From everyday questions to specialized queries, IDNLearn.com has the answers. Get prompt and accurate answers to your questions from our community of knowledgeable experts.
Sagot :
To find the distance between a [tex]$2 \, \text{kg}$[/tex] laptop and a [tex]$4 \, \text{kg}$[/tex] jar of pennies, given that the gravitational force between them is [tex]$3.42 \times 10^{-10} \, \text{N}$[/tex], we can use Newton's law of universal gravitation. The formula for the gravitational force [tex]\( F \)[/tex] between two masses [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] separated by a distance [tex]\( r \)[/tex] is:
[tex]\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \][/tex]
where [tex]\( G \)[/tex] is the universal gravitational constant [tex]\( 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)[/tex].
Let's isolate [tex]\( r \)[/tex] in the formula:
[tex]\[ r^2 = \frac{G \cdot m_1 \cdot m_2}{F} \][/tex]
Taking the square root of both sides will give us [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{G \cdot m_1 \cdot m_2}{F}} \][/tex]
Plugging in the given values:
- [tex]\( G = 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)[/tex]
- [tex]\( m_1 = 2 \, \text{kg} \)[/tex]
- [tex]\( m_2 = 4 \, \text{kg} \)[/tex]
- [tex]\( F = 3.42 \times 10^{-10} \, \text{N} \)[/tex]
[tex]\[ r = \sqrt{\frac{6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \times 2 \, \text{kg} \times 4 \, \text{kg}}{3.42 \times 10^{-10} \, \text{N}}} \][/tex]
Simplifying the expression under the square root:
[tex]\[ r = \sqrt{\frac{6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \times 8 \, \text{kg}^2}{3.42 \times 10^{-10} \, \text{N}}} \][/tex]
[tex]\[ r = \sqrt{\frac{5.33944 \times 10^{-10} \, \text{m}^3 \, \text{s}^{-2}}{3.42 \times 10^{-10} \, \text{N}}} \][/tex]
[tex]\[ r = \sqrt{1.56122 \, \text{m}^2} \][/tex]
[tex]\[ r \approx 1.25 \, \text{m} \][/tex]
Therefore, the distance between the laptop and the jar of pennies is closest to option A.
So, the correct answer is:
A. [tex]$1.25 \, \text{m}$[/tex].
[tex]\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \][/tex]
where [tex]\( G \)[/tex] is the universal gravitational constant [tex]\( 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)[/tex].
Let's isolate [tex]\( r \)[/tex] in the formula:
[tex]\[ r^2 = \frac{G \cdot m_1 \cdot m_2}{F} \][/tex]
Taking the square root of both sides will give us [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{G \cdot m_1 \cdot m_2}{F}} \][/tex]
Plugging in the given values:
- [tex]\( G = 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)[/tex]
- [tex]\( m_1 = 2 \, \text{kg} \)[/tex]
- [tex]\( m_2 = 4 \, \text{kg} \)[/tex]
- [tex]\( F = 3.42 \times 10^{-10} \, \text{N} \)[/tex]
[tex]\[ r = \sqrt{\frac{6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \times 2 \, \text{kg} \times 4 \, \text{kg}}{3.42 \times 10^{-10} \, \text{N}}} \][/tex]
Simplifying the expression under the square root:
[tex]\[ r = \sqrt{\frac{6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \times 8 \, \text{kg}^2}{3.42 \times 10^{-10} \, \text{N}}} \][/tex]
[tex]\[ r = \sqrt{\frac{5.33944 \times 10^{-10} \, \text{m}^3 \, \text{s}^{-2}}{3.42 \times 10^{-10} \, \text{N}}} \][/tex]
[tex]\[ r = \sqrt{1.56122 \, \text{m}^2} \][/tex]
[tex]\[ r \approx 1.25 \, \text{m} \][/tex]
Therefore, the distance between the laptop and the jar of pennies is closest to option A.
So, the correct answer is:
A. [tex]$1.25 \, \text{m}$[/tex].
Answer:
To find the distance between a
2
kg
2kg laptop and a
4
kg
4kg jar of pennies, given that the gravitational force between them is
3.42
×
1
0
−
10
N
3.42×10
−10
N , we can use Newton's law of universal gravitation. The formula for the gravitational force
F between two masses
1
m
1
and
2
m
2
separated by a distance
r is:
=
⋅
1
⋅
2
2
F=
r
2
G⋅m
1
⋅m
2
where
G is the universal gravitational constant
6.67430
×
1
0
−
11
m
3
kg
−
1
s
−
2
6.67430×10
−11
m
3
kg
−1
s
−2
.
Let's isolate
r in the formula:
2
=
⋅
1
⋅
2
r
2
=
F
G⋅m
1
⋅m
2
Taking the square root of both sides will give us
r :
=
⋅
1
⋅
2
r=
F
G⋅m
1
⋅m
2
Plugging in the given values:
-
=
6.67430
×
1
0
−
11
m
3
kg
−
1
s
−
2
G=6.67430×10
−11
m
3
kg
−1
s
−2
-
1
=
2
kg
m
1
=2kg
-
2
=
4
kg
m
2
=4kg
-
=
3.42
×
1
0
−
10
N
F=3.42×10
−10
N
=
6.67430
×
1
0
−
11
m
3
kg
−
1
s
−
2
×
2
kg
×
4
kg
3.42
×
1
0
−
10
N
r=
3.42×10
−10
N
6.67430×10
−11
m
3
kg
−1
s
−2
×2kg×4kg
Simplifying the expression under the square root:
=
6.67430
×
1
0
−
11
m
3
kg
−
1
s
−
2
×
8
kg
2
3.42
×
1
0
−
10
N
r=
3.42×10
−10
N
6.67430×10
−11
m
3
kg
−1
s
−2
×8kg
2
=
5.33944
×
1
0
−
10
m
3
s
−
2
3.42
×
1
0
−
10
N
r=
3.42×10
−10
N
5.33944×10
−10
m
3
s
−2
=
1.56122
m
2
r=
1.56122m
2
≈
1.25
m
r≈1.25m
Therefore, the distance between the laptop and the jar of pennies is closest to option A.
So, the correct answer is:
A.
1.25
m
1.25m
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. For trustworthy and accurate answers, visit IDNLearn.com. Thanks for stopping by, and see you next time for more solutions.
