GRAVITATION: Points to Remember ~ φsiks4u

Newton's law of gravitation

F α m₁m₂

F α 1/r²

F = G(m₁m₂/r²)

where G = Universal Gravitational const

G = 6.67 x 10^-11Nm²/kg²

Acceleration due to gravity

On the Surface

THE COIN AND FEATHER EXPERIMENT

g = GM/R²

Point to note : Acceleration due to gravity is independent of the mass of the object

Factors effecting acceleration due to gravity

Altitude

Depth

Rotation of Planet

Altitude

With increase in altitude the value of 'g' decreases

Depth

With increase in depth the value of 'g' decreases.

When the particle is at depth
'd' mass of only the shaded
part is considered

Assuming a planet to be a perfect sphere having a uniform density 'ρ' and radius 'R'

Acceleration due to Gravity at the surface = g = (4/3)πρGR

At a depth 'd' from the surface of the planet

Acceleration due to gravity will be = g_d = (4/3)πρG(R-d)

Note: 'g' at the centre of planet = 0

The plot shows variations in gravitational acceleration as we move away from center of Planet

Rotation of planet(Effect of latitude)

Assume the Earth to be a sphere of radius R and mass M rotating on its axis with an angular velocity w. Consider a particle of mass m at P such that OP makes an angle of ø with OE . Here ø is the latitude of the particle. The particle is moving in a circle of radius R' = R cos ø.

The net force pulling the particle towards the center of the Earth is

where F_N is the total force acting on the particle, F_G is the force due to gravity and F_c is the centrifugal force. The centrifugal force is given by

If g' represents the gravitational acceleration, then

Note : At poles ø = 90⁰ : g' = g
At equator ø = 0⁰ : g' = g - Rω²

It is assumed that a body say A, creates a gravitational field in the space around it. The field has its own existence and has energy and momentum. When another body B is placed in gravitational field of a body, this field exerts a force on it. The direction and intensity of the field is defined in terms of the force it exerts on a body placed in it.

The intensity of gravitational field vector E at a point is defined by the equation

E = F/mass

where F is the force vector exerted by the field on a body of mass m placed in the field. The intensity of gravitational field is abbreviated as gravitational field. Its SI unit is N/kg.

By the way they are defined, intensity of gravitational field and acceleration due to gravity have equal magnitudes and directions, but they are two separate physical quantities.

Gravitational potential

Gravitational potential at a point is equal to the change in potential energy per unit mass, as the mass is brought from the reference point to the given point.

Es = - (GM/r)

Gravitational Potental Energy

Es = - (GMm/r)

Kepler's Laws of Planetary Motion

1. All planets move in elliptical orbits with the sun at a focus.

2. The radius vector from the sun to the planet sweeps equal area in equal time.

3. The square of the time period of a plant is proportional to the cube of the semimajor axis of the ellipse.