Heaviside é ß 0 A > é ß < Ê
Table 1. The measurement of G for various values of m.
m (g) m
−1(kg
−1) G (m/s
2)
100 10 9.33 ± 0.015
50 20 8.93 ± 0.020
30 33 8.50 ± 0.012
20 50 7.91 ± 0.074
Ã
º(unit step function) Ð" f
θ(t) =
( 1 if t > 0,
0 if t < 0 (2) s
. η H { 9 & ñ ô Ç ¹ 1 ϧ 4 _ ß ¼l s ¦, η 0 H r ç ß l 2 ¤ > _ \ _ ô Ç ¹ 1 ϧ 4 _ ß ¼l Ð" f y _ t 5 Å q r ç ß τ 1 l xî ß { 9 & ñ ô Ç ° ú כ` ¦ ° ú H . Ä ºo H ½ + Ëo & h Ü ¼ Ð τ B Ä º Â
ú ª ¦ & ñ ½ + É Ã º e . t n (n = 1, 2, · · ·) É r n P : & h s
n y l r H r y s . & h _ Å Òl T s Ù ¼ Ð t 1 = 0 ¦ × þ ô Ç T = t n − t n −1 s ¦ t n = (n − 1)T s
. Õ ª Q 5 Å q ¸\ @ /ô Ç î r1 l x~ ½ Ó& ñ d _ K H
v(t) = v 0 + g − η
m
t − η 0 τ
m X ∞ n=1
nθ(t − t n )θ(t n+1 − t)
− η 0 m
∞
X
n=1
(t − t n − τ )θ(t − t n )θ(t n + τ − t) (3)
Ð Å Ò# Q . Fig. 1 É r s d _ Õ ªA á Ô\ ¦ & h { © ô Ç B > h Ã
º[ þ t \ @ / # Õ ª 2 ; כ s .
Ä
ºo T \ q # | r ç ß 1 l xî ß _ î r1 l x` ¦ ¦ 9ô Ç
, Å Òl & h ¹ 1 ϧ 4 É r ´ òõ & h Ü ¼ Ð ß ¼l η eff ( ≡ η 0 τ /T ) { 9
& ñ ô Ç ¹ 1 ϧ 4 Ü ¼ Ð H ½ + É Ã º e ¦ 5 Å q ¸ H v(t) ≈ v 0 + Gt Ð j þ t à º e . G H m −1 _ < Êà º Ð" f
G(m −1 ) = g − (η + η eff )m −1 (4) s
¦ g, η, η eff H © à º[ þ t s . z ´+ « >\ " f 8 £ ¤& ñ ) a 5 Å q ¸ G H ` ¦ É r × æ§ 4 5 Å q ¸ g m ¹ 1 Ï_ ´ òõ í < Ê
)
a ° ú כs . z ´+ « > X <s ' РÒ' × æ§ 4 5 Å q ¸ g\ ¦ % 3 l 0 A
# ª ô Ç Ó ü t ^ _ | 9 | ¾ Ó\ @ / # z ´+ « >` ¦ Ã º' ¦ G
@
/ m −1 _ Õ ªA á Ô\ ¦ Õ ª 9^ ¦ 9 כ ¹ e . Õ ª Q g_ ` ¦
É r ° ú כ É r s Õ ªA á Ô_ G » ¡ ¤ ] X ¼ # G(m −1 = 0)_ ° ú כÜ ¼
Ð Å Ò# Q . ¢ ¸ô Ç f _ d (4)_ l Ö ¦ l (m −1 _ > Ã º)
H ß ¼l η tot ( ≡ η + η eff ) 8 ú x Ä »´ ò ¹ 1 ϧ 4 s . כ ¹
, r ç ß l 2 ¤ > \ ¦ s 6 x # × æ§ 4 5 Å q ¸\ ¦ 8 £ ¤& ñ ½ + É M :\ r
ç ß l 2 ¤ > Ó ü t ^ _ î r1 l x \ p u H % ò ¾ Ó É r H & h Ü ¼ Ð ß
¼l η tot { 9 & ñ ô Ç ¹ 1 ϧ 4 ` ¦ H כ s .
