Uniform Circular Motion

Short answer

• Uniform circular motion is motion around a circle at a constant speed. The speed does not change, but the velocity does change, because velocity includes direction and the direction of motion changes at every point on the circle. That continuous change in direction is an acceleration, called centripetal acceleration, and it always points toward the center of the circle. The size of that acceleration is a sub c equals v squared divided by r, where v is the speed and r is the radius. It can also be written as r times omega squared, where omega is the angular velocity. By Newton's second law, a net force toward the center must produce this acceleration. That net inward force is called centripetal force, and its size is F sub c equals m times v squared divided by r, where m is the mass. Centripetal force is not an extra kind of force. It is a role that tension, gravity, friction, the normal force, or a combination of forces can fill. If the inward force disappears, the object does not fly outward. It continues in a straight line along the tangent to the circle at that point, exactly as inertia predicts.

By Emma Hudson

What Uniform Circular Motion Means

In this topic, the word uniform has a narrow meaning: the speed is constant. The path is a circle, the radius stays fixed, and the object covers equal distances along the arc in equal times. What changes, continuously, is the direction in which the object is heading.

A point on a propeller spinning at a steady rate is a good example, as is a clock hand moving steadily around the dial. A satellite in a nearly circular orbit is another example, but a real orbit is only approximately circular. The test is always the same: is the path circular, is the radius fixed, and is the speed constant?

The strict definition also sets a boundary: a car speeding up around a bend is in circular motion but not uniform circular motion, and a ball in a widening spiral is not uniform either, because its radius is changing. The formulas here apply in simplest form only when speed and radius are both constant.

Speed Versus Velocity: Why Constant Speed Still Means Acceleration

The confusion in this topic usually comes from mixing up speed and velocity. Speed is a number with no direction attached. Velocity is a vector, which means it has both a size and a direction. Acceleration is defined as a change in velocity, not merely a change in speed. An object can therefore accelerate by speeding up, by slowing down, or by changing direction while keeping the same speed. Uniform circular motion is the third case.

At each instant, velocity points along the tangent to the circle, the straight line that just touches the circle at that point in the direction of travel. As the object moves, that tangent turns with it, so velocity points in a slightly different direction a moment later, even though its length, representing speed, stays the same.

That is why an object in uniform circular motion is accelerating even though a speedometer would show a steady reading. The acceleration is real, and a real inward force must supply it if the object is to stay on its path. If the inward force were suddenly removed, the object would not shoot outward from the center. It would leave along the tangent at that instant and continue in a straight line, following directly from inertia.

The Direction Picture: Tangent, Radius, and Center

Three directions organize almost every circular motion problem. The velocity is tangential, running along the tangent at the object’s position. The centripetal acceleration is radial and inward, running along the radius toward the center. The centripetal force points in the same inward direction. Velocity and acceleration are therefore perpendicular to each other at every instant in uniform circular motion.

That right angle relationship explains why speed stays constant: a force at a right angle to motion changes direction without speeding the object up or slowing it down. The inward pull steers without adding speed.

Drawing these arrows correctly prevents many errors. Velocity should touch the circle and point forward along the path; acceleration should start at the object and point straight toward the center, not forward, backward, or outward. In a ground frame diagram, an outward force arrow is wrong unless explicitly labeled as an apparent effect in a rotating frame, explained later.

Centripetal Acceleration: The Formula and What It Means

For an object in uniform circular motion, the magnitude of centripetal acceleration is:

a sub c equals v squared divided by r

Here a sub c is the centripetal acceleration, v is the constant speed, and r is the radius. The direction is always toward the center. The units are meters per second squared when v is in meters per second and r is in meters.

The formula can be understood by picturing the velocity vector at two nearby points on the circle. Both vectors have the same length, because speed is constant, but they point in slightly different directions. The change between them, drawn with their tails together, points inward. Dividing that change by the small time interval gives an inward acceleration, which in the limit becomes v squared divided by r.

