Math solution on Newton’s gravitational law

Problem 1

Problem: Two masses, 5 kg and 10 kg, are 2 meters apart. Calculate the gravitational force between them.

Solution: $𝐹=𝐺\frac{{𝑚}_1{𝑚}_2}{{𝑟}^2}=6.674\times 10^{-11}\frac{5\times 10}{2^2}=6.674\times 10^{-11}\frac{50}{4}=8.3425\times 10^{-10}\text{N}$

Problem 2

Problem: The gravitational force between two masses is $3.34\times 10^{-9}\text{N}$. If one mass is 2 kg and the distance between them is 1 m, find the other mass.

Solution: $𝐹=𝐺\frac{{𝑚}_1{𝑚}_2}{{𝑟}^2}⟹{𝑚}_2=\frac{𝐹{𝑟}^2}{𝐺{𝑚}_1}=\frac{3.34\times 10^{-9}\times 1^2}{6.674\times 10^{-11}\times 2}=\frac{3.34\times 10^{-9}}{1.3348\times 10^{-10}}\approx 25\text{kg}$

Problem 3

Problem: Two masses of 3 kg each are placed 0.5 m apart. Calculate the gravitational force between them.

Solution: $𝐹=𝐺\frac{{𝑚}_1{𝑚}_2}{{𝑟}^2}=6.674\times 10^{-11}\frac{3\times 3}{0.5^2}=6.674\times 10^{-11}\frac{9}{0.25}=2.40264\times 10^{-9}\text{N}$

Problem 4

Problem: If the gravitational force between two 1 kg masses is $6.674\times 10^{-11}\text{N}$, what is the distance between them?

Solution: $𝐹=𝐺\frac{{𝑚}_1{𝑚}_2}{{𝑟}^2}⟹𝑟=\sqrt{\frac{𝐺{𝑚}_1{𝑚}_2}{𝐹}}=\sqrt{\frac{6.674\times 10^{-11}\times 1\times 1}{6.674\times 10^{-11}}}=1\text{m}$

Problem 5

Problem: Two 8 kg masses are 0.2 m apart. Find the gravitational force between them.

Solution: $𝐹=𝐺\frac{{𝑚}_1{𝑚}_2}{{𝑟}^2}=6.674\times 10^{-11}\frac{8\times 8}{0.2^2}=6.674\times 10^{-11}\frac{64}{0.04}=1.06784\times 10^{-8}\text{N}$

Problem 6

Problem: The gravitational force between two masses is $5\times 10^{-10}\text{N}$. If the distance between them is 3 m and one mass is 4 kg, find the other mass.

Solution: $𝐹=𝐺\frac{{𝑚}_1{𝑚}_2}{{𝑟}^2}⟹{𝑚}_2=\frac{𝐹{𝑟}^2}{𝐺{𝑚}_1}=\frac{5\times 10^{-10}\times 3^2}{6.674\times 10^{-11}\times 4}=\frac{5\times 10^{-10}\times 9}{2.6696\times 10^{-10}}\approx 16.88\text{kg}$

Problem 7

Problem: If the gravitational force between two 10 kg masses is $1.3348\times 10^{-8}\text{N}$, what is the distance between them?

Solution: $𝐹=𝐺\frac{{𝑚}_1{𝑚}_2}{{𝑟}^2}⟹𝑟=\sqrt{\frac{𝐺{𝑚}_1{𝑚}_2}{𝐹}}=\sqrt{\frac{6.674\times 10^{-11}\times 10\times 10}{1.3348\times 10^{-8}}}=\sqrt{\frac{6.674\times 10^{-9}}{1.3348\times 10^{-8}}}\approx 0.707\text{m}$

Problem 8

Problem: Two 6 kg masses are 1.5 m apart. Calculate the gravitational force between them.

Solution: $𝐹=𝐺\frac{{𝑚}_1{𝑚}_2}{{𝑟}^2}=6.674\times 10^{-11}\frac{6\times 6}{1.5^2}=6.674\times 10^{-11}\frac{36}{2.25}=1.06784\times 10^{-9}\text{N}$

Problem 9

Problem: The gravitational force between two 7 kg masses is $4.6729\times 10^{-10}\text{N}$. What is the distance between them?

Solution: $𝐹=𝐺\frac{{𝑚}_1{𝑚}_2}{{𝑟}^2}⟹𝑟=\sqrt{\frac{𝐺{𝑚}_1{𝑚}_2}{𝐹}}=\sqrt{\frac{6.674\times 10^{-11}\times 7\times 7}{4.6729\times 10^{-10}}}=\sqrt{\frac{3.26952\times 10^{-9}}{4.6729\times 10^{-10}}}\approx 0.836\text{m}$

Problem 10

Problem: If the gravitational force between two masses is $2.6696\times 10^{-9}\text{N}$ and the distance between them is 0.5 m, with one mass being 1 kg, find the other mass.

Solution: $𝐹=𝐺\frac{{𝑚}_1{𝑚}_2}{{𝑟}^2}⟹{𝑚}_2=\frac{𝐹{𝑟}^2}{𝐺{𝑚}_1}=\frac{2.6696\times 10^{-9}\times 0.5^2}{6.674\times 10^{-11}\times 1}=\frac{6.674\times 10^{-10}}{6.674\times 10^{-11}}=10\text{kg}$

These problems illustrate the application of Newton's Law of Universal Gravitation in calculating the force between masses, determining distances, and finding unknown masses.