Ä
ºo H ¨ î ¨ î ô Ç _ s ^ ¦ 0 A\ r ç ß l 2 ¤ > \ ¦ [ O u ¦ z ´ +
« >` ¦ Ã º' % i . Ó ü t ^ H r ç ß l 2 ¤ > \ z 0 > 7 á x s _ s
á Ô\ ) a Ê ê Ä »z î r1 l x \ Z ~ # . r ç ß l 2 ¤
>
H 7 á x s _ s á Ô\ ¦ M :o ¦ { 9 º _ & h [ þ t s n ) . Ä º o
H Ó ü t ^ _ | 9 | ¾ Ó` ¦ or v " f z ´+ « >` ¦ ì ø Í4 ¤ % i .
Ó ü
t ^ _ | 9 | ¾ Ó É r 20 g, 30 g, 50 g, 100 g Ü ¼ Ð × þ % i Ü ¼ 9 7
á
x s _ s á Ô_ | 9 | ¾ Ó É r Á ºr ½ + É Ã º e .
%
6 £ § Y > > h_ & h [ þ t É r d > 5 gt Ù ¼ Ð ¦ 9 t
· ú
§ ¤ ¦ Qt & h [ þ t É r 6 > hm Õ ªÒ ¨` ¦ t % 3 . r ç ß l 2
¤ > _ 1 l x à º 60 Hzs Ù ¼ Ð Å Òl T H 1/60 ís 9 6 Å
Òl _ r ç ß É r 0.1 ís . Ä ºo H & h [ þ t_ y Õ ªÒ ¨ \
@
/ô Ç 7 á x s _ s á Ô_ U ´s \ ¦ 8 £ ¤& ñ # 0.1 í 1 l xî ß _ r ç ß ç
ß ¨ î ç H 5 Å q ¸\ ¦ ½ ¨ % i . Õ ªo ¦ " f 5 Å q ¸ @ / r
ç ß _ Õ ªA á Ô\ @ /ô Ç f d ¹ 1 Ôl (linear fitting)\ ¦ : x
# y | 9 | ¾ Ó m\ @ /ô Ç Ó ü t ^ _ 5 Å q ¸ G\ ¦ Æ ÒØ ¦ % i .
Fig. 2 \ " f s X <s ' \ ¦ t ¦ : x > á Ô ÐÕ ªÏ þ ` ¦ 6 x ô
Ç f d ¹ 1 Ôl \ ¦ : x # G @ / m −1 _ Õ ªA á Ô\ ¦ Õ ªo ¦, G(m −1 = 0)_ ° ú כÜ ¼ Ð g\ ¦ ½ ¨ % i . 8 £ ¤& ñ ° ú כ` ¦ ¨ î ç H l
0 A # y | 9 | ¾ Ó\ @ / # z ´+ « >` ¦ [ j m ì ø Í4 ¤ % i ¦
õ H Table 1 \ ] jr ) a . Õ ª õ Ä ºo H þ j& h f d G = 9.66 − 0.035m −1 ` ¦ % 3 % 3 Ü ¼ 9 s d õ z ´+ « > X <s '
ÐÂ Ò' g = 9.66 ± 0.03 m/s 2 ` ¦ Æ ÒØ ¦ % i H X < s H z ´+ « >
s
à º' ) a ;\ " f_ × æ§ 4 5 Å q ¸ à а ú כ 9.80 m/s 2 õ 1.43 %_ ¸ \ ¦ Ð . s ¸ H z ´+ « >õ & ñ , 7 á x s _ s á
Ô ì r$ 3 õ & ñ , 7 á x s _ s á Ô_ | 9 | ¾ Ó Á ºr 1 p x \ _ # µ 1 ÏÒ q t ô
Ç כ Ü ¼ Ð Ò q ty ) a . × æ§ 4 5 Å q ¸_ à а ú כ É r 0 A ¸ü < K µ 1 Ï
¦ ¸_ < Êà º Ð Å Ò# Q × æ§ 4 5 Å q ¸ / B Nd [21]` ¦ : x # > í
ß ÷ &% 3 . Õ ªo ¦ r ç ß l 2 ¤ > \ _ ô Ç 8 ú x Ä »´ ò ¹ 1 ϧ 4 _ ß
¼l H η tot = 0.035 ± 0.002 Ne ` ¦ · ú Ã º e .
Ð: x_ â Ä ºü < ° ú s , ¹ 1 ϧ 4 _ ´ òõ \ ¦ Á ºr ¦ G_ 8
£
¤& ñ ° ú כ` ¦ × æ§ 4 5 Å q ¸ g_ ° ú כÜ ¼ Ð ç ß Å Ò 9½ + Ét ¸ É r .
