Circular Motion

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: Circular motion has captivated scientists and philosophers since antiquity. Although circular motion is described using distinct equations and terminology, these are all derived from Newton’s laws of motion. In particular, the interplay between inertia and centripetal force accounts for everything from planetary motion to the feeling of being pressed against the side of a car during a sharp turn.

PRINCIPAL TERMS

• angular momentum: the rotational momentum of an object around an axis, defined as the product of its moment of inertia and its angular velocity.
• centrifugal force: a perceived force that seems to act outward on a rotating object, pushing it away from the center of its circular path; commonly confused with centripetal force.
• centripetal force: the force that impels a rotating object inward, acting at a right angle to its momentum.
• Coriolis force: the effects of the spinning motion of a planet on objects at the planet’s surface, such as deflection in large-scale wind patterns.
• inertia: the principle that an object remains at rest or continues moving in the same direction at the same speed unless an outside force acts on it.
• radian: abbreviated rad, the International System of Units standard unit of angular measure, the length of the corresponding arc in a unit circle. A full circle is compose of 2π radians.

Circular Motion and Newton’s Laws

Since antiquity, scientists and philosophers have recognized circular motion, particularly in the context of planetary orbits, as distinct from linear motion. This is somewhat misleading, however. While there are distinct equations to describe the properties of circular motion, these are all derived from the laws of motion codified by English physicist Isaac Newton (1642–1727). The same rules that govern the orbits that interested Renaissance astronomer and physicist Galileo Galilei (1564–1642) also govern a car turning around a race track or a tennis ball rotating with backspin. Angular momentum, just like linear momentum (often simply "momentum"), is conserved in a system. The total momentum of that system before and after any impact or exchange of energy is constant.

Objects moving in a circular path are subject to inertia, as described by Newton’s first law. An object at rest will stay at rest, and an object in motion will continue moving in the same direction at the same speed, unless acted on by an outside force. In other words, objects "want" to move in a straight line. For an object to move in a curved path, a force must be acting on it. In circular motion, a centripetal force "pulls" or "pushes" the object toward the center of its circular path. Centripetal force simply describes the way a particular force acts. Gravity is the centripetal force in planetary orbits; the tug of a yo-yo string provides centripetal force during the "around the world" trick. The competing influences of inertia (straight-line tendency) and centripetal force (pull or push toward the center) result in the familiar arc of circular motion.

Uniform versus Nonuniform Circular Motion

Circular motion can be uniform or nonuniform. An object in uniform circular motion travels at the same speed throughout its path, with a constant rate of acceleration. It may seem strange at first to think that an object with constant speed is accelerating, but remember that acceleration is dependent on change in velocity, which in turn is dependent on speed and direction. If an object is maintaining a constant speed but constantly changing direction, it is still accelerating.

This constant acceleration is also predicted by Newton’s laws. Because an object moving in a circular path must always be acted on by a centripetal force, it must always be accelerating. By definition, a mass must be accelerating if a nonzero net force is acting on it. With uniform circular motion, an object will always be accelerating toward the center of the circle.

Many of the most familiar forms of circular motion are uniform, or close enough to be treated as such. Examples include planetary orbits, the spin of a Ferris wheel, and the rotation of Earth on its axis. Even objects that vary in speed over time, such as a windmill varying with wind speed, often exhibit periods of roughly uniform motion.

Nonuniform circular motion, by contrast, involves an object moving in a circular path with varying speed and acceleration. A car accelerating into and out of a turn is one real-life example. Because the speed changes over time, so does the acceleration. In nonuniform circular motion, the vector of acceleration at any given moment may not be aimed at the center.

Centrifugal and Centripetal Force

Anyone who has ever been a passenger in a car during a sharp turn can attest to the fact that there seems to be another force acting outward, away from the center of the arc. The same sensation can be felt during sharp turns in roller coasters and other amusement-park rides. This is commonly referred to as centrifugal force. However, it is not an actual force but a trick of perception.

