Answer :
Sure! Let's work through the solution step by step to find the net gravitational force exerted on a 64 kg mass placed midway between two larger masses.
Step 1: Understand the Problem and Given Information
- We have three masses: 215 kg, 579 kg, and 64 kg.
- The 64 kg mass is placed exactly midway between the 215 kg and 579 kg masses.
- The distance between the 215 kg and 579 kg masses is 0.493 meters.
- We are given the universal gravitational constant, [tex]\( G = 6.672 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \)[/tex].
Step 2: Determine the Distances
- Since the 64 kg mass is exactly in the middle, it is 0.493 m / 2 = 0.2465 m away from each of the larger masses.
Step 3: Calculate the Gravitational Force from the 215 kg Mass
The formula for gravitational force is:
[tex]\[
F = \frac{G \cdot m_1 \cdot m_2}{r^2}
\][/tex]
For the 215 kg mass:
- [tex]\( m_1 = 215 \, \text{kg} \)[/tex]
- [tex]\( m_2 = 64 \, \text{kg} \)[/tex]
- [tex]\( r = 0.2465 \, \text{m} \)[/tex]
Substituting the values:
[tex]\[
F_1 = \frac{6.672 \times 10^{-11} \cdot 215 \cdot 64}{0.2465^2}
\][/tex]
[tex]\[
F_1 \approx 1.5109 \times 10^{-5} \, \text{N}
\][/tex]
Step 4: Calculate the Gravitational Force from the 579 kg Mass
For the 579 kg mass:
- [tex]\( m_1 = 579 \, \text{kg} \)[/tex]
- [tex]\( m_2 = 64 \, \text{kg} \)[/tex]
- [tex]\( r = 0.2465 \, \text{m} \)[/tex]
Substituting the values:
[tex]\[
F_2 = \frac{6.672 \times 10^{-11} \cdot 579 \cdot 64}{0.2465^2}
\][/tex]
[tex]\[
F_2 \approx 4.0689 \times 10^{-5} \, \text{N}
\][/tex]
Step 5: Determine the Net Gravitational Force
- The forces [tex]\( F_1 \)[/tex] and [tex]\( F_2 \)[/tex] are in opposite directions because one mass is on one side of the 64 kg mass and the other is on the opposite side.
- The net gravitational force is the difference between these two forces because [tex]\( F_2 > F_1 \)[/tex].
Calculate the net force:
[tex]\[
\text{Net Force} = F_2 - F_1
\][/tex]
[tex]\[
\text{Net Force} = 4.0689 \times 10^{-5} - 1.5109 \times 10^{-5}
\][/tex]
[tex]\[
\text{Net Force} \approx 2.5580 \times 10^{-5} \, \text{N}
\][/tex]
So, the magnitude of the net gravitational force exerted on the 64 kg mass is approximately [tex]\( 2.5580 \times 10^{-5} \, \text{N} \)[/tex].
Step 1: Understand the Problem and Given Information
- We have three masses: 215 kg, 579 kg, and 64 kg.
- The 64 kg mass is placed exactly midway between the 215 kg and 579 kg masses.
- The distance between the 215 kg and 579 kg masses is 0.493 meters.
- We are given the universal gravitational constant, [tex]\( G = 6.672 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \)[/tex].
Step 2: Determine the Distances
- Since the 64 kg mass is exactly in the middle, it is 0.493 m / 2 = 0.2465 m away from each of the larger masses.
Step 3: Calculate the Gravitational Force from the 215 kg Mass
The formula for gravitational force is:
[tex]\[
F = \frac{G \cdot m_1 \cdot m_2}{r^2}
\][/tex]
For the 215 kg mass:
- [tex]\( m_1 = 215 \, \text{kg} \)[/tex]
- [tex]\( m_2 = 64 \, \text{kg} \)[/tex]
- [tex]\( r = 0.2465 \, \text{m} \)[/tex]
Substituting the values:
[tex]\[
F_1 = \frac{6.672 \times 10^{-11} \cdot 215 \cdot 64}{0.2465^2}
\][/tex]
[tex]\[
F_1 \approx 1.5109 \times 10^{-5} \, \text{N}
\][/tex]
Step 4: Calculate the Gravitational Force from the 579 kg Mass
For the 579 kg mass:
- [tex]\( m_1 = 579 \, \text{kg} \)[/tex]
- [tex]\( m_2 = 64 \, \text{kg} \)[/tex]
- [tex]\( r = 0.2465 \, \text{m} \)[/tex]
Substituting the values:
[tex]\[
F_2 = \frac{6.672 \times 10^{-11} \cdot 579 \cdot 64}{0.2465^2}
\][/tex]
[tex]\[
F_2 \approx 4.0689 \times 10^{-5} \, \text{N}
\][/tex]
Step 5: Determine the Net Gravitational Force
- The forces [tex]\( F_1 \)[/tex] and [tex]\( F_2 \)[/tex] are in opposite directions because one mass is on one side of the 64 kg mass and the other is on the opposite side.
- The net gravitational force is the difference between these two forces because [tex]\( F_2 > F_1 \)[/tex].
Calculate the net force:
[tex]\[
\text{Net Force} = F_2 - F_1
\][/tex]
[tex]\[
\text{Net Force} = 4.0689 \times 10^{-5} - 1.5109 \times 10^{-5}
\][/tex]
[tex]\[
\text{Net Force} \approx 2.5580 \times 10^{-5} \, \text{N}
\][/tex]
So, the magnitude of the net gravitational force exerted on the 64 kg mass is approximately [tex]\( 2.5580 \times 10^{-5} \, \text{N} \)[/tex].
