# Uniform circular motion: Wikis

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# Encyclopedia

Figure 1: Velocity v and acceleration a in uniform circular motion at angular rate ω; the speed is constant, but the velocity is always tangent to the orbit; the acceleration has constant magnitude, but always points toward the center of rotation
Figure 2: The velocity vectors at time t and time t + dt are moved from the orbit on the left to new positions where their tails coincide, on the right. Because the velocity is fixed in magnitude at v = r ω, the velocity vectors also sweep out a circular path at angular rate ω. As dt → 0, the acceleration vector a becomes perpendicular to v, which means it points toward the center of the orbit in the circle on the left. Angle ω dt is the very small angle between the two velocities and tends to zero as dt→ 0
Figure 3: (Left) Ball in circular motion – rope provides centripetal force to keep ball in circle (Right) Rope is cut and ball continues in straight line with velocity at the time of cutting the rope, in accord with Newton's law of inertia, because centripetal force is no longer there

In physics, uniform circular motion describes the motion of a body traversing a circular path at constant speed. The distance of the body from the axis of rotation remains constant at all times. Though the body's speed is constant, its velocity is not: 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 towards the axis of rotation. This acceleration is, in turn, produced by a centripetal force which is also constant in magnitude and directed towards the axis of rotation.

## Velocity

Figure 1 illustrates velocity and acceleration vectors for uniform motion at four different points in the orbit. Because the velocity v 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 a, 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 r, when an angle θ is swept out, the distance traveled on the periphery of the orbit is s = rθ. Therefore, the speed of travel around the orbit is

$v = r \frac{d\theta}{dt} = r\omega$,

where the angular rate of rotation is ω. (By rearrangement, ω = v/r.) Thus, v is a constant, and the velocity vector v also rotates with constant magnitude v, at the same angular rate ω.

## 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 dθ = ω dt the change in v is a vector at right angles to v and of magnitude v dθ, which in turn means that the magnitude of the acceleration is given by

$a = v\frac{d\theta}{dt} = v\omega = \frac{v^2}{r} \, .$

## Centripetal force

The acceleration is due to an inward-acting force, which is known as the centripetal force (meaning "center-seeking force"). It is the force that keeps an object in uniform circular motion. From Newton's second law of motion, the centripetal force Fc for an object in uniform circular motion is related to the object's acceleration by

$\mathbf{F}_{\mathrm{c}} = m\mathbf{a} \, ,$

where m is the mass of the object.

Since the magnitude of the acceleration is given by a = v2/r, the magnitude of the centripetal force is given by

$F_{\mathrm{c}} = m\frac{v^2}{r} \, .$

The centripetal force can be provided by many different things, such as tension (as in a sling), friction (as between tires and road for a turning car), or gravity (as between the Sun and the Earth).

Figure 3 shows an example of the role of centripetal force in maintaining a circular orbit: a mass tied to a rope and spinning around in a horizontal circle. The tension in the rope is the centripetal force, and it is the force keeping the object in uniform circular motion.

If the rope is cut at a particular time, the ball continues to move in the direction of its velocity at the moment of cutting, traveling tangent to the circular path.

In Figure 3, the rope holding the ball of mass m is cut about 34 of the way around the orbit. After the rope is cut, the tension force/centripetal force is no longer acting upon the object so there is no force holding the object in uniform circular motion. Therefore it continues going in the direction when it was last in contact with the force. This is commonly mistaken for Centrifugal Force.

Consider the example of a car racing in a circular track. Similar to the tension force, the radially directed component of frictional force between the tires of the car and the road provides the centripetal force keeping the car in the circle. If the road were a frictionless plane, the car would not be able to move in uniform circular motion, and would instead travel in a straight line. For example, if there is a slick spot on the track, the car leaves the track much in the manner shown in Figure 3, accompanied by some spinning about its own axis to conserve angular momentum.