Circular motion explained

In physics, circular motion is a movement of an object along the circumference of 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 (circular orbits), 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:$$\backslash omega\; =\; \backslash frac\; =\; 2\backslash pi\; f\; =\; \backslash frac$$ and the units are radians/second.

The speed of the object traveling the circle is:$$v\; =\; \backslash frac\; =\; \backslash omega\; r$$

The angle swept out in a time is:$$\backslash theta\; =\; 2\; \backslash pi\; \backslash frac\; =\; \backslash omega\; t$$

The angular acceleration,, of the particle is:$$\backslash alpha\; =\; \backslash frac$$

In the case of uniform circular motion, will be zero.

The acceleration due to change in the direction is:$$a\_c\; =\; \backslash frac\; =\; \backslash omega^2\; r$$

The centripetal and centrifugal force can also be found using acceleration:$$F\_c\; =\; \backslash dot\; \backslash mathrel\backslash overset\; ma\_c\; =\; \backslash frac$$

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$$\backslash mathbf\; =\; \backslash boldsymbol\; \backslash omega\; \backslash times\; \backslash mathbf\; r,$$which is a vector perpendicular to both and, tangential to the orbit, and of magnitude . Likewise, the acceleration is given by$$\backslash mathbf\; =\; \backslash boldsymbol\; \backslash omega\; \backslash times\; \backslash mathbf\; v\; =\; \backslash boldsymbol\; \backslash omega\; \backslash times\; \backslash left(\backslash boldsymbol\; \backslash omega\; \backslash times\; \backslash mathbf\; r\; \backslash right),$$which is a vector perpendicular to both and of magnitude and directed exactly opposite to .^[1]

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 1 metre per second.
• The inward acceleration is 1 metre per square second, .
• It is subject to a centripetal force of 1 kilogram metre per square second, which is 1 newton.
• The momentum of the body is 1 kg·m·s⁻¹.
• The moment of inertia is 1 kg·m².
• The angular momentum is 1 kg·m²·s⁻¹.
• The kinetic energy is 0.5 joule.
• The circumference of the orbit is 2 (~6.283) metres.
• The period of the motion is 2 seconds per turn.
• The frequency is (2)⁻¹ hertz.

In polar coordinates

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

r is the radial vector from the origin to the particle location:$$\backslash mathbf(t)\; =\; R\; \backslash hat\backslash mathbf\_R(t)\backslash ,,$$where \hat u _{R(t)} 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 \hat u _{R(t)} as well, namely \hat u _{\theta(t)} . It is customary to orient \hat u _{\theta(t)} to point in the direction of travel along the orbit.

The velocity is the time derivative of the displacement:$$\backslash mathbf(t)\; =\; \backslash frac\; \backslash mathbf(t)\; =\; \backslash frac\; \backslash hat\backslash mathbf\_R(t)\; +\; R\; \backslash frac\; \backslash ,\; .$$

Because the radius of the circle is constant, the radial component of the velocity is zero. The unit vector

\hat u _{R(t)} 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 r (t) . If the particle displacement rotates through an angle in time, so does \hat u _{R(t)} , describing an arc on the unit circle of magnitude . See the unit circle at the left of Figure 4. Hence:$$\backslash frac\; =\; \backslash frac\; \backslash hat\backslash mathbf\_\backslash theta(t)\; \backslash ,,$$where the direction of the change must be perpendicular to \hat u _{R(t)} (or, in other words, along \hat u _{\theta(t)} ) because any change d\hat u _{R(t)} in the direction of \hat u _{R(t)} would change the size of \hat u _{R(t)} . The sign is positive because an increase in implies the object and \hat u _{R(t)} have moved in the direction of \hat u _{\theta(t)} .Hence the velocity becomes:$$\backslash mathbf(t)\; =\; \backslash frac\; \backslash mathbf(t)\; =\; R\backslash frac\; =\; R\; \backslash frac\; \backslash hat\backslash mathbf\_\backslash theta(t)\; =\; R\; \backslash omega\; \backslash hat\backslash mathbf\_\backslash theta(t)\; \backslash ,\; .$$

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

\hat u _{\theta(t)} is found the same way as for \hat u _{R(t)} . Again, \hat u _{\theta(t)} is a unit vector and its tip traces a unit circle with an angle that is . Hence, an increase in angle by r (t) implies \hat u _{\theta(t)} traces an arc of magnitude, and as \hat u _{\theta(t)} is orthogonal to \hat u _{R(t)} , we have:$$\backslash frac\; =\; -\backslash frac\; \backslash hat\backslash mathbf\_R(t)\; =\; -\backslash omega\; \backslash hat\backslash mathbf\_R(t)\; \backslash ,,$$where a negative sign is necessary to keep \hat u _{\theta(t)} orthogonal to \hat u _{R(t)} . (Otherwise, the angle between \hat u _{\theta(t)} and \hat u _{R(t)} would decrease with an increase in .) 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:$$\backslash mathbf\_R(t)\; =\; -\backslash omega^2\; R\; \backslash hat\backslash mathbf\_R(t)\; \backslash ,,$$while the tangential component changes the magnitude of the velocity:$$\backslash mathbf\_\backslash theta(t)\; =\; R\; \backslash frac\; \backslash hat\backslash mathbf\_\backslash theta(t)\; =\; \backslash frac\; \backslash hat\backslash mathbf\_\backslash theta(t)\; =\; \backslash frac\; \backslash hat\backslash mathbf\_\backslash theta(t)\; \backslash ,\; .$$

