Circular motion


In kinematics, circular motion is movement of an object along a circle or rotation along a circular arc. It can be uniform, with a constant rate of rotation and constant tangential speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves the circular motion of its parts. The equations of motion describe the movement of the center of mass of a body, which remains at a constant distance from the axis of rotation. In circular motion, the distance between the body and a fixed point on its surface remains the same, i.e., the body is assumed rigid.
Examples of circular motion include: special satellite orbits around the Earth, a ceiling fan's blades rotating around a hub, a stone that is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.
Since the object's velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this 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 the motion of a body traversing a circular path at a constant speed. Since the body describes circular motion, its distance from the axis of rotation remains constant at all times. Though the body's speed is constant, its velocity is not constant: velocity, a vector quantity, depends on both the body's speed and its direction of travel. This changing velocity indicates the presence of an acceleration; this centripetal acceleration is of constant magnitude and directed at all times toward the axis of rotation. This acceleration is, in turn, produced by a centripetal force which is also constant in magnitude and directed toward the axis of rotation.
In the case of rotation around a fixed axis of a rigid body that is not negligibly small compared to the radius of the path, each particle of the body describes a uniform circular motion with the same angular velocity, but with velocity and acceleration varying with the position with respect to the axis.

Formula

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, is:
and the units are radians/second.
The speed of the object traveling the circle is:
The angle swept out in a time is:
The angular acceleration,, of the particle is:
In the case of uniform circular motion, will be zero.
The acceleration due to change in the direction is:
The centripetal and centrifugal force can also be found using acceleration:
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 as
which is a vector perpendicular to both and, tangential to the orbit, and of magnitude. Likewise, the acceleration is given by
which is a vector perpendicular to both and of magnitude and directed exactly opposite to.
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.
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 radial vector from the origin to the particle location:
where 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 as well, namely. It is customary to orient to point in the direction of travel along the orbit.
The velocity is the time derivative of the displacement:
Because the radius of the circle is constant, the radial component of the velocity is zero. The unit vector has a time-invariant magnitude of unity, so as time varies its tip always lies on a circle of unit radius, with an angle the same as the angle of. If the particle displacement rotates through an angle in time, so does, describing an arc on the unit circle of magnitude. See the unit circle at the left of Figure 4. Hence:
where the direction of the change must be perpendicular to because any change in the direction of would change the size of. The sign is positive because an increase in implies the object and have moved in the direction of.
Hence the velocity becomes:
The acceleration of the body can also be broken into radial and tangential components. The acceleration is the time derivative of the velocity:
The time derivative of is found the same way as for. Again, is a unit vector and its tip traces a unit circle with an angle that is. Hence, an increase in angle by implies traces an arc of magnitude, and as is orthogonal to, we have:
where a negative sign is necessary to keep orthogonal to. See the unit circle at the left of Figure 4. Consequently, the acceleration is:
The centripetal acceleration is the radial component, which is directed radially inward:
while the tangential component changes the magnitude of the velocity:

Using complex numbers

Circular motion can be described using complex numbers and Euler's Formula. Let the axis be the real axis and the axis be the imaginary axis. The position of the body can then be given as, a complex "vector":
where is the imaginary unit, and is the argument of the complex number as a function of time,.
Since the radius is constant:
where a dot indicates differentiation in respect of time.
With this notation, the velocity becomes:
and the acceleration becomes:
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 held constant, but whose direction also is always changing. The acceleration points radially inwards 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. Therefore, the speed of travel around the orbit is
where the angular rate of rotation is. Thus, is a constant, and the velocity vector also rotates with constant magnitude, at the same angular rate.

Relativistic circular motion

In this case, the three-acceleration vector is perpendicular to the three-velocity vector,
and the square of proper acceleration, expressed as a scalar invariant, the same in all reference frames,
becomes the expression for circular motion,
or, taking the positive square root and using the three-acceleration, we arrive at the proper acceleration for 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, which in turn means that the magnitude of the acceleration is given by

Non-uniform circular motion

In non-uniform circular motion, an object moves in a circular path with varying speed. Since the speed is changing, there is tangential acceleration in addition to normal acceleration.
The net acceleration is directed towards the interior of the circle.
The net acceleration may be resolved into two components: tangential acceleration and centripetal acceleration. Unlike tangential acceleration, centripetal acceleration is present in both uniform and non-uniform circular motion.
In non-uniform circular motion, the normal force does not always point to the opposite direction of weight.
The normal force is actually the sum of the radial and tangential forces. The component of weight force is responsible for the tangential force. The centripetal force is due to the change in the direction of velocity.
The normal force and weight may also point in the same direction. Both forces can point downwards, yet the object will remain in a circular path without falling down.
The normal force can point downwards. Considering that the object is a person sitting inside a plane moving in a circle, the two forces will point down only when the plane reaches the top of the circle. The reason for this is that the normal force is the sum of the tangential force and centripetal force. The tangential force is zero at the top. Since weight is perpendicular to the direction of motion of the object at the top of the circle and the centripetal force points downwards, the normal force will point down as well.
From a logical standpoint, a person travelling in that plane will be upside down at the top of the circle. At that moment, the person's seat is actually pushing down on the person, which is the normal force.
The reason why an object does not fall down when subjected to only downward forces is a simple one. Once an object is thrown into the air, there is only the downward gravitational force that acts on the object. That does not mean that once an object is thrown into the air, it will fall instantly. The velocity of the object keeps it up in the air. The first of Newton's laws of motion states that an object's inertia keeps it in motion; since the object in the air has a velocity, it will tend to keep moving in that direction.
A varying angular speed for an object moving in a circular path can also be achieved if the rotating body does not have a homogeneous mass distribution.
One can deduce the formulae of speed, acceleration and jerk, assuming that all the variables depend on :
Further transformations may involve and their corresponding derivatives: