The law of universal gravitation

Two steel balls of $1\ \text{kg}$ each are placed so that their centres are $1\ \text{m}$ apart, as shown in the diagram.

Apply the law of universal gravitation, $F = G\cdot\frac{M\cdot m}{r^2}$, with $G = 6,67\cdot10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$, and calculate the gravitational force attracting them. Does this seem like a large or small force? Comment on it.

In the law of universal gravitation, $F = G\cdot\frac{M\cdot m}{r^2}$:

1. a) State what each letter represents ($F$, $G$, $M$, $m$ and $r$) and its SI unit of measurement.
2. b) The value of $G$ is $6,67\cdot10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$, a very small number. Explain the consequence of this for the force between objects of ordinary mass.

Two classmates, with masses of $60\ \text{kg}$ and $70\ \text{kg}$, are sitting such that their bodies are on average $1\ \text{m}$ apart. Calculate the gravitational force of attraction between them ($G = 6,67\cdot10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$) and express it in standard form.

The Sun keeps the Earth in orbit by attracting it. Calculate this force using the following data: mass of the Sun $M_S = 1,99\cdot10^{30}\ \text{kg}$, mass of the Earth $M_T = 5,97\cdot10^{24}\ \text{kg}$, mean distance between their centres $r = 1,5\cdot10^{11}\ \text{m}$ and $G = 6,67\cdot10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$. Then, compare the result with the Earth–Moon force, of about $2\cdot10^{20}\ \text{N}$: how many times larger is it?

Calculate the gravitational force of attraction between the Earth and the Moon, as shown in the diagram.

Given: mass of the Earth $M_T = 5,97\cdot10^{24}\ \text{kg}$, mass of the Moon $M_L = 7,35\cdot10^{22}\ \text{kg}$, mean distance between their centres $r = 3,84\cdot10^{8}\ \text{m}$ and $G = 6,67\cdot10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$. Express the result in standard form, rounded to two decimal places.

Two twin oil tankers, each with a mass of $10^{6}\ \text{kg}$, are anchored with their centres of mass separated by $100\ \text{m}$.

1. a) Calculate the gravitational force attracting them ($G = 6,67\cdot10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$).
2. b) Compare this force with that needed to drag a sheet of paper across a table, of the order of $0,01\ \text{N}$. Will the ships notice this attraction? Justify your answer.

At what distance from the centre of the Earth is the gravitational force on a body reduced to half of that felt on the surface?

1. a) Set up the equality using the fact that the force decreases with $\frac{1}{r^2}$.
2. b) Rearrange for $r$ in terms of the Earth's radius $R_T$ (a $\sqrt{2}$ will appear).
3. c) Using $R_T = 6370\ \text{km}$, calculate at what height above the surface one must ascend, in kilometres. Do you think a commercial airplane, flying at a height of about $10\ \text{km}$, notices this decrease? Give reasons for your answer.

Let us deduce the value of $g$ on the surface of the Earth from the law of universal gravitation.

1. a) Write the force exerted by the Earth on a body of mass $m$ in two ways: as $F = m\cdot g$ and as $F = G\cdot\frac{M\cdot m}{r^2}$.
2. b) Equate the two expressions, cancel the mass $m$ and rearrange for $g$.
3. c) Calculate its value with $M = 5,97\cdot10^{24}\ \text{kg}$, $r = 6,37\cdot10^{6}\ \text{m}$ and $G = 6,67\cdot10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$. What well-known value do you obtain?

Two fixed masses, of $100\ \text{kg}$ and $400\ \text{kg}$, are $3\ \text{m}$ apart. We want to place a third mass on the line joining them, between the two, such that the net gravitational force on it is zero, as shown in the diagram.

1. a) If $x$ is the distance from the third mass to that of $100\ \text{kg}$, write down the equality between the two forces acting on it.
2. b) Solve the equation and calculate $x$ (hint: first simplify $G$ and the third mass, and then take square roots).
3. c) Does the result depend on the value of the third mass? What about $G$? Comment on this.

Newton tested his law by comparing the fall of objects at the surface with the motion of the Moon.

1. a) Calculate the force that the Earth exerts on a body of $1\ \text{kg}$ on its surface ($r = 6,37\cdot10^{6}\ \text{m}$, $M_T = 5,97\cdot10^{24}\ \text{kg}$, $G = 6,67\cdot10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$).
2. b) Calculate the force it would exert on that same body if it were at the Moon's distance ($r = 3,84\cdot10^{8}\ \text{m}$).
3. c) Verify that the second force is about $3600$ times smaller and relate that number to the ratio between the two distances, which is about $60$. What law is confirmed?