Two features of the formula deserve attention. First, acceleration grows with the square of speed. Doubling the speed at the same radius multiplies the centripetal acceleration by four, not by two. Second, acceleration falls as radius grows. Doubling the radius at the same speed halves the centripetal acceleration. That is why a wide curve at highway speed can feel gentle, while the same speed on a tight curve demands much more grip from the tires.

Centripetal acceleration is an acceleration, not a force. A force is still needed to cause it. Keeping the two distinct avoids adding a separate centripetal arrow to a diagram that already lists the real forces: in a correct diagram, the net inward component of the real forces equals mass times centripetal acceleration, with no extra arrow.

For scale, OpenStax gives the acceleration of Earth around the Sun as 5.93 times 10 to the power negative 3 meters per second squared, and the Moon around Earth as 2.73 times 10 to the power negative 3 meters per second squared. Those values are far smaller than in a spinning ride or a fast turn, because the radii involved are enormous.

Angular Velocity, Period, and Frequency

Linear speed is not the only way to describe circular motion. Often it is easier to describe how fast the object is turning. Angular velocity, written with the Greek letter omega, measures the angle swept out per unit time, in radians per second. One full revolution is 2 pi radians, so an object that makes one full turn in T seconds has an angular velocity of 2 pi divided by T.

The period, written as T, is the time for one complete revolution. The frequency, written as f, is the number of revolutions per second, measured in hertz. Period and frequency are reciprocals: f equals 1 divided by T, and T equals 1 divided by f. A shorter period therefore means a higher frequency and a higher angular velocity.

These quantities connect to linear speed through the radius:

v equals omega times r

omega equals 2 pi times f, which also equals 2 pi divided by T

v equals 2 pi times r divided by T

Using the relation between v and omega, centripetal acceleration can be rewritten as:

a sub c equals r times omega squared

Both forms describe the same acceleration: use the first when speed and radius are known, and the second when turning rate and radius are known. Checking both against each other catches unit errors quickly.

A clock hand illustrates period: a steadily moving second hand completes one revolution in 60 seconds, so its period is 60 seconds, its frequency is 1 divided by 60 hertz (about 0.0167 hertz), and its angular velocity is 2 pi divided by 60 radians per second (about 0.105 radians per second). The tip of a longer second hand moves faster in linear terms than a shorter one at the same angular velocity, because linear speed equals angular velocity times radius. Angle units need care: these formulas use radians, not degrees, so convert revolutions per minute or degrees to radians before substituting.

Centripetal Force: A Role, Not a New Kind of Force

Newton’s second law states that net force equals mass times acceleration. In uniform circular motion, the acceleration is the centripetal acceleration, so the net force toward the center must be:

F sub c equals m times a sub c

Substituting the expressions for centripetal acceleration gives two useful forms:

F sub c equals m times v squared divided by r

F sub c equals m times r times omega squared

The direction of this net force is toward the center of curvature, the same direction as the centripetal acceleration, and it is perpendicular to the velocity at every instant. As with acceleration, the square dependence on speed matters: doubling the speed at the same radius requires four times the inward force for the same mass.

The most important point in this section is that centripetal force is not a separate entry in the catalog of forces. Tension, gravity, friction, electrical attraction, the normal force, and other real forces can serve as the centripetal force, alone or in combination. OpenStax states this directly: any net force causing uniform circular motion is called a centripetal force, and any force or combination of forces can cause a centripetal or radial acceleration. The word centripetal therefore describes what a force is doing in a particular situation, not what kind of force it is.

That distinction changes how diagrams should be drawn. List the real forces, identify which forces or components point toward the center, and take their inward sum as the centripetal force. Do not draw an additional centripetal arrow alongside the real forces, or the effect will be counted twice.

Who Supplies the Inward Force? Four Common Cases

Tension is the inward force for a ball whirled on a string in a horizontal circle, in the idealized case where the string stays horizontal. The string pulls the ball toward the center, and that pull is the centripetal force. If the string is cut, tension disappears at once, and the ball leaves along the tangent. In a real vertical circle, gravity also contributes and tension varies around the circle, so the motion is generally not uniform. The horizontal case is the clean uniform example to learn first.

