Uniform circular motion

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 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 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.

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.

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 acceleration for some values of radius and magnitude of velocity

|v|   r		1 m/s 3.6 km/h 2.2 mph	2 m/s 7.2 km/h 4.5 mph	5 m/s 18 km/h 11 mph	10 m/s 36 km/h 22 mph	20 m/s 72 km/h 45 mph	50 m/s 180 km/h 110 mph	100 m/s 360 km/h 220 mph
		Slow walk		Bicycle		City car		Aerobatics
10 cm 3.9 in	Laboratory centrifuge	10 m/s² 1.0 g	40 m/s² 4.1 g	250 m/s² 25 g	1.0 km/s² 100 g	4.0 km/s² 410 g	25 km/s² 2500 g	100 km/s² 10000 g
20 cm 7.9 in		5.0 m/s² 0.51 g	20 m/s² 2.0 g	130 m/s² 13 g	500 m/s² 51 g	2.0 km/s² 200 g	13 km/s² 1300 g	50 km/s² 5100 g
50 cm 1.6 ft		2.0 m/s² 0.20 g	8.0 m/s² 0.82 g	50 m/s² 5.1 g	200 m/s² 20 g	800 m/s² 82 g	5.0 km/s² 510 g	20 km/s² 2000 g
1 m 3.3 ft	Playground carousel	1.0 m/s² 0.10 g	4.0 m/s² 0.41 g	25 m/s² 2.5 g	100 m/s² 10 g	400 m/s² 41 g	2.5 km/s² 250 g	10 km/s² 1000 g
2 m 6.6 ft		500 mm/s² 0.051 g	2.0 m/s² 0.20 g	13 m/s² 1.3 g	50 m/s² 5.1 g	200 m/s² 20 g	1.3 km/s² 130 g	5.0 km/s² 510 g
5 m 16 ft		200 mm/s² 0.020 g	800 mm/s² 0.082 g	5.0 m/s² 0.51 g	20 m/s² 2.0 g	80 m/s² 8.2 g	500 m/s² 51 g	2.0 km/s² 200 g
10 m 33 ft	Roller-coaster vertical loop	100 mm/s² 0.010 g	400 mm/s² 0.041 g	2.5 m/s² 0.25 g	10 m/s² 1.0 g	40 m/s² 4.1 g	250 m/s² 25 g	1.0 km/s² 100 g
20 m 66 ft		50 mm/s² 0.0051 g	200 mm/s² 0.020 g	1.3 m/s² 0.13 g	5.0 m/s² 0.51 g	20 m/s² 2 g	130 m/s² 13 g	500 m/s² 51 g
50 m 160 ft		20 mm/s² 0.0020 g	80 mm/s² 0.0082 g	500 mm/s² 0.051 g	2.0 m/s² 0.20 g	8.0 m/s² 0.82 g	50 m/s² 5.1 g	200 m/s² 20 g
100 m 330 ft	Freeway on-ramp	10 mm/s² 0.0010 g	40 mm/s² 0.0041 g	250 mm/s² 0.025 g	1.0 m/s² 0.10 g	4.0 m/s² 0.41 g	25 m/s² 2.5 g	100 m/s² 10 g
200 m 660 ft		5.0 mm/s² 0.00051 g	20 mm/s² 0.0020 g	130 m/s² 0.013 g	500 mm/s² 0.051 g	2.0 m/s² 0.20 g	13 m/s² 1.3 g	50 m/s² 5.1 g
500 m 1600 ft		2.0 mm/s² 0.00020 g	8.0 mm/s² 0.00082 g	50 mm/s² 0.0051 g	200 mm/s² 0.020 g	800 mm/s² 0.082 g	5.0 m/s² 0.51 g	20 m/s² 2.0 g
1 km 3300 ft	High-speed railway	1.0 mm/s² 0.00010 g	4.0 mm/s² 0.00041 g	25 mm/s² 0.0025 g	100 mm/s² 0.010 g	400 mm/s² 0.041 g	2.5 m/s² 0.25 g	10 m/s² 1.0 g

See also

• Example: Circular motion
• Circular motion
• Centripetal force
• Non-uniform circular motion
• Fictitious force
• Reactive centrifugal force
• Sling (weapon)

External links

• Physclips: Mechanics with animations and video clips from the University of New South Wales