Gravitational fields

Question 1

What is the strength of a gravitational field, g?

Answer 1

The force per unit mass of a small test mass placed in the field (test mass is small such that it doesn’t significantly affect the gravitational field)

= F/m

Also the acceleration of a freely falling object - one that only experiences gravitational force

Question 2

Radial vs uniform field

Answer 2

Radial - Line towards the centre of the body, g decreases with increasing distance

Uniform - magnitude of g is the same in magnitude and direction throughout the field - equally spaced and parallel

(Earth viewed from outer space vs viewed from surface)

Question 3

Gravitational potential energy W

Answer 3

Energy of an object due to its position in a gravitational field - position for 0 gpe is at infinity - object so far away that the gravitational force on it is negligible

Question 4

Gravitational potential (at a point) V

Answer 4

GPE per unit mass of a small test mass - work done per unit mass to move an object from infinity to that point

Question 5

What is an equipotential

Answer 5

A surface of constant potential - no work is done moving along an equipotential

Equipotentials for equal increases in potential are spaced further apart for a planet

Over a small region, equipotentials are horizontal (parallel to ground), as a gravitational field over a small region is uniform

Question 6

Potential gradient

Answer 6

Change in V/R for potential v over small distance r

also g=-V/R

Question 7

Show that potential gradient = -g

Answer 7

To move a test mass m a distance r in the opposite direction to the gravitational force on it - its GPE must increase by an equal and opposite force F acting over a distance r, and by an amount of energy = Fr

change in V = Fr/m – F=mV/r – as gravitational force is in opposite direction - Fgrav = -mV/r

rearrange to give g = -V/R

g acts in the opposite direction to the potential gradient

Question 8

Potential gradient contour analogy

Answer 8

Potential gradients are like contours on a map
Closer equipotentials - greater potential gradient and stronger field (closer contours steeper terrain), where equipotentials show equal spacing for equal changes of potential - potential gradient is constant (and g)

Potential gradient is at right angles to equipotential lines - so direction of gravitational force is always perpendicular to equipotentials

Question 9

Rules for gravitational force between 2 objects

Answer 9

Always attractive
Inversely proportional to the square of distance between objects
Proportional to mass of objects

Question 11

Gravitational field strength using law of gravitation

Answer 11

F=GMm/r^2 (M»m)

g = F/m = GM/r^2

Question 12

Magnitude of attractive forces for spherical masses

Answer 12

Force of attraction on a test mass m from a distance R from the centre is the same as if all of the mass of M was concentrated at the centre

Question 13

Relationship between g and r

Answer 13

g=g(s)R^2/r^2 where gs is g at the surface, and R is the radius of the body

inverse square relationship

Question 14

g inside a planet

Answer 14

Inside a planet, only the mass in the sphere of radius r contributes to g, and remainder pf mass outside gives no resultant force. So as r becomes smaller, the mass of M that contributes to g is also smaller

derive eq then

Question 15

Gravitational potential near a spherical planet

Answer 15

-GM/r

Question 16

deriving escape velocity for a planet mass M and radius R

Answer 16

1/2mv^2 > GMm/r

V^2 > 2GM/R or V^2 > 2GR as gR^2 = GM

Question 17

What is a geostationary orbit

Answer 17

One that is directly above the equator and has a time period of 24hrs

Question 18

What is a synchronous orbit

Answer 18

One with the same time period as the orbiting body
(geostationary specifically over equator)

Question 19

Significance of g=-V/R

Answer 19

g is in opposite direction of gradient.
r increases as you move away from the source of the gravitational field, but the force is in the opposite direction, ie towards the source of the gravitational field.

Question 20

What is a hypothesis

Answer 20

a scientific hypothesis is a suggestion,
prediction or untested idea.

Question 21

geosynchronous orbits vs low polar orbit

Answer 21

• It orbits over the Equator.
• It maintains a fixed position in relation to the
  surface of the Earth.
• It has a period of 24 hours (the same as the
  Earth’s period of rotation on its axis).
• The geosynchronous satellite enables
  uninterrupted communication between a
  transmitter and a receiver whereas the
  other satellite does not.
• Unlike the other satellite, the
  geosynchronous satellite does not require the
  use of a steerable dish.