Gravity is the inward force for a satellite in a nearly circular orbit and, in simplified treatments, for the Moon around Earth. Gravity pulls toward the center of the attracting body, which in a circular orbit is also the center of the circle. The satellite is continuously falling in the sense that its velocity is continuously redirected, yet its distance stays nearly constant because forward motion carries it around the curve.

Friction is the inward force for a car turning on a flat, level road: static friction acts horizontally toward the center of the turn, and the required friction grows with speed squared and falls as radius grows. On ice, where friction is very small, the car cannot get enough inward force and follows a straighter path.

The normal force supplies the inward force in banked turns and in rides where a wall pushes on the rider. On a banked road, the tilted surface gives the normal force a horizontal component toward the center; at the ideal banking angle for a given speed, that component alone can supply the needed force. On a spinning ride, the wall pushes inward on each rider, and that push is the centripetal force.

Often more than one force contributes: a car on a banked curve traveling faster than the ideal banking speed has a normal component and a friction component pointing partly inward (below that speed, friction points up the slope instead), and a tetherball at the top of a vertical circle has tension and gravity both pointing inward. In every case, resolve forces into inward and outward components and set the net inward force equal to mass times centripetal acceleration.

Worked Examples With Arithmetic Checked

Example 1: A point on a steadily turning wheel

A point on the edge of a wheel moves in a circle of radius 2.0 meters and completes one revolution in 4.0 seconds. Find its speed, angular velocity, and centripetal acceleration.

Angular velocity equals 2 pi divided by the period: 2 times 3.1416 divided by 4.0 seconds gives 1.5708 radians per second, which rounds to 1.57 radians per second. Linear speed equals angular velocity times radius: 1.5708 times 2.0 meters gives 3.1416 meters per second, which rounds to 3.14 meters per second. As a check, 2 pi times radius divided by period gives the same 3.14 meters per second.

Centripetal acceleration equals speed squared divided by radius: 3.1416 squared equals 9.8696, and 9.8696 divided by 2.0 meters gives 4.93 meters per second squared. Radius times angular velocity squared gives the same result: 2.0 times 1.5708 squared equals 4.93 meters per second squared, toward the center.

Example 2: A car on a flat curve

A car of mass 1,200 kilograms travels at a steady speed of 20 meters per second around a flat curve of radius 50 meters. Find its centripetal acceleration and the friction force required.

Centripetal acceleration equals speed squared divided by radius: 20 squared equals 400, and 400 divided by 50 meters gives 8.0 meters per second squared toward the center. The required centripetal force equals mass times acceleration: 1,200 kilograms times 8.0 meters per second squared gives 9,600 newtons. On a flat road, static friction must supply that full 9,600 newtons. If the tires and road cannot supply that much static friction, the car will not hold the 50 meter curve at 20 meters per second.

At double the speed, 40 meters per second, acceleration would be 1,600 divided by 50, or 32 meters per second squared, and force would be 38,400 newtons, four times larger. That square dependence is why safe cornering speeds fall quickly on tight curves.

Example 3: A ball on a string

A ball of mass 0.50 kilograms is whirled in a horizontal circle of radius 0.80 meters at a steady speed of 4.0 meters per second. Find its centripetal acceleration and the tension in the string, ignoring the small vertical effects that a real string would introduce.

Centripetal acceleration equals speed squared divided by radius: 4.0 squared equals 16, and 16 divided by 0.80 meters gives 20 meters per second squared toward the center. Tension equals mass times acceleration in this idealized horizontal case: 0.50 kilograms times 20 meters per second squared gives 10 newtons. If the speed rose to 8.0 meters per second at the same radius, the acceleration would rise to 80 meters per second squared and the tension to 40 newtons, again four times larger for double the speed.

