Study Material

What is elliptical motion?

In elliptical motion the mobile describes an ellipse, just as the planets do around the Sun , and the Moon and artificial satellites around the Earth, to name a few familiar examples. elliptical motion

The force that gives rise to this movement is the force of gravity, a central force. These kinds of forces are directed towards (or from) a fixed point O, and their modulus depends on the distance to that point. If r is the distance and r is the unit vector in the radial direction, the central force F is a vector function of the form:

F = F (r) r

what is elliptical motion? Diagram of the elliptical movement of a planet around the Sun. 
Figure 1. Diagram of the elliptical movement of a planet around the Sun.

With some mathematics it can be shown that the motion of an object under the action of gravity follows one of these four trajectories: ellipse, circumference, hyperbola or parabola.

Characteristics of the elliptical movement elliptical motionelliptical motion

Some of the main characteristics of elliptical movement under central force are:

-The angular momentum with respect to O is conserved, called L and which is calculated through the vector product between the position and velocity vectors: L = r × m v , where m represents the mass of the moving object.

-The elliptical orbit lies in the plane determined by the vectors r and v.

-From the conservation of angular momentum, the so-called law of areas is derived , which establishes that the mobile travels through equal areas in equal times. elliptical motion

-Mechanical energy is also conserved in elliptical motion, if there are no dissipative forces.

-The time it takes for the mobile to make an orbit and its total energy depend only on the length “a” of the ellipse’s semi-major axis.

Differences with circular motion

Although both in circular and elliptical motion the object moves in a closed and repetitive path, that is, periodic, there are obvious differences between one movement and another, such as:

The speed is maximum when the distance to the Sun is minimum..
Figure 2.- The speed is maximum when the distance to the Sun is minimum.

-In circular motion the mobile describes a circumference, whose radius (distance to the center of the path) is constant, while in elliptical motion it describes an ellipse, in which the distance to the center of the path is variable (see figure 1 ).

-In the case of uniform circular motion MCU, the mobile sweeps equal angles in equal times, but in planetary elliptical motion equal areas are swept in equal times. This is the law of areas, also known as Kepler’s second law of planetary motion.

Important equations of planetary elliptical motion elliptical motion


In elliptical motion derived from gravitational attraction, the period T of motion is the time it takes for the planet or satellite (m) to make an elliptical turn around the Sun or Earth (M). Applying the conservation of energy, it follows that it is proportional to the cube of the length of the semi-major axis of the ellipse:

Where G is the universal gravitation constant: 6.67 × 10 -11 N ∙ m 2 / kg 2 , M is the mass of the Sun, the Earth or the object causing the interaction on m and “a” is the length of the semi-major axis .

Mechanical energy elliptical motion

The total energy for the planet (m) – Sun (M) system is:

Angular momentum

The magnitude of the angular momentum at a point on the elliptical orbit also depends on the length of the semi-major axis, as well as on the eccentricity “e”, a dimensionless parameter that indicates how flattened the ellipse is. If e = 0, the ellipse becomes a circle.


The magnitude of the speed is given by the following equation:

elliptical motion

Where r is the distance between a point on the orbit (location of the planet) and the focus (Sun).

Examples of elliptical motion

Planetary movements

Kepler’s first law states that the movement of the planets around the Sun follows an elliptical path, with the Sun in one of the foci. Some comets that periodically visit Earth, such as Halley’s Comet, also follow an elliptical motion.

Apart from this elliptical translational movement and that of rotation around their axis, the planets have their own movements due to the complex gravitational interactions with the other planets and celestial bodies in the Solar System. In this way are the precession and nutation movements that the Earth possesses and that are due to the joint gravitational attraction of the Sun and the Moon.

In precession, the Earth’s axis describes a cone as it rotates around the axis perpendicular to the plan or the ecliptic. And in nutation, which is superimposed on precession, the Earth’s axis oscillates up and down in an elliptical loop every 18.6 years. In total it makes 1385 of these loops in 25,767 years, which is the period of the precession of the earth’s axis.

A particle of ocean water

In ocean waters, a particle performs an elliptical motion, with the ellipse becoming more and more flattened with increasing depth. On the other hand, when the waters are deep, the movement of the particles is circular. elliptical motion

What happens is that when the wave approaches the coast, friction forces appear thanks to its proximity to the bottom, and this friction tends to slow down the movement in the lower part of the trajectory, while the crest continues its movement.

The result is that the circumference flattens, and the effect is accentuated as the depth increases.

elliptical motion: Movement of ocean waves in: A) deep waters and B)
Figure 3. Movement of ocean waves in: A) deep waters and B)

Elliptical mode of oscillation in a physical pendulum

A physical pendulum consists of a rigid solid that can oscillate in a plane around an axis perpendicular to it. If the object is allowed to move freely, it can describe any angle around the axis that joins the center of mass with the suspension point, as well as rotate around it. elliptical motion

Thanks to the rotation of the Earth, the pendulum is able to describe orbits of approximately elliptical shape, which are known as elliptical mode of oscillation, characterized by an angular momentum other than 0.

There are also flat mode (angular momentum 0) and conical mode (angular momentum other than 0), the latter with a circular path on a horizontal plane. elliptical motion

Elliptical bikes elliptical motion

The elliptical movements described previously occur in nature, but can also be used to make useful gadgets, such as elliptical bikes, which are very popular machines for doing aerobics.

They are stationary bicycles that basically consist of a handlebar and two pedals that the person operates by pushing himself with his weight , describing an ellipse with his feet. It is a natural, low-impact movement that is beneficial because it moves many muscle groups throughout the body.

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