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A graph of force (F) versus elongation (x) shown in figure below. Find the spring constant!
Solution
Hooke's law formula :
k = F / x
F = force (Newton)
k = spring constant (Newton/meter)
x = the change in length (meter)
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Newton’s gravitational law These questions are intended to give you practice in using the gravitational law. They will give you a feeling for typical forces with a range of masses and also how sensitive force is to distance.
Useful data G =
–11
6.67 × 10
–
N m !g
&
"arth#s mass = $.%7 × 10 !g
'oon#s mass = 7.(& × 10 !g (0
)un#s mass is .0 × 10 !g 6
*adius of the 'oon = 1.6& × 10 m 6
*adius of the "arth = 6.(7 × 10 m $
"arth–'oon distance = (.+ × 10 !m +
"arth–)un distance = 1.$ × 10 !m
1.
,ou may may some someti time mes s find find itit diff diffic icul ultt to get get up up from from the the sof sofa a afte afterr watc watchi hing ng a Tprogramme. ssuming the force of gravity acts /etween the centre of your /ody and the centre of the sofa estimate the attraction /etween you and your sofa.
.
alcul alculate ate the si2e si2e of of the the gra gravit vitati ationa onall pull pull of a sph spher ere e of of mass mass 10 !g on a mass mass .0 !g when their centres are 00 mm apart. 3hat is the force of the .0 !g mass on the 10 !g mass4
(.
t wha whatt dist distan ance ce apar apartt woul would d two two equa equall mass masses es of of 1$0 1$0 !g need need to to /e /e pla place ced d for for the the –$
force /etween them to /e .0 × 10 N4
&.
alc alcul ulat ate e the the grav gravit itat atio iona nall pull pull of of the the "art "arth h on each each of of the the foll follow owin ing g /odi /odies es55
the 'oon
$.
×
and satellite mass +0 !g at a distance from the "arth#s centre +.0
10 m.
10 m
×
6
)how that the unit for G the universal gravitational constant can /e e8pressed ( –
as m s
6.
7
satellite with mass 100 !g at a distance from the "arth#s centre &.
–1
!g .
alculate the weight of an astronaut whose mass 9including spacesuit: is 7 !g on the 'oon4
3hat is the astronaut;s weight on "arth4
omment on the difference.
7.
)how that pull of the )un on the 'oon is a/out . times larger than the pull of the "arth on the 'oon.
+.
3hy then does the 'oon or/it the "arth4
The merican space agency N) plans to send a manned mission to 'ars later this century. 'ars has a mass 6.& 8 10 ( !g and a radius (.(+ 8 10 6 m. < = 6.67 8 1011 N m !g
%. 9a: The mass of a typical astronaut plus spacesuit is +0 !g. 3hat would /e the gravitational force acting on such an astronaut standing on the surface of 'ars4
9/: )tate whether an astronaut on 'ars would feel lighter or heavier than on "arth.
Practical advice This resource could /e used for either class wor! or homewor!.
Answers and worked solutions 1.
>or the values estimated in the answers5 F
.
GMm =
r
=
2
(6.67 × 10
11
−
2
−
N kg
m 2 ) × 60 kg × 100 kg
( 0.5 m )
−
=
1.6 × 10
=
3.3 × 10
2
6
N.
9
N.
?ull on the .0 !g mass
F
GMm =
r
=
2
(6.67 × 10
11
−
2
−
N kg
m 2 ) × 10 kg × 2.0 kg
( 0.200 m )
2
−
The pull on the 10 !g mass will /e equal /ut opposite in direction.
(. r
Gm =
F
=
m
G =
F
6.67
1$0 !g
×
10 −11 N !g − m
.0
=
10 −$ N
×
0.7 m.
&. F
=
r
F
r
F
=
9 6.67
r
=
$.
N kg
6.
'oon F
=
11
−
N !g
2
−
m2
=
r
×
×
=
=
−
m :
×
9$.%7
((.+ 10
−
11
N !g
−
m : × 9$.%7
10
−
11
N !g
−
kg m s
9 6.67
2
−
×
kg
10
2
−
11
−
10 7 m
×
)
m2
×
10 6 m
=
N !g
m3 s
−
)
2
−
m :
(1.6&
×
×
×
×
10 & !g:
10 + m
)
×
×
10 & !g:
×
100 !g
10 & !g:
×
+0 !g
kg
×
97.(&
1
−
×
10 !g:
m : × 9$.%7
(+.0
GMm
"arth
10
×
( &.
GMm =
9 6.67
GMm =
96.67
GMm
=
=
=
.0
( N.
$.0
×
10 N.
.
7 !g
10 6 m
)
×
97.(&
×
10 !g:
=
1.(
×
10 N.
×
10
F
7.
(6.67 × 10
GMm =
r
2
=
11
−
2
−
N kg
m 2 ) × 72 kg × (5.97 × 10 24 kg)
( 6.37
×
10 6 m )
2
=
7.1 × 10 2 N.
)un–'oon F
GMm =
r
=
9 6.67
×
10
−
11
N !g
−
m : × 9.0
(1.$
×
×
10 (0 !g:
10 11 m
)
×
97.(&
×
10 !g:
=
&.&
×
10 0
"arth–'oon F
GMm =
r
2
=
(6.67 × 10
ratio of attractions
11
−
N kg
2
−
m 2 ) × (5.97 × 10 24 kg) × (7.34 × 10 22 kg)
( 3.8 =
&.& × 10
0
N
.0 × 10
0
N
=
×
10 m ) 8
2
.
+.
The 'oon does of course or/it the )un as part of the "arth–'oon system. ,ou can thin! of the 'oon#s or/it of the "arth as superimposed on its or/it of the )un.
External references Auestions 1 +5 This is ta!en from dvancing ?hysics hapter 11 +03 Auestion %5 This is ta!en from )alters Borners dvanced ?hysics section )T additional sheet + and %
