State Newtons law of gravitation
gravitational force of attraction between two point masses is directly proportional to the product of their masses and inversely proportional to the square of the separation between their centres.
F = GMm/r^2
negative in graph but ignored for magnitude of force
Define Gravitational field strength
gravitational field strength at a point g is defined as the gravitational force per unit mass exerted on a small test mass
from g = f/m g = gmm/r^2/m
= gm/r^2 Nkg^-1
vector same direction a gravitational force
negative for graph
Why is value of g approximately constant near surface of earth
For distance h above earth surface, it is much smaller than radius of earth.
g=f/m = gMe/(Re + h)^2 approx = gMe/r^2
Define Gravitational Potential energy
The gravitational potential energy,U, of a mass m at a point in the gravitational field due to a mass M is work done by an external force in bring the mass m from infinity to that point
Work done is opposite to gravitational attraction, against gravity in bringing mass from infinity to that point
Derive U
integrate force external which is negative of the -ve gravitational force, Gmm/r^2
integrate from r to infinity.
Since at infinity U=0,
U= -GMm/r
Define gravitational potential
Work done per unit mass in bringing small test mass from infinity to that point
= -Gm/r
negative because in its WD, F and S in opposite directions
Gravitational Potential energy of a mass depends on both value of mass as well as position of mass. However gravitational potential is only dependent on position
Define equipotential lines
is formed by all the points that are at the same gravitational potential. in 2 dimensions. These points from a circle while in three dimensions these point for a sphere around a spherical massive object.
What can be derived from graphs
Negative Gradient at a point of potential-r graph is g
Area under g-r graph is magnitude change in potential
Derive escape speed
minimum energy at infinity to escape =
-gmm/r^2 + 1/2mv^2 = 0
Total energy at point =
-gmm/r^2 + 1/2mv^2
sub both to get v= root(2gr)
Derive relationship between T and r
Gravitation force provides for Fc
Gmm/r^2 = m(r)(omega)^2
since omega = 2pi/t
derive until get
r^3 = GM/4pi^2 T2
r^3 proportional to T^2
Explain Significance of - total energy of object in orbit
Bound to orbit cannot escape (0 or +ve)
To derive total energy of orbit
Gravitational force provides for centripetal force
gravitational force=mv^2/r
Derive 1/2mv^2 = 1/2gmm/r
then add with potential (rmb -ve) sign
