11.1 - Gravitational Fields

Question 1

What force underlies all current theories about the formation of galaxies, stars and planets and explain how this force works to do this

Answer 1

The current theories about the formation of the galaxies, stars and planets are all driven by gravity.

Every particle with mass attracts every other particle with mass. So, if the Big Bang spread tiny particles across space, they would all attract each other. The acceleration generated by these tiny forces might be exceedingly small, but the universe has plenty of time.

Slowly but surely, the particles move towards each other and clump together. This effect is more pronounced for particles nearer to each other, as they attract more strongly than those separated by larger distances. These clumps of matter will continue to attract nearby particles or lumps that have formed, and continue to accrete into larger and larger bodies of material. This collection of matter might become a planet, or if enough material gets packed together densely enough, nuclear fusion may start and it becomes a star.

Question 2

What’s similar about the causes of an electric field compared to a gravitational field

Answer 2

Electric fields are created by charged particles, a massive particle will also generate a radial gravitational field around itself. A particle that has mass will feel a force when it is in a gravitational field. Unlike electric fields, gravity is always attractive

Question 3

Is gravity an attractive force

Answer 3

YES always attractive 🤭😘

Question 4

What is a gravitational field, what’s the equation for force

Answer 4

A particle that has mass will feel a force when it is in a gravitational field. Gravity is always attractive.

The force that a body will feel is the strength of the gravitational field (g) multiplied by the amount of mass (m), as given by the equation:

F = mg

Question 5

How can we use F = mg to calculate acceleration

Answer 5

From this force equation, we can also see how quickly a massive particle would accelerate. From Newton’s second law we know F = ma, so we can equate the two equations

F = mg = ma

a = mg/m = g

Therefore, what we have previously referred to as the acceleration due to gravity is the same as the gravitational field strength, g

On earth these are 9.81 ms^-2 or 9.81 Nkg^-1

Question 6

What are the units for acceleration due to gravity

Answer 6

ms^-2

Question 7

What are the units for gravitational field field strength, g

Answer 7

9.81 Nkg^-1

Question 8

Who first published the law of the force of gravity between 2 objects

Answer 8

Sir Isaac Newton

Question 9

What is the equation that Newton first published that gives us the gravitational force between two masses, m(subscript 1) and m(subscript 2), which are separated by distance, r, between their centres of gravity

Answer 9

F = Gm(subscript 1) x m(subscript 2) / r^2

Wheee G is the gravitational constant, G = 6.67 x 10^-11 Nm^2 kg^-2

Question 10

How did we work out the mass of the earth

Answer 10

Even before space travel, it was possible to use data about the moons orbit to work out the mass of the earth. The time period of the moons orbit around the earth is 27.3 days, or T = 2.36 x 10^6. The average orbital radius for the moon is 384,000 km, or r = 3.84 x 10^8 m. From these data, we can calculate the mass of the earth:

Gravitational attraction between moon and earth, F = Gm(subscript E)m(subscript m) / r^2

Centripetal force required to keep moon in orbit, F = m(subscript m) x v^2 / r

Gravity is the cause of centripetal force so these are equal, rearrange to get m(subscript E) like so:

m(subscript E) = rv^2 / G

The speed of the moon comes from the time it takes to orbit:

v = 2 x pi x r / T

Then sub in values to find the mass of the earth as 6.02 x 10^24 kg

Question 11

Define gravitational field

Answer 11

A gravitational field is a region of spacetime which is curved. This curvature will cause particles to experience an accelerating force.

Question 12

What things generate a gravitational field and what is the field like

Answer 12

Any mass will generate a gravitational field, which will then exert a force on any mass within the field. As gravity is always attractive, the field produced by a point mass will be radial in shape and the field lines will always point towards the mass.

Question 13

Tell me about the gravitational field strength of a point mass

Answer 13

The radial field produced by a point mass naturally has its field lines closed together nearer the mass, as a result of its geometry. This means that the strength of the field decreases with increasing distance from the mass causing it. The decrease is significant, and in outer space, as far as it is possible to be from a galaxy or other particles, there will be regions where there is virtually no gravity. This can be explained mathematically by the formula which tells us the strength of a gravitational at a certain distance, r, from a mass, M. We have already seen the force on a mass, m, caused by a gravitational field is F = mg.