These examples use round numbers chosen for teaching, and each step was recomputed and cross checked with the second formula form where one exists. Keep extra digits through intermediate steps and round only at the end.

Uniform Versus Nonuniform Circular Motion

Uniform and nonuniform circular motion share a circular path, but they differ in whether speed changes. In uniform circular motion, speed is constant, tangential acceleration is zero, and the only acceleration is centripetal, pointing inward. In nonuniform circular motion, speed is changing, so there is also a tangential acceleration, which points along the tangent in the direction of motion when the object is speeding up, and opposite the motion when it is slowing down.

The total acceleration in nonuniform motion is the vector sum of the centripetal and tangential components. Because those components are perpendicular, the size of the total acceleration is found with the Pythagorean relation: total acceleration equals the square root of centripetal acceleration squared plus tangential acceleration squared. The total acceleration no longer points straight toward the center. It tilts toward the tangential direction, forward if speeding up and backward if slowing down.

A worked contrast makes the point concrete. Take the car from Example 2, with centripetal acceleration 8.0 meters per second squared, and suppose it is also speeding up with a tangential acceleration of 2.0 meters per second squared. Its total acceleration is the square root of 8.0 squared plus 2.0 squared, which is the square root of 64 plus 4, which is the square root of 68, giving about 8.25 meters per second squared. The total acceleration tilts away from the inward radial direction by an angle whose tangent is 2.0 divided by 8.0, which is 0.25, giving an angle of about 14 degrees toward the direction of motion. The speed is no longer constant, so this is nonuniform motion, even though the path is still circular.

The Centrifugal Force Misconception

The word centrifugal is often used in everyday speech for a force that seems to push objects outward during a turn. In physics, that outward effect must be described carefully, because its status depends on the frame of reference, meaning the viewpoint from which the motion is described.

From the ground frame, which is treated as nearly inertial for ordinary problems, there is no outward force on an object in uniform circular motion. The real net force points inward. What a passenger feels as being pushed outward is inertia: the passenger’s body tends to continue in a straight line, while the car or ride turns inward around them. The seat, belt, door, or wall then pushes the passenger inward, supplying the centripetal force that keeps the passenger turning with the vehicle. The sensation is real, but the explanation in the ground frame is inertia plus an inward push, not an outward force.

From a rotating frame, a viewpoint that turns with the object, the object appears at rest, so an extra apparent force must be introduced to balance the inward real force. That apparent outward effect is often called centrifugal force and is labeled in textbooks as a fictitious or inertial force, because it has no identifiable physical source such as a string, a surface, or a planet. OpenStax makes this distinction explicit for the merry-go-round, stressing that in Earth’s frame there is no force trying to throw the rider off.

For problem solving, work in the ground frame and draw only real forces, unless a question explicitly asks for a rotating frame analysis including fictitious forces. If a rotating frame is used, label the centrifugal term clearly as a frame effect, separate from the real forces. Mixing the two descriptions in one diagram is a common source of sign errors.

Centripetal and centrifugal forces do not form an action and reaction pair. The reaction to a centripetal force is the equal and opposite force the moving object exerts on whatever supplies the inward pull, for example the ball pulling outward on the string while the string pulls inward on the ball. Those reactions act on different objects and belong in different diagrams.

Common Mistakes and a Problem Solving Checklist

Several mistakes recur, and most can be caught with a short checklist. First, check whether speed is truly constant: if the object speeds up, slows down, or moves in a vertical circle under gravity, the uniform formulas alone are not enough. Second, check that the radius is the radius of the circular path, not a diameter or a height.

Third, check vector directions before calculating: velocity is tangential, acceleration is inward, and net force is inward in uniform motion. Fourth, check units: speeds in meters per second, radii in meters, and angular velocities in radians per second, converting kilometers per hour or revolutions per minute first.

