Force is a funny thing in physics. Its relationship to speed is far less intuitive than most people probably think. For example, in the absence of frictional (e.g., the road) and "drag" (e.g., the air) effects, it requires literally no force to keep a car moving at 100 miles an hour (161 km/hr), but it *does* require an outside force to slow that car even from 100 to 99 mi/hr.

**Centripetal force,** which is exclusive to the dizzying world of rotational (angular) motion, has a ring of that "funniness" to it. For example, even when you know precisely *why,* in Newtonian terms, a particle's centripetal force vector is directed toward the center of the circular path around which the particle is traveling, it still seems a little weird.

Anyone who has ever experienced a strong centripetal force might be inclined to mount a serious, and even plausible-sounding, challenge to the underlying physics based on her own experience. (By the way, more on all of those mysterious quantities soon!)

To call centripetal force a "type" of force, as one might refer to the force of gravity and a few other forces, would be misleading. Centripetal force is really a special case of force that can be analyzed mathematically using the same essential Newtonian principles as are used in linear (translational) mechanics equations.

## Overview of Newton's Laws

Before you can fully explore centripetal force, it's a good idea to review the concept of force and where it "comes from" in terms of how human scientists describe it. In turn, that provides a great opportunity to review all three of the laws of motion of the 17th- and 18th-century mathematical physicist Isaac Newton. These are, ordered by convention and not importance:

**Newton's first law,** also called the *law of inertia,* states that an object moving with constant velocity will remain in this state unless perturbed by an external force. An important implication is that force is not required for objects to move, no matter how fast, at constant velocity.

- Velocity is a
**vector quantity** (therefore **bolded** as **v**) and thus includes both **magnitude** (or speed in the case of this variable) and **direction**, an always-important point that will become critical in a few paragraphs.

**Newton's second law**, written

states that if a net force in a system exists, it will accelerate a mass m in that system with a magnitude and direction **a**. Acceleration is the rate of change of velocity, so again, you see that force is not required for motion per se, only to change motion.

**Newton's third law** states that for every force **F** in nature there exists a force **–F** that is equal in magnitude and opposite in direction.

- This should not be equated with a "conservation of forces" as no such law exists; this can be confusing because other quantities in physics (notably mass, energy, momentum and angular momentum) are in fact conserved, meaning that they can neither be created in the absence of that quantity in some form not destroyed outright, i.e., kicked into nonexistence.

## Linear vs. Rotational Kinematics

The laws of Newton provide a useful framework for establishing equations that describe and predict how objects move in space. For purposes of this article, *space* really means two-dimensional "space" described by **x** ("forward" and "backward") and **y** ("up" and "down") coordinates in linear motion, θ (angle measure, usually in radians) and **r** (the radial distance from the axis of rotation) in angular motion.

The four basic quantities of concern in kinematics equations are **displacement**, **velocity** (rate of change of displacement), **acceleration** (rate of change of velocity) and **time**. The variables for the first three of these differ between linear and rotational (angular) motion because of the different quality of the motion, but they describe the same physical phenomena.

For this reason, although most students learn to solve linear kinematics problems before seeing their associates in the angular world, it would be plausible to teach rotational motion first and then "derive" the corresponding linear equations from these. But for various practical reasons, this is not done.

## What Is Centripetal Force?

What makes an object take a circular path instead of a straight line? For example, why does a satellite orbit the Earth in a curved path, and what keeps a car moving around a curved road even at what seems like impossibly high speeds in some cases?

#### Tips

**Centripetal force** is the name for any type of force that causes an object to move in a circular path.

As noted, centripetal force is not a distinct kind of force in the physical sense, but rather a description of *any* force that is directed toward the center of the circle representing the object's path of motion.

- The word
*centripetal* literally means "**center-seeking**."

#### Tips

Do not confuse centripetal force with the mythical-yet-persistent "centrifugal force."

## Sources of Centripetal Force

Centripetal force can arise from various sources. For example:

• The **tension T** (which has units of *force divided by distance*) in a string or rope attaching the moving object to the center of its circular path. A classic example is the tetherball set-up found on U.S. playgrounds.

• The **gravitational attraction** between the center of two large masses (for example, the Earth and the moon). In theory, all objects with mass exert a gravitational force on other objects. But because this force is proportional to the mass of the object, in most cases it is negligible (for example, the infinitesimally small upward gravitational pull of a feather on the Earth as it falls).

