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Wikipedia / Circular motion

In kinematics, circular motion is the motion of an object along a circular path. Examples of this include a stone tied to a string, a car moving around a curve, and a point on a rotating wheel. Circular motion can be uniform, meaning the speed is constant, or non-uniform, meaning the speed changes. Even in uniform circular motion, the object is accelerating because its velocity changes direction. The object accelerates toward the center of the circle; this inward acceleration is called centripetal acceleration. A force toward the center, called centripetal force, is required to produce this acceleration. Circular motion is also used to describe the motion of points in a rigid body rotating about a fixed axis, in which case each point in the body moves in a circle fixed around the axis of rotation.

More examples of circular motion include special satellite orbits around the Earth (circular orbits), a ceiling fan's blades rotating around a hub, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.

Without centripetal acceleration, the object would move in a straight line, according to Newton's laws of motion.

Uniform circular motion

In physics, uniform circular motion describes motion along a circular path at constant speed. Although the speed is constant, the velocity changes because the direction of motion changes. Therefore, the object accelerates inward, toward the center of the circle. This inward acceleration is called centripetal acceleration.

In rotation about a fixed axis, each particle of a rigid body moves in a circle centered on that axis. All particles share the same angular displacement, angular velocity, and angular acceleration, while their linear speed and acceleration depend on their distance from the axis.

Basic formulas

The main quantities used to describe circular motion are the radius, the time for one revolution, the angular velocity, the linear speed, and the centripetal acceleration.

For motion in a circle of radius , the circumference of the circle is . If the period for one rotation is , the angular rate of rotation, also known as angular velocity, and the units are radians/second.

The angle swept out in a time

The angular acceleration,

In the case of uniform circular motion, will be zero.

The centripetal and centrifugal force can also be found using acceleration. For a body of mass , the centripetal force is, by Newton's second law

In the simplest case the speed, mass, and radius are constant.

Consider a body of one kilogram, moving in a circle of radius one metre, with an angular velocity of one radian per second.

In vector form

The vector relationships are shown in Figure 1. The axis of rotation is shown as a vector perpendicular to the plane of the orbit and with a magnitude . The direction of is chosen using the right-hand rule. With this convention for depicting rotation, the velocity is given by a vector cross product which is a vector perpendicular to both and , tangential to the orbit, and of magnitude which is a vector perpendicular to both and of magnitude and directed exactly opposite to .

In polar coordinates

In polar coordinates, circular motion is especially simple because the radius is constant. The object’s position changes only through the angle around the circle. As a result, the velocity is tangent to the circle, while the centripetal part of the acceleration points inward. During circular motion, the body moves on a curve that can be described in the polar coordinate system as a fixed distance from the center of the orbit taken as the origin, oriented at an angle from some reference direction. See Figure 4. The displacement vector is the unit vector parallel to the radius vector at time and pointing away from the origin. It is convenient to introduce the unit vector orthogonal to point in the direction of travel along the orbit.

has a time-invariant magnitude of unity, so as time varies its tip always lies on a circle of unit radius, with an angle . If the particle displacement rotates through an angle in time , describing an arc on the unit circle of magnitude . The sign is positive because an increase in

is a unit vector and its tip traces a unit circle with an angle that is . Hence, an increase in angle traces an arc of magnitude would decrease with an increase in

The centripetal acceleration while the tangential component changes the magnitude

Using complex numbers

Circular motion can be described using complex numbers and Euler's formula. Let the where is the imaginary unit is the argument of the complex number as a function of time, .

where a dot indicates differentiation in respect of time.

The first term is opposite in direction to the displacement vector and the second is perpendicular to it, just like the earlier results shown before.

Velocity

Figure 1 illustrates velocity and acceleration vectors for uniform motion at four different points in the orbit. Because the velocity is tangent to the circular path, no two velocities point in the same direction. Although the object has a constant speed, its direction is always changing. This change in velocity is caused by an acceleration , whose magnitude is (like that of the velocity) held constant, but whose direction also is always changing. The acceleration points radially inwards (centripetally) and is perpendicular to the velocity. This acceleration is known as centripetal acceleration.

For a path of radius , when an angle is swept out, the distance traveled on the periphery of the orbit is where the angular rate of rotation is . (By rearrangement, .) Thus, is a constant, and the velocity vector also rotates with constant magnitude , at the same angular rate .

Relativistic circular motion

Acceleration

The left-hand circle in Figure 2 is the orbit showing the velocity vectors at two adjacent times. On the right, these two velocities are moved so their tails coincide. Because speed is constant, the velocity vectors on the right sweep out a circle as time advances. For a swept angle the change in is a vector at right angles to and of magnitude

Non-uniform circular motion

In non-uniform circular motion, an object moves along a circular path with changing speed. Because the speed changes, the object has tangential acceleration in addition to centripetal acceleration.

The acceleration can be separated into two perpendicular components. The radial component, or centripetal acceleration, points toward the center of the circle and changes the direction of motion. The tangential component points along the tangent to the circle and changes the speed.

The radial acceleration has magnitude

where is the instantaneous speed and is the radius of the circular path. The tangential acceleration has magnitude

  1. .

The total acceleration is the vector sum of the radial and tangential components. In uniform circular motion, the tangential component is zero because the speed is constant. In non-uniform circular motion, both components may be present.

Applications

Solving applications involving circular motion often involves force analysis. In uniform circular motion, the speed is constant, so the acceleration is entirely radial. In non-uniform circular motion, there is also tangential acceleration because the speed changes.

When drawing a free-body diagram, centripetal force is usually not drawn as a separate force. Instead, the real forces acting on the object are resolved into radial and tangential components.

See also

References