A satellite in low polar orbit has a fairly
short time period, scanning the Earth
several times during the day.

Question 22

An astronaut floats in a spacecraft in circular orbit around the earth, are they weightless

Answer 22

• the force of gravity on the astronaut is still
  mg, where g is the local value of the field
  strength within the spacecraft;
• this force provides the centripetal force to
  keep the astronaut in orbit;
• the astronaut is in free fall, as is the
  spacecraft;
• the astronaut appears weightless because
  he or she is not supported

Weight from sensation of support

Question 23

What is a force field?

Answer 23

A region in which an object experiences a non contact force, arising from the interaction between mass, static charges and between moving charges

Question 24

Definition of free fall

Answer 24

Only acted upon by gravity

Question 25

Gravitational potential in a radial field

Answer 25

-GM/r (m is mass of body)

Question 26

Why gravitational potential + energy is “negative”

Answer 26

Consider 2 masses at a distance r in a gravitational field. In order to separate them more, work must be done on the system, so U must increase. In order to separate them an infinite distance, work will also have to be done, however the system will have 0 potential energy, as a system of 2 bodies that don’t interact have 0 potential energy (also eq as r tends to inf). Which means that the potential energy of the system must INCREASE to zero, and is therefore always negative

when interaction is attractive, potential in negative, when interaction is repulsive, potential is positive, when potential is 0, bodies are infinitely spaced apart

pulling stuff apart increases the potential energy (making it less negative)

Question 27

Explanation for g=-V/R using eq

Answer 27

consider a mass moving a distance r in the opposite direction to the gravitational force Fg on it. Energy must be increased by an equal and opposite F through r, and = Fr. Change in potential V = W/m =Fr/m

so F = Mv/r, therefore Fg = -Mv/r

g = Fg/m = -v/r

Question 28

Explanation for g=-V/R using energy kinda

Answer 28

ay you start out at rest, and there are no forces acting on you except for gravity. What direction do you move? In the direction of the gravitational field, of course! Now, because you’re now moving, that means you have kinetic energy, which you didn’t before. But, energy is conserved, so where did this energy come from? It was converted from (gravitational) potential energy! This “uses up” some of the gravitational potential energy, so you must now be in a location with lower gravitational potential than where you started.

Unless you go far enough that the gravitational field around you changes significantly, gravity will keep accelerating you faster and faster, which means more and more kinetic energy and less and less potential energy, i.e., you keep accelerating in the direction of falling potential.

In contrast, the gradient of a potential points in the direction of rising potentials, more or less by definition, because of how derivatives work

Question 29

Uniform field

Answer 29

V/R constant - g constant

also potential gradient always orthogonal to equipotentials, so direction of gravitational force is at right angles to equipotentials

Question 30

Why g decreases inside a planet

Answer 30

Only the mass in the sphere radius r contributes to g, all other mass provides no resultant force. So g is zero at the centre

Question 31

Area under a graph of “Force required to move 1kg mass” against distance from centre of mass

Answer 31

Change in potential, as change in potential between any 2 points is work done per unit mass from one point to another

Question 32

Equipotentials and field lines

Answer 32

Field lines perpendicular to equipotentials, and point from higher to lower potential
Field is stronger where equipotentials and field lines closest together (dense)

Question 33

acthually equipotentials aren’t lines theyre

Answer 33

Surfaces

Question 34

Where could the escape velocity theoretically take you

Answer 34

To infinity, if ignoring the effect of air resistance and other masses

Question 35

Why must the total energy of an orbiting satellite be negative

Answer 35

As it doesn’t have enough energy to escape (to infinity)
total ek = -1/2(GMm/r)

equal in magnitude to Ek, but ek is positive and total E is negative

In order to escape orbit (not ground), a satellite needs to have Ek equal to GPE at that height. Alr has necessary Ek, so would need to gain a further equal amount of Ek equal to initial amount to escape the planet (minimum)

Question 37

Explain what happens to speed of a satellite when it moves to an orbit closer to the body

Answer 37

Potential energy decreases, gain kinetic. V increases

Alternatively v² proportion to 1/r

Question 38

Difference in wording w potential

Answer 38

Work done per UNIT mass rather than test mass

Question 39

Be able to sketch graphs of force, potential etc vs radius

Answer 39

year