The "force of gravity" (or properly, the acceleration owing to gravity) **g** near Earth's surface is 9.8 m/s^{2}.

• **Friction.** A typical example of a frictional force in introductory physics problems is that between the tires of a car and the road. But perhaps an easier way to view the interplay between friction and rotational motion is to imagine objects that are able to "stick" to the outside of a rotating wheel better than others can at a given angular velocity because of the greater friction between the surfaces of these objects, which remain in a circular path, and the wheel's surface.

## How Centripetal Force Causes a Circular Path

The angular velocity of a point mass or object is completely independent of what else might be going on with that object, kinetically speaking, at that point.

After all, angular velocity is the same for all points in a solid object, regardless of distance. But since there is also a tangential velocity **v _{t}** in play, the matter of tangential acceleration arises or does it? After all, something moving in a circle yet accelerating would simply have to break free of its path, all else held the same. Right?

The physics basics prevent this apparent quandary from being a real one. Newton's second law (**F** = m**a**) requires that the centripetal force is an object's mass m times its acceleration, in this case centripetal acceleration, which "points" in the direction of the force, that is, toward the center of the path.

You would be right to ask: "But if the object is accelerating toward the center, why doesn't it move that way?" The key is that the object has a linear velocity **v _{t}** that is directed tangentially to its circular path, described in detail below and given by

**v**.

_{t}= ωrEven if that linear velocity is constant, its direction is always changing (thus it must be experiencing acceleration, which is a change in velocity; both are vector quantities). The formula for centripetal acceleration is given by:

- Based on Newton's second law, if
**v** is centripetal acceleration, then what must be the expression for centripetal force _{t}^{2}/r**F**? (Answer below.)_{c}

## Around the Bend

A car entering a turn with constant *speed* serves as a great example of centripetal force in action. For the car to remain on its intended curved path for the duration of the turn, the centripetal force associated with the car's rotational motion must be balanced or exceeded by the frictional force of the tires on the road, which depends on the car's mass and the intrinsic properties of the tires.

When the turn ends, the driver makes the car go in a straight line, the direction of the velocity stops changing, and the car stops turning; there is no more centripetal force from friction between the tires and the road directed orthogonally (at 90 degrees) to the velocity vector of the car.

## Centripetal Force, Mathematically

Because the centripetal force

is directed tangentially to the object's motion (i.e., at 90 degrees), it cannot do any work on the object horizontally because none of the net force component is in the same direction as the object's motion. Think of poking directly at the side of a train car as it whizzes horizontally past you. This will neither speed the car along nor slow it down one bit, unless your aim is not true.

#### Tips

The horizontal component of the net force on the object in such an instance would be (F)(cos 90°) which equals zero, so the forces are balanced in the horizontal direction; according to Newton's first law, the object will therefore stay in motion at a constant velocity. But because it does have an inward acceleration, this velocity must be changing, and thus the object moves in a circle.

## Centripetal Force and Non-Uniform Circular Motion

So far, only uniform circular motion, or motion with constant angular and tangential velocity, has been described. When, however, there is non-uniform tangential velocity, there is by definition **tangential acceleration**, which has to be added (in the vector sense) to centripetal acceleration to get the net acceleration of the body.

In this case, the net acceleration no longer points toward the center of the circle and solving for the problem's motion becomes more complex. An example would be a gymnast hanging from a a bar by her arms and using her muscles to generate enough force to ultimately start swinging around it. Gravity is clearly aiding her tangential velocity on the way down but slowing it on her way back up.

## An Example of Vertical Centripetal Force

Building on the previous velocity of vertically oriented centripetal force, imagine a roller coaster with mass M completing a circular path with radius R on a "loop the loop" style ride.

In this case, for the roller coaster to remain on the tracks owing to centripetal force, the net centripetal force must at east equal the weight (= M**g** = 9.8 M, in newtons) of the roller coaster at the very top of the turn, or else the force of gravity will pull the roller coaster off its tracks.

This means that M**v _{t}^{2}**/R must exceed M

**g**, which, solving for v

_{t}, gives a minimum tangential velocity of:

Thus the mass of the roller coaster actually doesn't matter, only its speed!

#### References

- Stanford University: Newton's Three Laws of Motion
- Brigham Young University Physics: A Comparison of Translational and Rotational Mechanics
- Georgia State University Hyperphysics: Centripetal Force
- The Education University of Hong Kong: Non-Uniform Circular Motion
- B.C. Open Campus Textbooks: Centripetal Force