Fifth, remember the square: acceleration and force scale with speed squared, so a small speed change has a larger effect than expected. Sixth, do not double count centripetal force. Seventh, for a maximum safe speed on a flat curve, use maximum static friction as the limiting force; mass cancels there only because friction is proportional to weight, not in general.

A reliable working order is: draw the circle and mark the center, draw velocity tangential and acceleration inward, list real forces, resolve them into inward and tangential components, write net inward force equals mass times v squared divided by r, substitute numbers with units, and check the answer for sense. An implausibly large acceleration or friction force should prompt a second look at radius, speed, and units.

Frequently Asked Questions

What is uniform circular motion in simple terms?

Uniform circular motion is movement around a circle at a steady speed. The object keeps the same speed and distance from the center but keeps changing direction, so it accelerates toward the center even though its speed never changes.

If speed is constant, how can there be acceleration?

Acceleration means a change in velocity, and velocity includes direction as well as speed. In circular motion, direction changes continuously, so velocity changes continuously even when speed stays the same, producing acceleration toward the center.

What is the formula for centripetal acceleration?

Centripetal acceleration equals speed squared divided by radius, written as a sub c equals v squared divided by r. It can also be written as radius times angular velocity squared, or a sub c equals r times omega squared. The direction is always toward the center of the circle. Doubling speed multiplies the acceleration by four at the same radius, while doubling radius halves it at the same speed.

What is centripetal force, and is it a real force?

Centripetal force is the net inward force causing centripetal acceleration, given by F sub c equals m times v squared divided by r. It is supplied by real forces, but it is not a new kind of force: tension, gravity, friction, the normal force, or a combination can act as the centripetal force. Do not add a separate centripetal arrow to a diagram that already lists the real forces.

What is the difference between uniform and nonuniform circular motion?

In uniform motion, speed is constant and the only acceleration is centripetal, pointing inward. In nonuniform motion, speed is also changing, so there is an additional tangential acceleration along the path, and total acceleration no longer points straight toward the center.

Is centrifugal force real?

In the ground frame used for most problems, there is no outward centrifugal force on the moving object. The outward sensation comes from inertia, while a real inward force keeps the object turning. In a rotating frame, physicists introduce an apparent outward inertial effect, often called centrifugal force, which must be labeled as a frame effect, not a real interaction.

What happens if the centripetal force is removed?

The object leaves along the tangent at the point where the force stopped and continues in a straight line at the same speed, unless another force acts on it. It does not fly straight outward from the center.

How are period, frequency, and angular velocity related?

Period is the time for one revolution, frequency is revolutions per second, and angular velocity is radians per second. Frequency equals 1 divided by period, angular velocity equals 2 pi divided by period (also 2 pi times frequency), and linear speed equals angular velocity times radius.

Keep reading: Transverse Wave vs Longitudinal Wave: How Each One Moves, and How to Tell Them Apart

Sources & references

• OpenStax, University Physics Volume 1, Section 4.5 Uniform Circular Motion, via LibreTexts. Used for the definition of uniform circular motion as motion in a circle with constant speed, for centripetal acceleration as a sub c equals v squared divided by r toward the center, for the position, velocity, and acceleration relations including v equals omega times r and a sub c equals r times omega squared, for the distinction between centripetal and tangential acceleration in nonuniform motion, and for the typical centripetal acceleration values for Earth around the Sun and the Moon around Earth. https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_%28OpenStax%29/04%3A_Motion_in_Two_and_Three_Dimensions/4.05%3A_Uniform_Circular_Motion
• OpenStax, University Physics Volume 1, Section 6.3 Centripetal Force. Used for centripetal force as any net force causing uniform circular motion, for F sub c equals m times a sub c and the forms F sub c equals m times v squared divided by r and F sub c equals m times r times omega squared, for tension, gravity, friction, and banked roadway forces as examples that can supply centripetal force, for the flat curve friction calculation method, for ideal banking, and for the discussion of inertial forces and noninertial frames including the statement that centrifugal force in the rotating frame discussion does not exist as a real force in Earth's frame. https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force