Gravitation Flashcards
Gravitation
Force of attraction between any two bodies in the universe
Gravity
Force of attraction between the earth and any object lying on or near its surface
Newton’s law of gravitation
Every partice in the universe attracts every other particle with a force which is directly proportional to the produce of their masses and inversely proportional to the square of the distances between them
F = Gm1 m2 / r^2
G value
6.67 * 10 ^ -11 Nm^2 / kg^2
Gravitation in vector format
F = -G m1 m2 * (r cap) / r^2
Principle of Superposition of gravitational forces
Gravitational force between two masses acts independently and uninfluenced by the presence of other bodies. Hence, the resultant gravitational force acting on a particle due to a number of masses is the vector sum of the gravitational forces exerted by the individual masses on the given particle
Shell Theorem
- If a point mass lies outside a uniform spherical shell with a spherically symmetric internal mass distribution, the shell attracts the point mass as if the entire mass of the shell were concentrated at its centre
- If a point mass lies inside a uniform spherical shell, the gravitational force on the point mass is zero. But if a point mass lies inside a homogeneous solid sphere, the force on the point mass acts towards the centre of the sphere.
Mass of earth in terms of density
M = ro * V = ro * 4/3 pi * r^3
Acceleration due to g on different points on earth
- Above earth: F = GMm / r^2
- Below earth’s surface: F = GMm / R^3 * r
- At points on earth’s surface: F = GMm / R^2
Relation between g and G
g = GMm / R^2
Variation of g with altitude derivation
gh = g (1 - 2h / R)
Variation of g with depth
g = g(1 - d / R)
Formulas of g variation
With altitude:
1. g = g * R^2 / (R + h)^2
2. g = g (1 - 2h / R)
With depth:
1. g = g (1 - d / R)
With radius
1. Inversely proportional
Variation of g with shape of Earth
- Flat at poles and bulges at equatior
- Since g is inversely proportional to R^2
- Re > Rp or ge < gp
Gravitational Field
Space surrounding a material body within its gravitational force of attraction can be experienced
Gravitational Field intensity
At any point in the gravitational field due to a given mass is defined as the force experienced by a unit mass placed at that point providede the presence of unit mass does not disturb the original gravitational field
gravitational field intensity formula
E = F / m = GM / r^2 = g
Gravitational Potential Energy
Energy associated with it due to its position in the gravitational field of another body and is measured by the amount of work done in bringing a body from infinity to a given point in the gravitational field of the other
Gravitational potential energy derivation
U = -GMm / r
Gravitational Potential
Potential energy associated with a unit mass due to its position in the gravitational field of another body
V = -GM / r
Total energy of a body in a gravitational field
E = KE + PE = 1/2 mv^2 + (-GMm / R)
Escape Velocity
Minimum velocity with which a body must be projected vertically upwards in order that it may just escape the gravitational field of the earth
Escape velocity formulas and derivations
v = root (2 GM / R) = root (2gR) = root (8pi ro G R^2 / 3)
Escape velocity at a height h
ve = root (2g (re)^2 / (re + h))
re = rad. of earth
Orbital Velocity
Velocity required to put the satelleite into its orbit around the earth
Derivation of orbital velocity and formulas
v = root (GM / R + h) = root (gR^2 / R + h) = R root (g / R + h)
Relation between orbital and escape velocity
ve = (root 2) (vo)
Derive time period and all for satellite
T = 2pi [Root (R + h)^3 / gR^2]
Total energy of satellite
E = GMm / 2r {-Gmm / r + 1/2 Gmm / r}
Binding energy of a satellite
Gmm / 2r {energy required by a satellite to leave its orbit around the earth and escape to infinity}
Kepler’s Laws of Motion
- dL / dt = 0 (L is constant)
- Delta A / Delta T = constant
- T1^2 / T2^2 = R1^3 / R2^3
