Gravitational and strong interactions

What Nour says

1. 1. Universal law of gravitation

   Today, the whole chapter: "Gravitational and strong interactions", Physics, Year 3, Mathematics section.

   We will look at the force between masses, then the fields they create, and finally the force that holds the nucleus together.

   Each part has a clear idea, a drawing, and one worked example like in the bac.

   Let's start with the universal law of gravitation.

   This law describes the force of attraction between any two masses in the universe.

   In other words, any two objects with mass pull on each other, even planets and apples.

   Let's draw two point masses, A with mass m_A and B with mass m_B.

   They are separated by a distance d between their centers.

   Each mass attracts the other with a force along the line joining them.

   Mass B pulls mass A toward it, and mass A pulls mass B back with equal strength.

   The two forces are equal in size but point in opposite directions.

   The law gives the force's magnitude: F equals G times m_A times m_B, divided by d squared.

   Here G is the universal gravitation constant, the same number everywhere in the universe.

   The masses m_A and m_B are measured in kilograms, and the distance d in meters.

   The force F is measured in newtons, just like any other force we studied before.

   This force is always attractive, it never pushes two masses apart.

   Here is why the formula makes sense physically.

   The force grows when the masses are bigger, since more matter means more attraction.

   The force shrinks fast when the distance grows, because of the square in the denominator.

   If you double the distance, the force becomes four times weaker, not just two times.

   This is why gravity from a far planet feels so weak compared to Earth's pull.

   Now let's work through an example, just like in the bac exam.

   The Moon has a mass of 7,35 times 10 to the power 22 kilograms.

   The distance between their centers is 3,84 times 10 to the power 8 meters.

   We want the gravitational force between the Earth and the Moon.

   We use the formula: F equals G times m_T times m_L, divided by d squared.

   We plug in the numbers: G, the Earth's mass, the Moon's mass, and the distance squared.

   This gives a force of about 1.98 times ten to the power 20 times 10 to the power 20 newtons.

   This huge force is what keeps the Moon orbiting around the Earth.

   Notice the force acts along the straight line joining the Earth and the Moon.

   Also notice the Earth pulls the Moon exactly as strongly as the Moon pulls the Earth.

   This is the same action-reaction pair we saw with the forces F_{A/B} and F_{B/A}.

   Let's state the key result clearly now.

   The universal law of gravitation links force, two masses, and the distance between them.

   The force is always attractive, grows with mass, and shrinks fast with distance.

   This single law explains why apples fall, and why planets stay in orbit.

   Keep this formula in mind, because we will use it again for the gravitational field.

   What happens to the gravitational force if the distance between two masses doubles?

2. 2. Gravitational field

   Let's move to the gravitational field.

   A mass creates a gravitational field around it, even with nothing else nearby.

   In other words, the field is the influence a mass spreads through space around itself.

   Let's draw a mass A, the one creating the field.

   At any point M near it, we can measure the field's strength.

   The field vector at point M always points toward the mass A, pulling inward.

   The field's strength is given by G_A equals G times m_A, divided by r squared.

   Here m_A is the source mass in kilograms, and r is the distance from A to M in meters.

   The field G_A is measured in newtons per kilogram.

   Here is why we use a field instead of just a force.

   The field exists at every point in space, whether or not another mass is actually there.

   A mass m placed at M then feels a force equal to m times the field G_A.

   Far from the source, the field becomes very weak, almost zero.

   Now let's solve an example, in the style of the bac.

   Earth's mass is 5,97 times 10 to the power 24 kilograms.

   Earth's radius is 6,37 times 10 to the power 6 meters.

   The satellite flies 400 kilometers above the surface, so we add that to the radius.

   The total distance from Earth's center is r equals R_T plus 4 times 10 to the power 5 meters.

   We plug this r into the field formula, with Earth's mass and the constant G.

   This gives a field strength of about 8.68 newtons per kilogram.

   This is close to the value of 9,8 newtons per kilogram we feel on the ground.

   It is slightly smaller because the satellite is farther from Earth's center.

   Let's state the key result now.

   The gravitational field describes the pull a mass creates at every point around it.

   Its strength depends only on the source mass and the distance from it.

   Next we will connect this idea to the field we feel every day on Earth.

   Does the gravitational field at point M depend on a mass placed at M?

3. 3. Gravity field (champ de pesanteur)

   Now let's talk about the gravitational field (champ de pesanteur).

   That is the field created by a whole planet, felt by objects near its surface.

   Let's draw the Earth as a big circle, with its center O.

   Earth's mass is M_T, and its radius is R_T, measured from the center to the surface.

   Let's place a point M at altitude h above the surface.

   At point M, gravity pulls any object straight down, toward the center O.

   The gravity field g has the same formula we just saw for the gravitational field.

   Here R_T plus h is the full distance from Earth's center to point M.

   The gravity field g is measured in newtons per kilogram, same as before.

   On the ground, when h is zero, we get the familiar value g_0.

   Here is why the gravity field matters so much for us.

   This field is what gives every object its weight, pulling it toward the ground.