Consider a passenger in a car driving in a straight line at a uniform speed. The car and the passenger are both moving forward at the same rate; the passenger perceives no force. When the car accelerates, the passenger feels as though he or she is being pressed backward into the seat. In fact, the seat, along with the rest of the car, is accelerating forward and pushing the person forward. The inertia of the passenger’s body resists, creating the illusion of a backward force. Likewise, centrifugal force is a trick of perception caused by the inertia of an object in motion resisting inward acceleration caused by centripetal force.

The equations for uniform circular motion are all derived from Newton’s second law of motion, which relates force (F) to the mass (m) and acceleration (a) of an object:

F = ma

For linear motion, acceleration is expressed in terms of change in velocity (v) over time. Using the relationship of a circle’s radius (r) to its circumference, the acceleration of an object in circular motion (a_c) can be written as follows, where r is the radius of the circular path:

a_c = v² / r

The velocity component of this equation can be further broken down. It can be thought of as the circumference of the circular path (2πr) divided by the period of rotation or revolution (T):

v = 2πr / T

This works because velocity is equal to the absolute distance traveled (displacement) over the time it took to travel said distance. For circular motion, angular velocity is often more useful. Angular velocity measures the distance an object travels relative to the amount of a circle it completes in terms of radians. By definition, a full circle (360 degrees) is equal to 2π radians of rotation. Because radians, as a unit, are based on the relationship between the angle of rotation in a circle and the corresponding proportion of the circumference covered, they are the only unit of measurement for angles that works with these equations.

Plugging these expressions into the Newton’s second law gives a formula for the force acting on an object in circular motion, which always act inward toward the center of the circular path:

Note that force is measured in newtons (N). One newton is the amount of force it takes to accelerate one kilogram of mass at a rate of one meter per second per second—that is, one kilogram-meter per second squared (kg∙m/s²). When using this formula for F_c, the values must be in terms of kilograms, meters, and seconds in order to produce an accurate calculation of force in newtons.

Sample Problem

Imagine a ride called the Rotor at a local carnival. It is a large upright metal cylinder, approximately 3 meters across, attached to a massive motor concealed below. Tim, a young man weighing about 50 kilograms, walks in and lines up with his friends along the inner walls of the cylinder. When the ride turns on, the cylinder begins to spin, increasing in speed until it is spinning at a rate of 33 revolutions per minute. The force applied to the riders is great enough that when the floor drops away, Tim and his friends remain pinned to the inside walls of the cylinder. After a few seconds at top speed, the floor comes back up, and the ride slows to a stop. Given that information, calculate the centripetal force acting on Tim while the ride was at maximum speed.

Answer:

To begin, identify which variables are needed to solve for F_c in the formula for centripetal force:

Mass (m) and the radius (r) and period (T) of the circular motion are needed. The mass value of the object in question (Tim) is 50 kilograms. The radius of the circular path of motion can be inferred from the diameter for the cylinder: half of 3 meters is 1.5 meters. The period can be derived, logically, from the revolutions per minute. If the ride completes 33 revolutions in one minute, it must complete one revolution in 1/33 of a minute, or approximately 1.82 seconds.

Next, plug these values into the equation and calculate:

The centripetal force acting on Tim is about 893.88 newtons. This is enough to overwhelm his weight (the force of gravity acting on his mass) and to pin him to the inner wall of the ride while it is at maximum speed, but it is not enough to do him any harm.

Circular Motion around the World

Examples of circular motion abound. Many motors, whether in electric toothbrushes or cars, generate circular motion from an external energy source, such as electricity or fuel combustion. Merry-go-rounds spin children on playgrounds. Indeed, the entire planet is spinning, as are the solar system, the galaxy, and even, it seems, the entire universe.

Understanding circular motion not only helps understand motion at an everyday, human scale but also influences aspects of life at the planetary level. Understanding the motion of the moon has helped humans understand tides and navigate the seas. Understanding the interaction between Earth’s rotation (faster at the equator than the poles) and its atmosphere explains the Coriolis effect, which causes winds that would normally travel across the planet in a straight line to bend clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere.

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