Chapter 13Chapter 13Gravitation Gravitation

PhysicsI 2048PhysicsI 2048

Newton’s law of gravitation

Besides the three laws of motion, Newton also discovered the universal law of gravitation.

The force of gravity between two point object of mass m1 and m2 is attractive and of magnitude

where r is the distance between the masses andG is a constant referred as the universal gravitationconstant.

Newton’s law of gravitation

GravitationGravitation The law of gravity applies to all objects small or large. G is very small (=6.67x10-11 N·m2kg-

2) The force is inverse proportional to distance. Satisfies superposition.

Gravitation and the Principle of Gravitation and the Principle of SuperpositionSuperposition

Where F1,netis the net force on particle 1 due to n particles

Gravitation Near Earth's SurfaceGravitation Near Earth's Surface Newton was able to show that the net force exerted by the sphere on a point mass m is the same as if all the mass of the sphere were concentrated at its center.

For a mass is near the surface of earth:

Variation of Variation of aagg with Altitudewith Altitude

Gravitation Near Earth's SurfaceGravitation Near Earth's Surface

Earth's mass is not uniformly Earth's mass is not uniformly distributed.distributed.

Gravitation Near Earth's SurfaceGravitation Near Earth's Surface

Earth is not a sphereEarth is not a sphere.. Earth is Earth is approximately an ellipsoid, flattened approximately an ellipsoid, flattened at the poles and bulging at the at the poles and bulging at the equator.equator.

Earth is rotatingEarth is rotating.. The rotation axis The rotation axis runs through the north and south runs through the north and south poles of Earth. poles of Earth.

Gravitation Near Gravitation Near Earth's SurfaceEarth's Surface

Gravitational Potential EnergyGravitational Potential Energy

Proof of the gravitational potential Proof of the gravitational potential energy equationenergy equation

Let us shoot a baseball directly away from Earth along the path in Figure

Potential Energy and ForcePotential Energy and Force

This is Newton's law of gravitation. This is Newton's law of gravitation. The minus sign indicates that the The minus sign indicates that the force on mass force on mass mm points inward, points inward, toward mass toward mass MM

Escape Speed Escape Speed When the projectile reaches infinity, it When the projectile reaches infinity, it stops and thus has no kinetic energy. It stops and thus has no kinetic energy. It also has no potential energy because an also has no potential energy because an infinite separation between two bodies is infinite separation between two bodies is zero potential energy zero potential energy

Some Escape SpeedsSome Escape Speeds

Kepler’s law of orbital motion

Kepler’s three laws(1) Planets follow elliptical orbits, with the Sun at one focus of the ellipse.

Kepler’s law of orbital motion

(2) As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time.

Kepler’s law of orbital motion

(3) The period of a planet increases as its mean distance from the Sun, r raised to the 3/2 power

Kepler’s law of orbital motionHere we will show that the Kepler’s third law can be derived from the definition of centripetal acceleration and the universal gravitation law.

Satellites: Orbits and EnergySatellites: Orbits and Energy The potential energy of the system is given by The potential energy of the system is given by EquationEquation

we write Newton's second law (we write Newton's second law (FF = = mama) as ) as

Where a is the ellipsis semimajor axisWhere a is the ellipsis semimajor axis

The mean diameters of planets M and The mean diameters of planets M and E are 6.9 × 10E are 6.9 × 1033 km and 1.3 × 10 km and 1.3 × 1044 km, km, respectively. The ratio of the mass of respectively. The ratio of the mass of planet M to that of planet E is 0.11. planet M to that of planet E is 0.11. (a)(a) What is the ratio of the mean density of What is the ratio of the mean density of M to that of E? M to that of E? (b)(b) What is the ratio of What is the ratio of the gravitational acceleration on M to the gravitational acceleration on M to that on E? that on E? (c)(c) What is the ratio of What is the ratio of escape speed on M to that on E?escape speed on M to that on E?

b-

a-

C-

Two neutron stars are separated by Two neutron stars are separated by a distance of 1.0 x 10a distance of 1.0 x 101010 m. They each m. They each have a mass of 1.0 x 10have a mass of 1.0 x 103030 kg and a kg and a radius of 1.0 x 10radius of 1.0 x 1055 m. They are m. They are initially at rest with respect to each initially at rest with respect to each other. As measured from that rest other. As measured from that rest frame, how fast are they moving frame, how fast are they moving when when (a)(a) their separation has their separation has decreased to one-half its initial value decreased to one-half its initial value and and (b)(b) they are about to collide? they are about to collide?

(a) Use the principle of conservation (a) Use the principle of conservation of energy. The initial potential energy of energy. The initial potential energy is. is.

The initial kinetic energy is zero since The initial kinetic energy is zero since the stars are at rest.the stars are at rest.

The final potential energy is. The final potential energy is.

(b) Now the final separation of the (b) Now the final separation of the centers is centers is