Circular motion

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In physics, circular motion is rotation along a circle: a circular path or a circular orbit. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. We can talk about circular motion of an object if we ignore its size, so that we have the motion of a point mass in a plane.

Contents 1 Constant speed 1.1 Mathematical description 1.2 Using complex numbers 1.3 Deriving the centripetal force 1.4 Kepler's third law 2 Variable speed 3 See also

Constant speed

In the simplest case the speed is constant. It is one of the simplest cases of accelerated motion.

Circular motion involves acceleration of the moving object by a centripetal force which pulls the moving object towards the center of the circular orbit. Without this acceleration, the object would move inertially in a straight line, according to Newton's first law of motion. Circular motion is accelerated even though the speed is constant, because the velocity of the moving object is constantly changing.

Examples of circular motion are: an artificial satellite orbiting the Earth in geosynchronous orbit, a stone which is tied to a rope and is being swung in circles (cf. hammer throw), a racecar turning through a curve in a racetrack, an electron moving perpendicular to a uniform magnetic field, a gear turning inside a mechanism.

A special kind of circular motion is when an object rotates around its own center of mass. This can be called spinning motion, or rotational motion.

Circular motion is characterized by an orbital radius r, a speed v, the mass m of the object which moves in a circle, and the magnitude F of the centripetal force. These quantities all relate to each other through the equation

<math> F = {m v^2 \over r} </math>

which is always true for circular motion.

Since <math> v = r \omega\ </math>, the above equation can be expressed as <math> F = m r \omega\ ^2</math>

Mathematical description

Circular motion can be described by means of parametric equations, viz.

<math> x(t) = R \, \cos \, \omega t, \qquad \qquad (1) </math>

<math> y(t) = R \, \sin \, \omega t, \qquad \qquad (2) </math>

where R and ω are coefficients. Equations (1) and (2) describe motion around a circle centered at the origin with radius R. The quantity ω is the angular velocity, and t is the time.

The derivatives of these equations are

<math> \dot{x}(t) = - R \omega \, \sin \, \omega t, \qquad \qquad (3) </math>

<math> \dot{y}(t) = R \omega \, \cos \, \omega t. \qquad \qquad (4) </math>

The vector <math>\ (x,y)</math> is the position vector of the object undergoing the circular motion. The vector <math> (\dot{x},\dot{y}) </math>, given by equations (3) and (4), is the velocity vector of the moving object. This velocity vector is perpendicular to the position vector, and it is tangent to the circular path of the moving object. The velocity vector must be considered to have its tail located at the head of the position vector. The tail of the position vector is located at the origin.

The derivatives of equations (3) and (4) are

<math> \ddot{x}(t) = - R \omega^2 \, \cos \, \omega t, \qquad \qquad (5) </math>

<math> \ddot{y}(t) = - R \omega^2 \, \sin \, \omega t. \qquad \qquad (6) </math>

The vector <math> (\ddot{x},\ddot{y}) </math>, called the acceleration vector, is given by equations (5) and (6). It has its tail at the head of the position vector, but it points in the direction opposite to the position vector. This means that circular motion can be described by differential equations, thus

<math> \ddot{x} = - \omega^2 x, </math>

<math> \ddot{y} = - \omega^2 y, </math>

or letting x denote the position vector, then circular motion can be described by a single vector differential equation

<math> \ddot{\mathbf{x}} = - \omega^2 \mathbf{x}. </math>

Using complex numbers

Defining the complex position

<math>\ z=x+iy </math>

and using Euler's formula gives the modern description of circular motion around the zero point.

<math>\ z(t) = R e^{i\omega t}</math>

The derivative of the position is the complex velocity:

<math> \dot{z}(t) = i\omega R e^{i\omega t}</math>

The factor i means that the velocity is perpendicular to the position.

The derivative of the velocity is the complex acceleration:

<math> \ddot{z}(t) = - \omega^2 R e^{i\omega t}</math>

The minus sign implies that its direction is opposite to that of the complex position.

The second order differential equation of the circular motion is

<math> \ddot{z} = - \omega^2 z </math>

The first order differential equation of the circular motion is

<math> \dot{z} = i \omega z </math>

Deriving the centripetal force

From equations (5) and (6) it is evident that the magnitude of the acceleration is

<math> a = \omega^2 R. \qquad \qquad (7) </math>

The angular frequency ω is expressed in terms of the period T as

<math> \omega = {2 \pi \over T}. \qquad \qquad (8) </math>

The speed v around the orbit is given by the circumference divided by the period:

<math> v = {2 \pi R \over T}. \qquad \qquad (9) </math>

Comparing equations (8) and (9), we deduce that

<math> v = \omega R. \qquad \qquad (10) </math>

Solving equation (10) for ω and substituting into equation (7) yields

<math> a = {v^2 \over R}. \qquad \qquad (11) </math>

Newton's second law of motion is usually expressed as

<math> F = m a \,</math>

which together with equation (11) implies that

<math> F = {m v^2 \over R},\qquad \qquad (12) </math>

(QED).

Kepler's third law

For satellites tethered to a body of mass M at the origin by means of a gravitational force, the centripetal force is also equal to

<math> F = {G M m \over R^2} \qquad \qquad (13) </math>

where G is the gravitational constant, 6.67·10⁻¹¹ N·m²·kg⁻². Combining equations (12) and (13) yields

<math> {G M m \over R^2} = {m v^2 \over R} </math>

which simplifies to

<math> G M = R v^2. \qquad \qquad (14) </math>

Combining equations (14) and (10) then yields

<math> \omega^2 R^3 = G M \ </math>

which is a form of Kepler's harmonic law of planetary motion.

Variable speed

In the general case, circular motion requires that the total force can be decomposed into the centripetal force required to keep the orbit circular, and a force tangent to the circle, causing a change of speed.

The magnitude of the centripetal force depends on the instantaneous speed.

In the case of an object at the end of a rope, subjected to a force, we can decompose the force into a radial and a lateral component. If the radial component is either outward, or inward but not more than the required centripetal force, then the stress in the rope provides the difference, provided that the rope is strong enough. If the radial component is inward and more than the required centripetal force, then circular motion is not maintained.

An example is a rotation of an object at the end of a rope, in a vertical plane. If the speed is large enough, circular motion is maintained.

In the case of a rigid body with a hinge, the motion is circular anyway, because the stress can both pull and push.

See also

• angular momentum
• simple harmonic motioncs:Pohyb po kružnici

ko:등속원운동 it:Moto circolare pl:Ruch obrotowy pt:Movimento circular sl:Kroženje fi:Pyörimisliike