Centrifugal/centripetal and coriolis accelerations

Type of physical science: Classical physics

Field of study: Mechanics

The centripetal force operates on an object when that object describes a curved path; it is responsible only for a change in direction and not a change in speed. The centrifugal force is a responsive force that is produced by an object that follows a curved path. The Coriolis effect is a deflective force that ultimately arises from Earth's rotation on its axis.

Overview

The centrifugal and centripetal accelerations are viewed often as closely related phenomena. In fact, they are quite different, and there is good reason to argue that the centrifugal acceleration is a superfluous concept. On the other hand, the Coriolis acceleration is an important concept that can be described in the context of the centripetal acceleration. The terms "force" and "acceleration" will be used interchangeably. According to Sir Isaac Newton's second law of motion (force equals mass times acceleration), a force per unit mass is numerically equivalent to an acceleration. Nevertheless, it should be remembered that a change in velocity (an acceleration) is actually the consequence of a net force.

Centripetal acceleration may be illustrated by a simple demonstration. A rock is tied to a string and whirled about so that the rock describes a circular orbit of constant radius. The string is then cut and the rock flies off in a straight line that is tangent to its original circular path. The behavior of the rock illustrates Newton's first law of motion; that is, an object in straight-line, unaccelerated motion remains that way unless acted upon by an unbalanced force. Prior to being cut, the string exerts a net force on the rock by confining it to a curved (circular) path. The net force is directed inward, perpendicular to the direction of motion, and toward the center of the circular orbit. For this reason, the net force is known as the centripetal ("center-seeking") force.

When the string is cut, the centripetal force no longer operates and the rock follows a straight path.

A net force causes an acceleration. One usually thinks of an acceleration as a change in speed, as when an automobile speeds up or slows down. Acceleration is a vector quantity; that is, it has both magnitude and direction. Hence, an acceleration may consist of a change in either speed or direction, or both. In the rock-on-a-string example, the centripetal force is responsible only for a continuous change in the direction of the rock (a curved rather than a straight path); the rock neither speeds up nor slows down. It can be demonstrated that the acceleration imparted to a unit mass by the centripetal force is directly proportional to the square of the (tangential) velocity and inversely proportional to the radius of curvature.

The centripetal force is not itself an independent force; rather, it arises from the action of other forces and may be the consequence of imbalances in other forces. In the rock-on-a-string example, the string is responsible for the centripetal force. In the case of an automobile rounding a curve, the frictional resistance of the tires against the road provides the centripetal force.

Consider another example: As each person rotates with the earth, a complete circle is described once every twenty-four hours. The centripetal force that confines one to a circular path arises from the gravitational attraction between humans and the planet.

Newton's third law of motion helps put the centrifugal force in perspective. For every action, there is an equal and opposite reaction. The centrifugal force is equal to the centripetal force in magnitude but opposite in direction; that is, the centrifugal force is a response to the centripetal force. In the rock-on-a-string example, the centripetal force acts on the rock (to confine the rock to a circular path), while the rock exerts a responsive (centrifugal) force on the string. Hence, the centrifugal force differs from the centripetal force in that it does not affect the motion of the rock.

The centrifugal force is a superfluous concept. When a rock is whirled about, one feels an outward directed pull of the string. This pull is the centrifugal force as transmitted by the string. Actually, what is felt is the inertial tendency of the rock to follow a straight path. When a car rounds a corner, a force is felt that pushes one outward from the curve; it is commonly described as the centrifugal force. Actually, the passenger is experiencing the inertial tendency of his or her body to continue in a straight path while the car is following a curved path. Hence, it is usually sufficient to visualize any object following a curved path to be influenced by a centripetal force alone without invoking a centrifugal force.

