Oliver Gorton Darrione, Jose, David, Kate Lab 4: Free Fall: Find Acceleration due to Gravity (g) 2!4"2"!#
Ob$ective %&is lab &ad t'o ob$ectives: (!) to calculate t&e acceleration o an ob$ect in ree all, and (2) to calculate t&e reaction tie o our lab *artners against t&e constant o acceleration" %'o dierent activities 'ere conducted to eet t&ese ob$ectives" %o acco*lis& (!), a ass 'as attac&ed to t&e end o a lengt& o *a*er ta*e, and allo'ed to dro* '&ilst a ta*e tier ar+ed t&e *a*er ta*e at regular intervals: at a rate o -." %&e dots *rinted on t&e *a*er ta*e resulted in a *osition versus tie gra*& ro '&ic& acceleration 'as e/tra*olated" %o acco*lis& (2), a eter stic+ 'as dro**ed bet'een t&e ingers o lab *artners, and t&e distance *assed beore &e or s&e 'as able to catc& t&e eter stic+ 'as recorded" %&e tie ela*sed during t&is interval o ree all 'as calculated using t&e +no'n constant g, and t&e easured dis*laceent t&at occurred during ree all"
A**aratus 01ui*tent
%a*e tier 'it& *a*er ta*e 3ass 2g %able cla* and rod 3as+ing ta*e 3eter stic+
Activity !
Activity 2
s 9 gt62 t "4; s6";, 7&ere D is dis*laceent in centieters and t is tie in seconds ( t&e reaction tie)" g 2s 5 t62 v(t) gt, 7&ere g is acceleration due to gravity, s is dis*laceent (distance ro eac& ar+ on t&e *a*er ta*e ro t&e origin,) t is tie in seconds (in increents o !5-.,) and v is velocity" 8aety considerations: 7atc& your toes or alling asses"
%&eory Ob$ects accelerate to'ards 0art& because o t&e gravitational orce bet'een t&e ob$ect and 0art&" 7&en t&e orce o gravity is t&e only orce acting on t&e ob$ect < naely '&en air resistance is negligible < it is said to be in reeall" Ob$ects in ree all accelerate to'ards eart& at ="#5s62" %&e irst o t&e t'o ob$ectives 'as addressed by e/aining t&e relations&i* bet'een t&e distance bet'een evenly tied ar+s on *a*er ta*e '&ic& re*resent t&e dis*laceent o t&e attac+ed 'eig&t ater eac& *eriod o (!5) seconds" Kineatics tells us t&at acceleration is t&e rate at '&ic& velocity c&anges, and velocity is t&e rate at '&ic& dis*laceent c&anges" %&ereore 'e ind t&at dis*laceent s ";gt62" 8ince gravity is t&e only orce acting on t&e 'eig&t (assuing riction is negligible, see conclusion), 'e can 'rite g 2s5t62" 8o t&e gra*& o t&e 'eig&ts dis*laceent to t&e tie s1uared *rovides t&e constant o acceleration g (g is double t&e slo*e o t&e gra*&) %&e *osition versus tie data collected in activity ! can be used 'it& t&is orula to calculate an e/*eriental value or g < by *lotting *osition as a unction o tie s1uared and inding double t&e rate o c&ange < t&us acco*lis&ing ob$ective (!)" %&e sae orula (s ";gt62) can be rearranged to obtain t "4;s6";, '&ere t is tie, s is dis*laceent, and g is t&e constant o gravity at =# c5s62" %&is derivation o t&e e1uation 'as used to analy.e t&e data collected in activity 2> to calculate t&e r eaction tie ro a given easured distance< allen by t&e eter stic+, s, and t&e +no'n constant o acceleration g"
%able o sybols: s dis*laceent ?centieters@ t tie ?seconds@ a acceleration ?centieters *er second s1uared@ g acceleration due to eart&s gravity ?=# c5s62@
Data %ables Data table !, Activity !: Dis*laceent versus tie ela*sed s1uared" Dot nuber
Dis*laceent (s) ?c"@
%ie ela*sed s1uared ?s62@
Dot nuber
Dis*laceent (s) ?c"@
%ie ela*sed s1uared ?s62@
!
";
"2BB"""
!4
2="B
";44"""
2
!"!
"!!"""
!;
CC"#
"2;
C
2"
"2;
!
C#"2
"B!!"""
4
C"2
"44"""
!B
42"=
"#2BB"
;
4"
"=44"""
!#
4B"#
"=
"C
"!
!=
;C"
"!2BB"""
B
#"4
"!C!!"""
2
;#";
"!!"""
#
!"
"!BB"""
2!
4"2
"!22;
=
!C"!
"22;
22
B"2
"!C44"""
!
