Question

Describe the differences between circular motions and elliptical motions. Explain how Newton’s universal gravitational law can be applied to study the interplanetary motions.

Answer 1

circular motions

circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.

Examples of circular motion include: an artificial satellite orbiting the Earth at constant height, a stone which is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.

Since the object's velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion.

elliptical motions

elliptic orbit is a Kepler orbit with the eccentricity less than 1; this includes the special case of a circular orbit, with eccentricity equal to zero. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1 (thus excluding the circular orbit). In a wider sense it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

In a gravitational two-body problem with negative energy both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Also the relative position of one body with respect to the other follows an elliptic orbit.

Examples of elliptic orbits include: Hohmann transfer orbit, Molniya orbit and tundra orbit.

Newton’s universal gravitational law can be applied to study the interplanetary motions.

Using Kepler’s third law and his own second law, Newton found that the amount of the attractive force, called gravity, between a planet and Sun a distance d apart is Force = kp × (planet mass) / (d)2, where kp is a number that is the same for all the planets. In the same way he found that the amount of the gravity between the Sun and a planet is Force = ks × (Sun mass) / (d)2. Using his third law of motion, Newton reasoned that these forces must be the same (but acting in the opposite directions).

The term G is a universal constant of nature. If you use the units of kilograms (kg) for mass and meters (m) for distance, G = 6.672 × 10-11 m3 /(kg sec2). If you need a refresher on exponents, square & cube roots, and scientific notation, then please study the math review appendix.

Spherically symmetric objects (eg., planets, stars, moons, etc.) behave as if all of their mass is concentrated at their centers. So when you use Newton’s Law of Gravity, you measure the distance between thecenters of the objects.

The simplest way to travel between the planets is to let the Sun’s gravity do the work and take advantage of Kepler’s laws of orbital motion. A fuel efficient way to travel is to put the spacecraft in orbit around the Sun with the Earth at one end of the orbit at launch and the other planet at the opposite end at arrival. These orbits are called “Hohmann orbits” after Walter Hohmann who developed the theory for transfer orbits. The spacecraft requires only an acceleration at the beginning of the trip and a deceleration at the end of the trip to put it in orbit around the other planet.