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.

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 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 object's velocity vector is constantly changing direction.

1 Constant speed
2 Variable speed
3 See also
4 External links

[edit] Constant speed

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.

The speed is one metre per second
The inward acceleration is one metre per second per second.
It is subject to a centripetal force of one kilogram metre per second per second, which is one newton.
The momentum of the body is one kg·m·s⁻¹.
The moment of inertia is one kg·m².
The angular momentum is one kg·m²·s⁻¹.
The kinetic energy is 1/2 joule.
The circumference of the orbit is 2π (~ 6.283) metres.
The period of the motion is 2π seconds per turn.
The frequency is (2π)⁻¹ hertz.
From the point of view of quantum mechanics, the system is in an excited state having quantum number ~ 9.48×10³⁵.

Then consider a body of mass m, moving in a circle of radius r, with an angular velocity of ω.

The speed is v = r·ω.
The centripetal (inward) acceleration is a = r·ω ² = r ⁻¹·v ².
The centripetal force is F = m·a = r·m·ω ² = r⁻¹·m·v ².
The momentum of the body is p = m·v = r·m·ω.
The moment of inertia is I = r ²·m.
The angular momentum is L = r·m·v = r ²·m·ω = I·ω.
The kinetic energy is E = 2⁻¹·m·v ² = 2⁻¹·r ²·m·ω ² = (2·m)⁻¹·p ² = 2⁻¹·I·ω ² = (2·I)⁻¹·L ² .
The circumference of the orbit is 2·π·r.
The period of the motion is T = 2·π·ω ⁻¹.
The frequency is f = T ⁻¹ . (Instead of letter f, the frequency is often denoted by the Greek letter ν, which however is almost indistinguishable from the letter v used here for velocity).
The quantum number is J = 2·π·L h⁻¹.

[edit] 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. The radial component is either outward or inward.