Gravitational Fields Flashcards
Gravitational field
Region in which an object with mass experiences a force
Field lines
Arrows indicating the direction in which a small test mass would experience a force of attraction
Closer field lines
Stronger gravitational field
Equipotentials
Lines drawn to indicate points of equal potential
Cross so perpendicular to the field lines
How much work is done moving a test mass along an equipotential
None
Uniform field
Field lines are parallel to each other
Equipotentials are parallel to each other
Field lines and equipotentials are perpendicular to each other
Newton’s law of gravitation
The attractive force between two point masses is directly proportional to the product of their masses
And inversely proportional to the square of their separation
Mathematically express newtons law of gravitation
F∝Mm/r^2
F=GMm/r^2
G; Gravitational constant
M; Mass of one object
m; Mass of the other object
r; Distance between the centre of masses of each body
Technically should be negative
Units for G
Gravitational constant
Nm^2kg^-2
Assumptions when dealing with planets
Spherical
Uniform density so C.O.M is physical centre
Gravitational field strength
The force acting on a body per unit of mass
Equation for gravitational field strength
g=F/m
Where F is the force acting on the body
And m is the mass of the body in the field
Gravitating body
Any object/mass which creates a gravitational field
Test mass
Body with negligible size and mass
Which placed within the field will experience a force towards the gravitating body
Derive the equation for gravitational field strength and separation
F=gm
gm=GMm/r^2
g=GMm/r^2m
g=GM/r^2
How do you find the net gravitational field strength at a point if there are multiple gravitating bodies
Vector sum of the gravitational field strengths due to each of the gravitating bodies
Neutral point
Where the effective gravitational field strength is zero
Both gravitational field strengths have the same magnitude and are in opposite directions
Neutral point equation
m1/m2 = r1^2/r2^2
What happens when an object moves against the field
Moving away from the centre
Object does work against the field
Increasing its potential energy but decreasing its kinetic energy
What happens when an object moves along a field
Moves towards the centre
Field does work on the object
Attraction
Decreasing it potential energy but increases its kinetic energy
Absolute potential energy/potential energy
Work done in moving an object with mass from infinity to that point in the field
Equation for Potential energy in a uniform field
Ep=mgh
For a radial gravitational field where is the potential defined as zero
At infinity
Equation for potential energy in a radial field
Ep=-GMm/r
Is gravitational potential energy a scalar or vector
Scalar
How do you get the absolute potential energy for more than one pair of gravitating bodies
Add
Why is there a negative sign
Infinity defined as zero potential energy
When a body moves away from the gravitating body it gains potential energy
So it gets closer to zero
So it gets less negative
Absolute gravitational potential/potential
The work done per unit mass in moving a body from infinity to that point in the gravitational field
Is absolute potential a scalar or vector
Scalar
How do you convert absolute potential to potential energy
x mass
Explain V=-GM/r
The m is the mass of the object creating the gravitational field
Since it is the work done per unit mass to get the gravitational potential, the mass of the object cancel out on the top and on the bottom
V in Jkg^-1
How do you calculate the work done in moving an object (Ep)
mass x change in potential
How do you get the absolute potential at a point where two gravitational fields act
The sum of the gravitational potential of the two gravitating bodies
Gravitational field strength given change in potential and radius
g=-V/r
Force given the change in potential energy and the change in radius
F=-E/r
Force and separation equation
F=GMm/r^2
Force and separation ratio
F1r1^2 = F2r2^2
Field strength and separation ratio
g1r1^2 = g2r2^2
Potential and separation ratio
V1r1 = V2r2
Potential energy and separation ratio
Ep1r1 = Ep2r2
How does the gravitational constant change when going from earth to the moon
It doesn’t
It is the same everywhere
Area under a force separation graph
Work done moving between the two points
Change in Ep
Area under a field strength and separation graph
Change in potential
Work done per unit mass
Gradient of potential separation graph
Magnitude of gravitational field strength
Gradient of potential energy separation graph
Gravitational force
Gravitational field strength in a sphere of uniform density aka planet
g=4/3 x πpGr
Where p is density and r is radius
Gravitational potential inside a sphere of uniform density aka planet
V=-2/3 x πpGr^2
Where p is density and r is radius
Orbital time period equation
KEPLERS THIRD LAW
T^2 = 4π^2r^3/GM
Proportionality for orbital time period
KEPLERS THIRD LAW
T^2 ∝ r^3
Kepler’s Third Law
The orbital time period squared is directly proportional to the radius of orbit squared
Ratio for Kepler’s third law
T1^2/r1^3 = T2^2/r2^3
How do you derive the equation for orbital time period
- Equate gravitational force equation to centripetal force equation in terms of angular velocity
- Cancel m
- Convert w into time (w=2π/T)
- Cancel and rearrange to get an equation for T^2
Orbital velocity equation
v=√GM/r
Escape velocity equation
v=√2GM/r
How do you derive the equation for orbital velocity
Equating gravitational force and centripetal force in terms of v
How do you derive the escape velocity
Equate kinetic energy to gravitational potential energy
What is escape velocity
For something to escape its orbit its kinetic energy must be greater than the increase in potential energy moving from position to infinity
When is an orbit stable
Total energy is negative
Magnitude of KE
Kinetic energy of orbit
GMm/2r
How is the kinetic energy of orbit derived
Subbing in the orbital velocity into the kinetic energy formula
Total energy of orbit
Ke+Pe = -GMm/2r
Potential energy of orbit
-GMm/r
Orbital velocity and separation ratio
v1√r1 = v2√r2
Escape velocity equation
v1√r1 = v2√r2
What doesn’t escape velocity apply to
Rockets or objects that have an engine to generate kinetic energy
Why can’t light escape a black hole
A black holes escape velocity is greater than the speed of light
Explain energy changes in an increasing orbit
Distance from the centre of the planet increases
Potential energy increases to get less negative and closer to zero
Kinetic energy decreases
Work done against field
Total energy increases so less negative
Velocity decreases
Explain energy changes in a decreasing orbit
Distance from the centre of the planet decreases
Potential energy decreases to get more negative and further from zero
Kinetic energy increases
Work done on object by field
Total energy decreases so more negative
Velocity increases
Polar orbit
A satellite moving from north to south pole and back
Typical orbit of 2 hours
What uses do satellites with polar orbits have
Can be positioned above any point on the Earth’s surface
Meteorology, weather, espionage, spying
Low earth orbit
Used by ISS and hubble telescope Most man made objects Orbital period of less than 128 minutes Orbital radius of less than 8400km Close to earth so goo for observation and spy satellites (detailed images)
Middle earth orbit
Anything above LEO but below geostationary
Orbital period between 2 and 24 hour (12 hours most common)
Used for GPS and galileo navigation satellites and communication satellites for communication in high latitude regions (near poles using polar orbits)
High earth orbit
Anything with an orbital radius greater than geostationary
Geosynchronous
24 hours to complete a cycle in same direction as earth is rotating
Geostationary
Takes 24 hours to complete a cycle in the same direction as earth
Fixed to a point on its equator
Advantage of geostationary
Ground dishes
Antennae
Aerials don’t need repositioning
Geostationary satellite uses
Communications of television l
Mobile phones
Sat nava
Issue with putting more satellites in orbit
Increased risk of collision
Orbital radius for geostationary
42200km
*all geostationary orbits are at the same height
Derive the equation for gravitational field strength in a sphere of uniform density
State density equation Rearrange for m Sub in volume of a sphere Rearrange the gravitational field strength equation Rearrange to get
g=4/3piGpr