Fig. 2. The graph of G as a function of m −1 . For each
falling mass (m = 20, 30, 50, 100 g), the value of ac-
celeration was measured three times. The dots are the
mean values of the acceleration for each mass. By lin-
ear fitting to experimental data, we obtain the solid line
G = 9.66 − 0.035m −1 .
s
â Ä º\ Table 1\ " f ^ ¦ Ã º e H ü < ° ú s z H Ó
ü
t ^ _ | 9 | ¾ Ó ms Ü ¼ ` ¦ à º2 ¤ G_ ° ú כ É r × æ§ 4 5 Å q
¸_ à а ú כÜ ¼ РÒ' 8¹ ¡ ¤ 8 ß ¼> # Á # Qè ß . Ä ºo ¹ 1 Ï
§
4 _ ´ òõ \ ¦ y èr v l 0 A # Á º î r ( \ V\ ¦ [ þ t # Q, | 9
|
¾ Ós 100 g ) Ó ü t ^ \ ¦ 6 xô Ç # ¸ 8 £ ¤& ñ ° ú כ G H # y
g_ à а ú כÜ ¼ РÒ' © { © y # Á # Qè ß . Õ ª QÙ ¼ Ð × æ§ 4
5 Å
q ¸_ 8 £ ¤& ñ \ " f ø @½ + É ë ß ô Ç õ \ ¦ % 3 Ü ¼ 9 r ç ß l 2
¤ > _ ¹ 1 ϧ 4 _ ´ òõ \ ¦ ¦ 9 H כ s 9 à º& h e ` ¦ · ú Ã
º e .
: r 7 Hë H \ " f Ä ºo H × æ§ 4 5 Å q ¸\ ¦ 8 £ ¤& ñ H ª ô Ç
~
½ ÓZ O [ þ t \ ' a # ¶ ú ( R : r Ê ê, r ç ß l 2 ¤ > \ ¦ 6 x # Ä
»z î r1 l x_ × æ§ 4 5 Å q ¸\ ¦ Ð 8 & ñ S X > 8 £ ¤& ñ ½ + É Ã
º e H ~ ½ ÓZ O ` ¦ ½ ¨ % i . r ç ß l 2 ¤ > _ ¹ 1 ϧ 4 _ ´ ò õ
\ ¦ Á ºr Ó ü t ^ _ | 9 | ¾ Ó\ _ > r H H ¸ \ ¦ x ½ + É Ã
º \ O . s ½ ¨\ " f Ä ºo H g_ & ñ S X ô Ç ° ú כ` ¦ % 3 l 0 A
#
r ç ß l 2 ¤ > \ _ K K t H ¿ º 7 á x À Ó_ ¹ 1 ϧ 4 ({ 9 & ñ ô
Ç ¹ 1 ϧ 4 õ Å Òl & h ¹ 1 ϧ 4 )` ¦ ¦ 9 % i . & h _ Å Òl
Ð s ` | r ç ß 1 l xî ß _ î r1 l x` ¦ ¦ 9½ + É M :\ s ¹ 1 Ï
§
4 [ þ t_ 8 ú x ´ òõ H Fig. 1 \ " f ^ ¦ Ã º e H ü < ° ú s { 9 & ñ ô
Ç Ä »´ ò ¹ 1 ϧ 4 (−η tot ) õ H & h Ü ¼ Ð 1 l x1 p x < Ê` ¦ · ú à º e
.
Y c
p w à U Ø ô
[1] A. Peters, K. Y. Chung, and S. Chu, Nature 400, 849 (1999); Metrologia 38, 25 (2001).
[2] M. Fattori et al., Phys. Lett. A 318, 184 (2003).
[3] R. Poggiani, Class. Quantum Grav. 20, 567 (2003).
[4] ftp://ftp.pasco.com/manuals/English/ME/ME- 9207B/012-05760B/012-05760B.pdf;
ftp://ftp.pasco.com/manuals/English/ME/ME- 9202C/012-05762B/012-05762B.pdf.
[5] J. D. Wilson, Physics Laboratory Experiments, 5th ed. (Houghton Mifflin, New York, 1998).
[6] PASCO scientific, Physics Labs with Computers, Volume 1: Student Workbook (PASCO scientific, Roseville, 1999).
