Fields

force field

• region in which a body experiences a non-contact force

• can be represented as a vector

field line

• the path which a test mass in a gravitational field field would follow

radial field

• field lines always directed towards centre for gravitational field

• straight

• converge to or diverge from a single point

uniform field

• field is same strength and direction at every point

• parallel field lines

gravity

• universal attractive force acting between all matter

gravitational field strength, g

$$g=\frac{F}{m}$$

• the force per unit mass on a small test mass placed in the field at a particular position in the field where it is only acted on by gravitational force

• negative of gravitational potential gradient

• Nkg^-1

gravitational potential (V)

• the work done per unit mass to move a small test mass from infinity to a particular point in a gravitational field

• zero value at infinity

• negative everywhere within the field

• energy required per unit mass to remove that mass from the gravitational field (work must be done by the mass against gravity to leave the field)

• V=W/m

gravitational potential difference

• the work done in moving a mass m between two points in a gravitational field

• ∆W=m∆V

gravitational potential energy

• work done to move a small test mass from infinity to a particular point in a gravitational field

• also negative as 0 gpe at infinity and work must be done by gravitational force moving object from that point in field to infinity

• work is done by the field so work supplied is negative

Equipotential

• a line or surface in a field along which the electric or gravitational potential

  is constant

• no work is done when moving along an equipotential surface

• the direction of the field is perpendicular to a point on the line

Why equipotentials get more spaced apart

• at an increased distance from the mass, the graviatational field gets weaker so the gain of gpe per unit distance (height) decreases

• for an equal increase in potential, the equipotentials must become more spaced apart since that same work is done by mass (height)against a smaller force so a greater distance must be moved

• the closer they are, the greater the potential gradient and the stronger the field is

the potential gradient

change of potential per unit change in distance along the field at a point in a gravitational field

• Jkg^-1m^-1 or Nkg^-1

$$\frac{\Delta V}{\Delta r}$$

graph of gravitational potential V against distance from centre of planet

• values for V all negative as gravitational potential is always negative

• $$V=-\frac{GM}{r}$$

• V is inversely proportional to -r

• g ( gravitational field strength) is the gradient at any point

newton’s law of gravitation

assumes that the force between two objects with mass is

• always attractive

• proportional to the product of the masses of the two objects

• inversely proportional to the square of their separations

$$F=\frac{Gm1m2}{r^2}$$

Gravitational constant G

• 6.67×10^-11 Nm²kg^-2

• Same for all masses

gravitational force on a test mass mt in a gravitational field between two bodies

• test mass experiences attractive forces from both objects

• point in the field between the masses at which the test mass experiences a resultant force of 0 as the forces on it will be equal and opposite

• Force from m1 = $-\frac{Gm1m3}{d^2}$

• Force from m2 =$\frac{Gm2m3}{\left(D-d\right)^2}$

• where D is the distance between the centres of both bodies and d is the distance from centre of m1 to centre of test mass

gravitational force against distance

gravitational field strength from law of gravitation

• can determine the gravitational field strength at any point in the field of a mass

• a spherical body of mass can be treated as if it was a point mass acting from a point in the centre (same field lines)

• M is the mass of the body producing the field

• $$g=\frac{GM}{r^2}$$

variation of g for spherical planet as you move a test mass beyond radius

• g drops off rapidly as it is inversely proportional to square of distance from centre

variation of g with distance from centre less than radius R ( within body of mass)

• g is 0 at centre and increases linearly with distance until it reaches radius (surface)

• g is directly proportional to r

gravitational potential from law of gravitation

$$V=-\frac{GM}{r}$$

escape velocity

• minimum velocity an object must be given to escape from a planet when projected vertically from the surface

• needs sufficient kinetic energy to overcome gravitational force of attraction towards earth

• $$v=\sqrt{\frac{2GM}{R}}$$

radial speed of orbit

$$v=\sqrt{\frac{GM}{r}}$$

• derived from equating centripetal force equation to gravitational force

kepler’s third law

$$\frac{r^3}{T^2}=\frac{GM_{S}}{4\pi^2}$$

assumptions for calculating radial speed

• point masses

• circular orbit

geostationary satellite

• satellite stays above same point on earth’s equator

• 24 hr time period

• orbits in same direction as earth’s direction of rotation

• communications

• avoids dish having to track / needs to remain at same point above the earth

energy of an orbiting satellite

• sub radial speed equation into kinetic energy equation $\frac{GMm}{2r}$

• gravitational potential equation x mass $-\frac{GMm}{r}$

• $$E_{_{total}}=\frac{GMm}{2r}-\frac{GMm}{r}$$

• total energy always negtive

• for any object in fixed circular orbit, kinetic and potential energy constant

• elliptical orbit - total energy constant but potential and kinetic energy vary such that as one increases, other decreases