The Coriolis effect refers to the apparent deflection of a moving object that arises from the rotation of Earth. To an observer on Earth, the object is deflected by a force acting at right angles to the object's instantaneous direction. The Coriolis effect influences moving objects that are detached from Earth and is most significant for objects that travel long distances. The Coriolis deflection is also latitude- and velocity-dependent, increasing with both latitude and velocity of the object. Imagine that planet Earth is being observed from some fixed location in space. Over many hours, the path of a storm system is followed, clearly outlined by a swirling mass of clouds. From one's perspective in space, the storm center appears to follow a straight path at constant speed. At the same time, an observer on Earth's surface is tracking the storm and concludes that the storm center is describing a curved trajectory. Surprisingly, both descriptions of the storm's movement are correct. The two descriptions of the storm's trajectory are correct because the two observers used different frames of reference (coordinate systems) in tracking the storm. The earthbound observer's frame of reference is the familiar north-south, east-west coordinate system that rotates with Earth. To the earthbound observer, it is not obvious that this coordinate system is rotating because it, the observer, and Earth rotate together. Viewed from space, however, the earthbound coordinate system actually rotates as Earth rotates. It is as if Earth and the coordinate system rotate under the storm, thus giving the appearance of a curved trajectory. Meanwhile, from the observer's vantage point in space, the storm's movement is followed with respect to a nonrotating (inertial) coordinate system, fixed in space. Hence, the difference in storm track (curved versus straight) is caused by the difference in coordinate systems (rotating versus nonrotating). For this reason, the Coriolis effect is often described as more apparent than real.

Newton's first law of motion and the centripetal force aids the understanding of the Coriolis effect. Curved motion implies that a net force is operating, whereas unaccelerated, straight-line motion implies a balance of forces. If this law is applied to the storm track example, it can be concluded that a net force operates when the earthbound rotating coordinate system is used, whereas forces are balanced when the nonrotating coordinate system is used. Hence, changing the frame of reference (coordinate system) from nonrotating to rotating gives rise to a net force that is responsible for curved motion, that is, the Coriolis effect.

Another way to visualize the Coriolis effect is in terms of variations in tangential velocity of points at different latitudes on Earth. All points revolve in an easterly direction about the Earth's rotational axis but at different tangential speeds depending on their latitude. Although all points describe a 360-degree circle once every twenty-four hours, the actual distance covered (the circumference of the circle) declines with latitude. In twenty-four hours, points at lower latitudes travel greater distances than points at higher latitudes and hence have a greater tangential velocity. The tangential velocity varies from a minimum of 0 kilometers per hour at the poles to 800 kilometers per hour at 60 degrees latitude, 1,400 kilometers per hour at 30 degrees latitude, to a maximum of more than 1,600 kilometers per hour at the equator.

Suppose a rocket is propelled from a launching pad in the Northern Hemisphere subtropics directly northward toward a target in mid-latitudes. As it leaves the launching pad, the rocket has two components of motion: eastward with the launching pad (moving with the rotating Earth) and northward as a result of its thrust. The target is also moving eastward with the rotating Earth but at a slower pace than the launching pad. Hence, the rocket appears to an observer at the target to veer off to the right, thus describing a northeasterly rather than northerly track and missing the target. Viewed from the perspective of the inertial coordinate system in space, however, the rocket actually followed a straight path.

The Coriolis effect can be experienced at first hand on a carousel. While the carousel is in motion, attempt to walk from the outer rim to the inner rim in a straight line (along a radius).

A pull will be felt to the right. The tangential velocity of the outer rim (analogous to low latitudes) is greater than that of the inner rim (analogous to higher latitudes). The Coriolis effect significantly influences the large-scale motion of any object (a rocket, global-scale winds, ocean currents, for example) traveling in any direction over Earth's surface. It deflects objects to the right of their initial direction in the Northern Hemisphere and to the left of their initial direction in the Southern Hemisphere. In the Northern Hemisphere, for example, the Coriolis effect causes large-scale winds that are blowing directly from the south to veer toward the east. Because the Coriolis effect ultimately stems from Earth's rotation on its axis, its magnitude varies with latitude. Rotation of the north-south, east-west frame of reference and, hence, the magnitude of the Coriolis effect, varies with latitude from zero at the equator to a maximum at the poles. This latitudinal variation can be understood by visualizing the daily rotation of towers situated at different latitudes. In a twenty-four-hour day, Earth completes one rotation, as does a tower situated at either the North or South pole. In the same period, a tower at the equator does not rotate at all; instead, it describes an end-over-end motion. For a tower located at any latitude in between, some rotation of a tower occurs as Earth rotates but not as much as at the poles. In other words, the Coriolis effect is latitude-dependent, varying (as the sine of the latitude) from zero at the equator to a maximum at the poles.