   The weight of an object equals its mass times the gravity field at its location.

   As altitude h grows, the distance from the center grows, so g gets smaller.

   This means a satellite far above Earth feels a weaker gravity field than we do.

   Now let's solve an example in the style of the bac.

   Earth's mass stays the same, 5,97 times 10 to the power 24 kilograms.

   Earth's radius is 6,37 times 10 to the power 6 meters.

   The altitude h is 600 kilometers, which is 6 times 10 to the power 5 meters.

   We add R_T and h together to get the total distance from the center.

   We plug this distance, squared, into the gravity field formula with G and M_T.

   This gives a gravity field of about 8.21 newtons per kilogram at that altitude.

   This value is smaller than the 9,8 newtons per kilogram we feel on the ground.

   It makes sense, since point M is farther from Earth's center than the surface.

   Let's state the key result now.

   The gravity field is created by a whole planet and gives weight to nearby objects.

   It decreases as altitude increases, since the distance to the center grows.

   This field connects directly to the force each object feels, called its weight.

   Keep in mind this gravity field is just the gravitational field, applied to a whole planet.

   Why is g smaller for a satellite in orbit than for someone standing on the ground?

4. 4. Comparing gravitational and electric interaction

   Now let's compare the gravitational interaction with the electric interaction we studied earlier.

   You already know the electric force between two charges, from an earlier chapter.

   Let's draw two point masses, A and B, separated by a distance d.

   Both masses attract each other with the gravitational force we wrote before.

   If A and B also carry electric charges, they feel an electric force too.

   Both formulas look alike, with a constant, two quantities, and the distance squared.

   Here is why the two forces behave so differently in practice.

   Gravitational force is always attractive, since mass is always positive.

   Electric force can attract or repel, depending on the sign of each charge.

   The gravitational constant G is extremely small compared to the electric constant k.

   This means gravitational force is far weaker than electric force, for similar quantities.

   Now let's compare them with real numbers, as asked in bac exercises.

   Each electron has a mass of 9,1 times 10 to the power minus 31 kilograms.

   Each electron carries a charge of 1,6 times 10 to the power minus 19 coulombs.

   We compute the ratio F_g over F_e, and the distance d cancels out completely.

   This ratio comes out to about 2.4 times 10 to the power minus 43.

   This number is incredibly tiny, showing gravity is almost nothing next to electric force.

   That is why gravity only matters for huge masses, like planets and stars.

   At the scale of atoms, electric force completely dominates over gravity.

   Let's state the key result now.

   Gravitational and electric forces share the same mathematical form, with distance squared in the denominator.

   But gravity is always attractive and incredibly weak compared to the electric force.

   This difference explains why gravity rules the cosmos, while electric force rules the atom.

   Which force is stronger between two electrons, the gravitational or the electric one?

5. 5. Strong interaction and nuclear cohesion

   Let's finish with the strong interaction and nuclear cohesion.

   The nucleus of an atom holds protons and neutrons packed tightly together.

   Let's draw a small nucleus with protons and neutrons close to each other.

   Protons are positively charged, so the electric force pushes them apart from each other.

   Yet the nucleus stays together, so another force must hold it, much stronger than repulsion.

   This is the strong interaction, that is the force that binds protons and neutrons in the nucleus.

   It acts only at extremely short range, inside the nucleus itself.

   Its range is about 10 to the power minus 15 meters, the size of a nucleus.

   Beyond that tiny distance, the strong interaction disappears completely.

   Here is why this force is essential for matter to exist.

   Without it, electric repulsion would push all protons apart, and no nucleus could form.

   The strong interaction attracts protons and neutrons together, overpowering the electric repulsion between protons.

   It acts equally between two protons, two neutrons, or a proton and a neutron.

   This is what we call nuclear cohesion, that is the stability the strong force gives to the nucleus.

   Without this cohesion, every atom's nucleus would instantly break apart.

   Now let's look at an example, in the style of the bac.

   The carbon nucleus has 6 protons and 6 neutrons, packed in a tiny volume.

   Each pair of protons repels through the electric force, since both carry positive charge.

   Yet the distance between nucleons is about 10 to the power minus 15 meters.

   At this tiny distance, the strong interaction is far more intense than the electric repulsion.

   So the strong force wins, and the carbon nucleus stays stable and bound together.

   This balance explains why the carbon nucleus does not fly apart on its own.

   The same reasoning applies to every stable nucleus in nature, from helium to iron.

   Let's state the key result now.

   The strong interaction is a short-range force that binds protons and neutrons inside the nucleus.

   It overcomes electric repulsion between protons, giving the nucleus its cohesion and stability.

   This force, unlike gravity, has nothing to do with mass or distance in the usual sense.

   With this, we have covered gravitation and the strong interaction, the two forces shaping matter.

   Why does the strong interaction keep protons together despite their electric repulsion?

   That completes our chapter on gravitational and strong interactions.

   We moved from the force between two masses, to fields, and finally to the nucleus itself.

   Review the formulas and the two exercises to be ready for the bac.