Revision Note 9: Gravitation

Newton’s Law of Gravitation:

F_G = Gm₁m₂/r²

Close to earth’s surface:

F_g = GmM/r_E² = weight of mass

So acceleration due to gravity:

g = F_g /m = GM/r_E² = constant, independent of m

For orbit computations: (circular orbit of radius = R, planet mass = m, Star mass = M)

F_G = GmM/R² = mv²/R

GM = v²R = constant for star system

Using v = ωR and T = 2π/ω,

T² is proportional to R³

Kepler’s Laws:

Law 1: Planet moves in an elliptical orbit, with the star in one of the foci.

Law 2: A line drawn from the star to the planet sweeps out equal areas in equal times.

Law 3: If semi-major axis = R (or radius = R for circular approximation), T²/R³ is constant for a star system.

Gravitational Potential Energy:

Close to earth’s surface:

PE = mgh, PE = 0 at surface

For any distance = r:

U = -GMm/r, U = 0 at infinity

Escape velocity:

Gravitational force inside a shell:

Gravitational force inside a shell is 0, since the mass inside is pulled from all sides.

Gravity inside mines is less than that at the surface. F_G when a particle mass, m, is x distance away from the center of the earth (mass = M, radius = r_E) is:

Problem Solving Tips:

Tip 9.1: Weight in an elevator

Weight on a scale in an elevator with acceleration = a (downwards +ve):

W_elev = m(g –a)

When a = g (free fall), W_elev = 0.

When a is –ve, then W_elev is greater than mg.

Tip 9.2: Accelerometer

If the pendulum makes an angle θ with the vertical:

a = g.tan θ