Gravitational fields Flashcards
What is the strength of a gravitational field, g?
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
Radial vs uniform field
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)
Gravitational potential energy W
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
Gravitational potential (at a point) V
GPE per unit mass of a small test mass - work done per unit mass to move an object from infinity to that point
What is an equipotential
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
Potential gradient
Change in V/R for potential v over small distance r
also g=-V/R
Show that potential gradient = -g
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
Potential gradient contour analogy
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
Rules for gravitational force between 2 objects
Always attractive
Inversely proportional to the square of distance between objects
Proportional to mass of objects
Gravitational field strength using law of gravitation
F=GMm/r^2 (M»m)
g = F/m = GM/r^2
Magnitude of attractive forces for spherical masses
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
Relationship between g and r
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
g inside a planet
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
Gravitational potential near a spherical planet
-GM/r
deriving escape velocity for a planet mass M and radius R
1/2mv^2 > GMm/r
V^2 > 2GM/R or V^2 > 2GR as gR^2 = GM
What is a geostationary orbit
One that is directly above the equator and has a time period of 24hrs
What is a synchronous orbit
One with the same time period as the orbiting body
(geostationary specifically over equator)
Significance of g=-V/R
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.
What is a hypothesis
a scientific hypothesis is a suggestion,
prediction or untested idea.
geosynchronous orbits vs low polar orbit
- 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.
An astronaut floats in a spacecraft in circular orbit around the earth, are they weightless
- 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
What is a force field?
A region in which an object experiences a non contact force, arising from the interaction between mass, static charges and between moving charges
Definition of free fall
Only acted upon by gravity
Gravitational potential in a radial field
-GM/r (m is mass of body)
Why gravitational potential + energy is “negative”
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)
Explanation for g=-V/R using eq
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
Explanation for g=-V/R using energy kinda
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
Uniform field
V/R constant - g constant
also potential gradient always orthogonal to equipotentials, so direction of gravitational force is at right angles to equipotentials
Why g decreases inside a planet
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
Area under a graph of “Force required to move 1kg mass” against distance from centre of mass
Change in potential, as change in potential between any 2 points is work done per unit mass from one point to another
Equipotentials and field lines
Field lines perpendicular to equipotentials, and point from higher to lower potential
Field is stronger where equipotentials and field lines closest together (dense)
acthually equipotentials aren’t lines theyre
Surfaces
Where could the escape velocity theoretically take you
To infinity, if ignoring the effect of air resistance and other masses
Why must the total energy of an orbiting satellite be negative
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)
Explain what happens to speed of a satellite when it moves to an orbit closer to the body
Potential energy decreases, gain kinetic. V increases
Alternatively v² proportion to 1/r
Difference in wording w potential
Work done per UNIT mass rather than test mass
Be able to sketch graphs of force, potential etc vs radius
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