!;"=
"2BB"""
2C
B";
"!4=44"""
!!
!=
"CC!!"""
24
#C
"!
!2
22"C
"4
2;
#="=
"!BC!!"""
!C
2;"=
"4=44"""
2
="=
"!#BB"""
%ie ela*sed s1uared ((!5)i)62, '&ere i is t&e dot nuber ro t&e origin" (e/a*le: %ie ela*sed (!) (!(!5))62 "2BB) Eest<it line or gra*& o s to t62: s=512.48x+1.3, '&ere / is t62" g 2s5t62 2;!2"4# 1024.96 cm/s^2 ercent error: (!24"=<=#)5=# 4. 6% error. Data table 2, Activity 2: eaction ties %rial ! (c)
%rial 2 (c)
%rial C (c)
ersonal Average (c)
%ie (sec")
Oliver
#
!2
!
!2
"!;
DJ
2
2;
!;
2
"22
David
!;
!C
!
!;
"!BC
Jose
!
!2
!B
!;
"!BC
Kate
!2
!#
!
!;
"!BC
Grou* average:
15
0.173
n
Average
1/ n
∑ x ❑
i
, 7&ere n is t&e nuber o trials, and
1
(e/a*le: grou* average (! 5 ;)(!2H2H!4"BH!;H!;"C) !4") %ie (sec") t "4;s6"; (e/a*le: %ie (oliver) "4;(!2I";) "!; seconds)
x ❑i
is t&e value or eac& trial"
onclusion %&e t'o ob$ectives o t&is lab < (!) to calculate t&e acceleration o an ob$ect in ree all, and (2) to calculate t&e reaction tie o our lab *artners against t&e constant o acceleration < 'ere et" %&e irst o t&e t'o ob$ectives 'as addressed by e/aining t&e relations&i* bet'een t&e distance bet'een evenly tied ar+s on *a*er ta*e '&ic& re*resent t&e dis*laceent o t&e attac+ed 'eig&t ater eac& *eriod o (!5) seconds" %&e constant g 'as calculated to be !24"= c5s62, an error o 4" above t&e acce*ted value o =# c5s62" A *ositive error 'as not e/*ected and is diicult to e/*lain since sources o error suc& as air resistance and riction bet'een t&e *a*er ta*e and t&e ta*e tier 'ould in&erently cause t&e ob$ect to accelerate at a slo'er rate, not at a &ig&er one as observed"%&is error could be cause b y an inaccurate at&eatical odel or t&e data, an error in t&e ta*e tier ac&ine (it 'ould &ave been ar+ing too slo'ly, causing t&e ga*s bet'een dots to be elongated,) or rando error" A ore accurate e/*erient ay be conducted by: increasing t&e nuber o data *oints by increasing t&e re1uency o t&e ta*e tier> increasing t&e accuracy o easureent devices used> increasing t&e density o t&e ass to reduce t&e eect o air resistance" During t&e e/*erient t&e ollo'ing ste*s 'ere ta+en to reduce error: t&e ta*e 'as su**orted above t&e ta*e tier as it ed t&roug& t&e ta*e tier in order to reduce riction5*ull, only t&e irst eter o dots 'ere collected as data *oints to reove error caused by a lac+ o tension in t&e ta*e ater t&e 'eig&t &ad reac&ed t&e ground (slig&tly ore t&an a eter a'ay,) t&e ass 'as ta*ed to t&e ta*e *a*er rat&er t&an being attac&ed via a ass &older, and t&e *a*er ta*e 'as ta*ed to a eter stic+ ater it 'as ar+ed in order to ac&ieve accurate and consistent dis*laceent easureents" %&e ive grou* ebers reaction ties 'ere eac& easured using t&e eter stic+ et&od described in t&e Ob$ective, and t&e data yielded an average r eaction tie o "!BC seconds, or !; centieters" %&is eans t&at on average, t&e grou* too+ "!BC seconds to react and gras* t&e alling eter stic+ < ater it &ad allen a corres*onding !; centieters" %&e greatest source o error in t&is e/*erient 'as caused by t&e *artici*ants '&o did not +ee* t&eir &ands in a stationary *osition '&ilst t&e eter stic+ 'as dro**ed or t&e to catc&" %&is led to t&e tendency o *artici*ants to ove t&eir &ands do'n'ards as t&ey gras*ed or t&e alling eter stic+" %&is could &ave been avoided b y *lacing *artici*ants ars on a surace suc& as a des+, 'it& t&eir &and and 'rist e/tended over t&e edge o t&e surace" Anot&er source o error 'as t&e inaccuracy o easuring t&e initial and inal *ositions o t&e eter stic+" 3easureents 'ere rounded to t&e nearest '&ole centieter"