Question 14

The gravitational force on a mass, m, because of another mass, M, is given by Newton’s expression:

Answer 14

F = GMm/r^2

Question 15

Newton’s expression and F = mg are equal so

Answer 15

F = GMm/r^2 (Newton’s expression) and F = mg are calculating the same force, so themselves must be equal:

GMm/r^2 = mg = F

so, g = GM/r^2

I.e the field strength is independent of the object being acted upon, eg a satellites mass doesn’t affect the field strength of the earth

Question 16

What is potential energy generally

Answer 16

Potential energy is that possessed by an object by virtue of its position. The potential at a point in any type of field is the general concept of the amount of energy needed to get to that position in the field for any object affected by the field.

Question 17

What is gravitational potential

Answer 17

For a gravitational field, the potential energy is expressed as an amount of energy per unit mass (Jkg^-1), as mass is what is affected by the field. The definition of gravitational potential is the amount of work done per unit mass to move an object from an infinite distance to that point in the field. (Infinite distance means potential energy is zero)

It is always negative

Question 18

Why is gravitational potential always a negative quantity

Answer 18

As gravitational fields are always attractive, objects will always gain energy on moving into a point in the field, and so gravitational potential is always a negative quantity. Hence why there is a minus sign in the equation

Question 19

What’s the equation for gravitational potential

Answer 19

Gravitational potential can be calculated at a distance r from a mass M from the equation:

V(subscript grav) = -GM/r

Question 20

The gravitational field strength tells us how quickly the potential is changing over distance, and the mathematical connection between the two is:

Answer 20

g = -dV(subscript grav) / dx

Where d means change in (usually means small change) whereas triangle symbol means large change.

Over small distances, in which g does not change significantly, this can be calculated using the actual distance change

g = - triangle V(subscript v) / triangle x

g x triangle x = -triangle V(subscript grav)

As potential was defined per unit mass, and is already a negative value, the magnitude of the actual gravitational potential energy change for an object m will be given by:

E = m x triangle V(subscript grav) = mg x triangle x

We have previously used

E(subscript p) = mg x triangle h

Question 21

Tell me about the similarities between electric and gravitational field equations

Answer 21

Mathematically, gravitational field equations are of exactly the same form as those we used to calculate the force between charged particles, electric potential and electric field strength.

The only differences are the symbols we use to represent the quantities causing the fields, and the constants of proportionality

The similarities come from the fact that both types of field are radial from a point. The constants of proportionality depend upon the way the forces interact with the fabric of the universe, and with the unit system we use in the calculations.

Question 22

For force, what gravitational field equation is similar to the electrical field equation:

F = kq(subscript 1) x q(subscript 2) / r^2

Answer 22

F = Gm(subscript 1)m(subscript 2) / r^2

Question 23

For field strength, tell me the gravitational field equation similar to the electrical field equation;

E = kq/r^2

Answer 23

g = Gm/r^2

Question 24

For potential, tell me the gravitational field equation similar to the electrical field equation;

V = kq/r

Answer 24

V(subscript grav) = -Gm/r

Question 25

How do the equations for gravitational and electrical fields differ

Answer 25

Although they share the same form, they do differ in some significant ways.

Whilst gravitational forces are always attractive, electric forces are not. As electrical forces can be either positive or negative, the electric field can be in either direction to or from a charge. A charged particle can be shielded from an electric field, but a massive particle cannot be shielded from a gravitational field. In addition, the electromagnetic force is significantly stronger than the gravitational force.

Gravity affects all things with mass, not just things that are charged.

Question 26

Define gravitational potential

Answer 26

Gravitational potential is the amount of work done per unit mass to move an object from an infinite distance to that point in the field.

Question 27

Why is gravitational potential negative

Answer 27

At a separation (say to Earth) of infinity (r=infinity), the energy of a mass is 0. 0 is an important number, so this separation is an important separation and why we use it in the definition. At a separation of the surface of Earth (r=6400km) gravity wants pull the test mass closer and closer. Its gravitational potential energy wants to pull it closer and closer. To do the opposite (ei, move a test mass from a point TO infinity) we need positive amounts of energy (like when we lift something up onto a shelf, or shoot a rocket out into space really far). But as we said what the gravitational potential energy wants to do is the opposite! So the work done by gravity is NEGATIVE. The gravitational potential energy is negative because us trying to do the opposite of what gravity wants needs positive energy.