Patel (ppp285) – HW06 – TSOI – (58160)
1
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001
(part 1 oF 2) 10.0 points
Three 6 kg masses are located at points in
the
xy
plane as shown.
37 cm
53 cm
What is the magnitude oF the resultant
Force (caused by the other two masses) on
the mass at the origin? The universal gravita
tional constant is 6
.
6726
×
10
−
11
N
·
m
2
/
kg
2
.
Correct answer: 1
.
95196
×
10
−
8
N.
Explanation:
Let :
m
= 6 kg
,
x
= 37 cm = 0
.
37 m
,
y
= 53 cm = 0
.
53 m
,
and
G
= 6
.
6726
×
10
−
11
N
·
m
2
/
kg
2
.
The Force From the mass on the right points
in the
x
direction and has magnitude
F
1
=
G
mm
x
2
=
Gm
2
x
2
=
(6
.
6726
×
10
−
11
N
·
m
2
/
kg
2
) (6 kg)
2
(0
.
37 m)
2
= 1
.
75466
×
10
−
8
N
.
The other Force points in the
y
direction
and has magnitude
F
2
=
(6
.
6726
×
10
−
11
N
·
m
2
/
kg
2
) (6 kg)
2
(0
.
53 m)
2
= 8
.
55157
×
10
−
9
N
.
F
2
F
1
F
θ
The magnitude oF the resultant Force is
F
=
r
F
2
1
+
F
2
2
=
b
(1
.
75466
×
10
−
8
N)
2
+ (8
.
55157
×
10
−
9
N)
2
B
1
/
2
=
1
.
95196
×
10
−
8
N
.
002
(part 2 oF 2) 10.0 points
At what angle From the positive
x
axis will the
resultant Force point? Let counterclockwise
be positive, within the limits
−
180
◦
to 180
◦
.
Correct answer: 25
.
9828
◦
.
Explanation:
The angle
θ
shown is
θ
= arctan
p
f
2
f
1
P
= arctan
p
8
.
55157
×
10
−
9
N
1
.
75466
×
10
−
8
N
P
=
25
.
9828
◦
.
003
(part 1 oF 3) 10.0 points
Given:
G
= 6
.
67259
×
10
−
11
N m
2
/
kg
2
A uniForm solid sphere oF mass
m
2
= 289 kg
and radius
R
2
= 1
.
22 m is inside and concen
tric with a spherical shell oF mass
m
1
= 207 kg
and radius
R
1
= 2
.
38 m (see the fgure).
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View Full DocumentPatel (ppp285) – HW06 – TSOI – (58160)
2
R
R
a
b
c
m
m
1
1
2
2
Find the magnitude of the gravitational
force exerted by the sphere and spherical shell
on a particle of mass 2 kg located at a dis
tance 0
.
52 m from the center of the sphere
and spherical shell.
Correct answer: 1
.
10445
×
10
−
8
N.
Explanation:
In this case the distance
a
= 0
.
52 m from
the particle of mass
m
= 2 kg to the center
of the sphere (whose mass is
m
2
= 289 kg) is
smaller than the radius
R
2
= 1
.
22 m of the
sphere. Thus, the particle is inside the sphere
and the gravitational force
F
a
exerted on it
increases linearly with the distance from the
particle to the center of the sphere
F
a
=
G
m
2
m
R
3
2
a
=
(289 kg) (2 kg)
(1
.
22 m)
3
(0
.
52 m)
×
6
.
67259
×
10
−
11
N m
2
/
kg
2
= 1
.
10445
×
10
−
8
N
.
004
(part 2 of 3) 10.0 points
Find the magnitude of the gravitational force
exerted by the sphere and spherical shell on
a particle of mass 2 kg located at a distance
1
.
84 m from the center of the sphere and
spherical shell.
Correct answer: 1
.
13917
×
10
−
8
N.
Explanation:
In this case the distance
b
= 1
.
84 m from
the particle of mass 2 kg to the center of the
sphere is greater than the radius
R
2
= 1
.
22 m
of the sphere, but less than the radius
R
1
=
2
.
38 m of the spherical shell.
First, the shell does not a±ect the particle
and, second, we can regard the sphere as a
point mass and apply Newton’s law of gravity
in its simplest version
F
b
=
G
m
2
m
b
2
=
(289 kg) (2 kg)
(1
.
84 m)
2
×
6
.
67259
×
10
−
11
N m
2
/
kg
2
= 1
.
13917
×
10
−
8
N
.
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 Spring '08
 Turner
 Physics, Force, Mass, Potential Energy, General Relativity

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