The reversal in Coriolis deflection between the two hemispheres is related to the difference in an observer's sense of Earth's rotation in the two hemispheres. For example, to an observer at the North Pole, Earth rotates in a counterclockwise sense, whereas to an observer at the South Pole, Earth rotates clockwise. This rotational reversal translates into a reversal in Coriolis deflection between the two hemispheres.

Applications

Earth-orbiting satellites provide an application of centripetal acceleration. Satellites move through space at constant speeds in orbits that are very nearly circular. Because friction is negligibly small, the principal force acting on a satellite is gravitation. Gravitation is the attractive force between two objects and is directly proportional to the product of the masses of the two objects and inversely proportional to the square of the distance between the two objects.

The gravitational attraction between Earth and a satellite confines the satellite to its circular orbit.

Hence, the gravitational force is numerically equivalent to the centripetal force. This discussion is another example of how the centripetal force is not an independent force but rather arises from other forces (in this case, the gravitational force).

It can be shown that the orbital speed of a satellite is inversely proportional to its altitude. Thus, in order to remain in orbit, a satellite at low altitudes must travel faster than a satellite at high altitudes. For purposes of weather observation, it is desirable for satellites to orbit Earth at the same rate as Earth rotates. In this way, instruments on board the satellite always observe the same portion of Earth. Geosynchronous satellites operated by the National Weather Service have orbital speeds of about 11,000 kilometers per hour at altitudes of about 36,000 kilometers.

The Coriolis effect has important applications in predicting the trajectories of long-range projectiles such as missiles and in understanding large-scale atmospheric and oceanic circulation systems. For example, large-scale horizontal winds above an altitude of about 1,000 meters (where friction is negligibly small) are affected by the centripetal, Coriolis, and pressure gradient forces. The pressure gradient force is always directed from regions of relatively high atmospheric pressure toward regions of relatively low atmospheric pressure (and perpendicular to isobars). In a Northern Hemisphere anticyclone (a high-pressure system), the pressure gradient force acts radially outward from the center of highest pressure and is opposed by the inward-directed Coriolis effect. The Coriolis effect is slightly greater than the pressure gradient force, and this difference (the centripetal force) confines the horizontal wind to a curved clockwise path around the area of highest pressure. In a Northern Hemisphere cyclone (a low-pressure system), the pressure gradient force acts radially inward toward the center of lowest pressure and is opposed by the outward-directed Coriolis effect. The pressure gradient force is slightly greater than the Coriolis effect, and this difference (the centripetal force) confines the horizontal wind to a curved counterclockwise path around the area of lowest pressure.

The Coriolis effect combines with friction (because of surface roughness) and horizontal air pressure gradients to shape surface winds in cyclones and anticyclones. In the Northern Hemisphere, surface winds in cyclones spiral in a counterclockwise and inward direction, whereas surface winds in an anticyclone spiral in a clockwise and outward direction. In the Southern Hemisphere, the corresponding surface circulation patterns are clockwise and inward in cyclones and counterclockwise and outward in anticyclones.

An important prerequisite for hurricane formation is a significant Coriolis effect; that is, the influence of Earth's rotation must be strong enough to induce and sustain a cyclonic circulation. The minimum latitude where the Coriolis effect is sufficient for hurricane formation is about 4 degrees, and most hurricanes form in the 10- to 20-degree latitude belt.

A popular misconception holds that the rotational motion that accompanies the draining of water from a sink or bathtub is consistently in one direction (clockwise or counterclockwise) in one hemisphere and in the opposite direction in the other hemisphere. Presumably, the difference is caused by the Coriolis effect. At the very small scale represented by a sink or bathtub, however, the magnitude of the Coriolis effect is simply too small to influence rotational direction. Drainage direction is more likely a consequence of some residual motion of the water when the sink or bathtub was first filled with water or the geometry of the sink or tub, and may be either clockwise or counterclockwise.