[7] PASCO scientific, Physics Labs with Computers, Volume 1: Teacher’s Guide (PASCO scientific, Ro- seville, 1999).
[8] http://www.picotech.com/experiments/gravity acceleration.
[9] P. Laws and H. Pfister, Phys. Teach. 36, 282 (1998).
[10] PASCO scientific, PASCO 2004 PHYSICS World- wide Catalog and Experiment Guide (PASCO scien- tific, Roseville, 2004), pp. 94-95.
[11] e.g. PASCO scientific, Instruction Manual and Ex- periment Guide for the PASCO scientific Model ME-9283, Tape Timer (PASCO scientific, Roseville, 1993).
[12] http://www.pa.msu.edu/courses/1997fall/PHY251/
freefall1.pdf.
[13] http://physicsx.pr.erau.edu/Courses/CoursesF2002/
PS253%20pages/Gravitational Acceleration.pdf.
[14] http://www.uccp.org/docs/Lab%20Facilitator%20 Manuals/UCCP%20AP%20Physics/Lab%203%20%
20%20%20Atwood’s%20Machine.pdf.
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lab7.pdf.
[16] http://www.as.ysu.edu/˜mcrescim/1501/217l8.html.
[17] http://heebok.kongju.ac.kr/experiment/dynamics/
fm7.html.
[18] http://polaris.phys.ualberta.ca/users/austen/Phys29x/
Manual/20KaterPendulum99.pdf.
[19] K. E. Jesse, Am. J. Phys. 48, 785 (1980).
[20] See Eq. (7.12) in M. W. Denny, Air and Water (Princeton University Press, Princeton NJ, 1993).
It gives the drag on a sphere of the radius r for any Reynolds number less than about 10 5 : drag =
1 2 ρ f πr 2
0.4v 2 + 12νv r + 6v
21+ √
2rv/ν
. Here ρ f is the density of the fluid, v the relative velocity between object and fluid, and ν the kinematic viscosity of the fluid. πr 2 is the frontal area of the sphere. For dry air at 20 ◦ C and one atmosphere, ρ f = 1.205 kg/m 3 and ν = 1.51 × 10 −5 m 2 /s. But we use cylindrical bod- ies as freely falling probes, which are released from the rest. For the biggest cylinder with m = 100 g and the circular radius r = 1.25 cm, the speed at which it reaches the ground is about 3.25 m/s . In this case the drag on the circular cylinder is about three times as large as that on the sphere [22]. The maximum drag on cylindrical probes (i.e. the drag on the biggest cylinder at highest speed) is about 0.0045 N. In fact, the air drag during most of the probes’ falling time is much smaller than this value.
Hence the air resistance is negligible compared with
the total effective frictional force due to the tape
timer which is exerted uniformly with the magni-
tude of 0.035 N during our experiment.
[21] http://www.npl.co.uk/pressure/faqs/altgrav.html.
An approximate value for g, at a given latitude and height above sea level, may be calculated from the formula: g = 9.7803184(1 + A sin 2 L − B sin 2 (2L)) − 3.086 × 10 −6 H, where A = 0.0053024, B = 0.0000059, L is the latitude, and H is the
height in metres above sea level. The uncertainty in the value of g so obtained is generally less than ±5 parts in 10 5 .
[22] B. R. Munson, D. F. Young, and T. H. Okiishi, Fun- damentals of Fluid Mechanics, 3rd ed. (John Wiley
& Sons, New York, 1998), p. 600.
Measurement of the Acceleration Due to Gravity by Considering the Frictional Effect of a Tape Timer
Seok-In Hong ∗
Department of Science Education, Gyeongin National University of Education, Incheon 407-753 Kang Young Lee
Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701 (Received 16 February 2004)
When we use a tape timer to measure the acceleration due to gravity, the falling body experiences a periodic frictional force due to hitting of the timer, as well as a constant frictional force due to the contact of the paper tape with the timer through the tape attached to the body. The total effect approximates a constant effective frictional force of η
tot. The falling acceleration G is given by G = g − η
tot/m, which depends upon the mass m of the body. By fitting the experimental data for G versus m
−1, we obtain the acceleration due to gravity as g = 9.66 ± 0.03 m/s
2(with an error of 1.43 %) and the total effective friction as η
tot= 0.035 ± 0.002 N.
PACS numbers: 01.50.Pa, 01.55.+b, 06.30.Gv
Keywords: Acceleration due to gravity, Tape timer, Friction
∗