As a general rule, for atmospheric circulation systems smaller than a thunderstorm, the Coriolis effect is negligibly small. Thus, for example, winds within a tornado theoretically could rotate in either a clockwise or counter-clockwise direction, although the latter seems to dominate in Northern Hemisphere tornadoes. The counterclockwise bias may be linked to the circulation within the tornado's parent thunderstorm.

Context

The centripetal and centrifugal accelerations are ideas in classical mechanics that follow Newton's laws of motion, first stated in 1687. Indeed, Newton's findings form the foundation for an understanding of mechanical and dynamical phenomena. As noted earlier, centripetal acceleration is a direct consequence of Newton's first law of motion, and centrifugal acceleration is a response to centripetal acceleration, as described by Newton's third law of motion. Of the two, centripetal acceleration is the most useful. Because the centrifugal acceleration really describes inertia, it is a superfluous concept.

The Coriolis effect is named for Gaspard-Gustave de Coriolis, the French engineer-mathematician who first described the phenomenon mathematically in 1835. Newton's laws of motion hold only when discussing an inertial frame of reference that is fixed with respect to a distant point in space. Coriolis demonstrated that whenever Newton's laws of motion are applied within a rotating coordinate system, then the equations of motion must include an inertial force that acts to the right of the direction of motion for counterclockwise rotation of the coordinate system (and to the left for clockwise rotation).

The Coriolis effect is an important consideration in stellar dynamics (rotation of sunspots), ballistics (orbiting of space vehicles), meteorology (winds), and oceanography (currents). It is an essential element in any mathematical model that seeks to describe large-scale circulation within the atmosphere or ocean. With increasing emphasis on numerical weather and climate prediction, the Coriolis effect will remain a key component in realistic mathematical models.

Principal terms

ACCELERATION: a change in speed and/or direction

GEOSYNCHRONOUS SATELLITE: a platform that orbits Earth at the same rate as Earth rotates so that the platform is always positioned over the same point on Earth's surface

GRAVITATION: the force by which every object attracts and is attracted by every other object

INERTIA: the tendency of an object at rest to remain at rest, or, if moving, to continue moving in the same direction

INERTIAL COORDINATE SYSTEM: a reference frame that is fixed with respect to some distant point in space, such as a star

NET FORCE: forces that are unbalanced, acting on an object so that the object accelerates

NEWTON'S FIRST LAW OF MOTION: an object at rest remains at rest and an object in straight-line unaccelerated motion remains that way unless acted upon by unbalanced forces

NEWTON'S SECOND LAW OF MOTION: force equals mass times acceleration

NEWTON'S THIRD LAW OF MOTION: to every action, there is an equal and opposite reaction

PRESSURE GRADIENT FORCE: force directed perpendicular to isobars from high pressure to low pressure

Bibliography

Bauman, R. "A Coriolis Paradox." THE PHYSICS TEACHER 21 (1983): 461. Explores some quantitative means of understanding the Coriolis effect.

Blackadar, Alfred K. "Simple Motions on the Rotating Earth." WEATHERWISE 39 (1986): 99-103. Provides a clear explanation of the Coriolis effect in weather systems.

Higbie, J. "Simplified Approach to Coriolis Effects." THE PHYSICS TEACHER 18 (1980): 459-460. Provides alternate methods to demonstrate the Coriolis effect.

Moran, Joseph M., and M. D. Morgan. METEOROLOGY: THE ATMOSPHERE AND THE SCIENCE OF WEATHER. New York: Macmillan, 1991. Chapters 9 through 14 demonstrate the role of centripetal acceleration and the Coriolis effect in atmospheric circulation systems. For a wide audience.

Stommel, H. M., and D. W. Moore. AN INTRODUCTION TO THE CORIOLIS FORCE. New York: Columbia University Press, 1989. A rather sophisticated quantitative treatment of the physical basis of the Coriolis effect.

Centripetal vs. centrifugal force

The Physics of Weather