Using complex numbers

Circular motion can be described using complex numbers. Let the axis be the real axis and the

y axis be the imaginary axis. The position of the body can then be given as z , a complex "vector":$$z\; =\; x\; +\; iy\; =\; R\backslash left(\backslash cos[\backslash theta(t)]\; +\; i\; \backslash sin[\backslash theta(t)]\backslash right)\; =\; Re^\backslash ,,$$where is the imaginary unit, and \theta(t) is the argument of the complex number as a function of time, .

Since the radius is constant:$$\backslash dot\; =\; \backslash ddot\; R\; =\; 0\; \backslash ,,$$where a dot indicates differentiation in respect of time.

With this notation, the velocity becomes:$$v\; =\; \backslash dot\; =\; \backslash frac\backslash left(R\; e^\backslash right)\; =\; R\; \backslash frac\backslash left(e^\backslash right)\; =\; R\; e^\; \backslash frac\; \backslash left(i\; \backslash theta[t]\; \backslash right)\; =\; iR\backslash dot(t)\; e^\; =\; i\backslash omega\; R\; e^\; =\; i\backslash omega\; z$$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 (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, 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$$v\; =\; r\; \backslash frac\; =\; r\backslash omega,$$where the angular rate of rotation is . (By rearrangement, .) 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,$$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; 0.$$and the square of proper acceleration, expressed as a scalar invariant, the same in all reference frames,$$\backslash alpha^2\; =\; \backslash gamma^4\; a^2\; +\; \backslash gamma^6\; \backslash left(\backslash mathbf\; \backslash cdot\; \backslash mathbf\backslash right)^2,$$becomes the expression for circular motion,$$\backslash alpha^2\; =\; \backslash gamma^4\; a^2.$$or, taking the positive square root and using the three-acceleration, we arrive at the proper acceleration for circular motion:$$\backslash alpha\; =\; \backslash gamma^2\; \backslash frac.$$

Acceleration

See main article: 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$$a\_c\; =\; v\; \backslash frac\; =\; v\backslash omega\; =\; \backslash frac$$

Centripetal acceleration for some values of radius and magnitude of velocity! colspan="2" rowspan="2" !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/s^2 1.0 g	40 m/s^2 4.1 g	250 m/s^2 25 g	1.0 km/s^2 100 g	4.0 km/s^2 410 g	25 km/s^2 2500 g	100 km/s^2 10000 g
20 cm 7.9 in		5.0 m/s^2 0.51 g	20 m/s^2 2.0 g	130 m/s^2 13 g	500 m/s^2 51 g	2.0 km/s^2 200 g	13 km/s^2 1300 g	50 km/s^2 5100 g
50 cm 1.6 ft		2.0 m/s^2 0.20 g	8.0 m/s^2 0.82 g	50 m/s^2 5.1 g	200 m/s^2 20 g	800 m/s^2 82 g	5.0 km/s^2 510 g	20 km/s^2 2000 g
1 m 3.3 ft	Playground carousel	1.0 m/s^2 0.10 g	4.0 m/s^2 0.41 g	25 m/s^2 2.5 g	100 m/s^2 10 g	400 m/s^2 41 g	2.5 km/s^2 250 g	10 km/s^2 1000 g
2 m 6.6 ft		500 mm/s^2 0.051 g	2.0 m/s^2 0.20 g	13 m/s^2 1.3 g	50 m/s^2 5.1 g	200 m/s^2 20 g	1.3 km/s^2 130 g	5.0 km/s^2 510 g
5 m 16 ft		200 mm/s^2 0.020 g	800 mm/s^2 0.082 g	5.0 m/s^2 0.51 g	20 m/s^2 2.0 g	80 m/s^2 8.2 g	500 m/s^2 51 g	2.0 km/s^2 200 g
10 m 33 ft	Roller-coaster vertical loop	100 mm/s^2 0.010 g	400 mm/s^2 0.041 g	2.5 m/s^2 0.25 g	10 m/s^2 1.0 g	40 m/s^2 4.1 g	250 m/s^2 25 g	1.0 km/s^2 100 g
20 m 66 ft		50 mm/s^2 0.0051 g	200 mm/s^2 0.020 g	1.3 m/s^2 0.13 g	5.0 m/s^2 0.51 g	20 m/s^2 2 g	130 m/s^2 13 g	500 m/s^2 51 g
50 m 160 ft		20 mm/s^2 0.0020 g	80 mm/s^2 0.0082 g	500 mm/s^2 0.051 g	2.0 m/s^2 0.20 g	8.0 m/s^2 0.82 g	50 m/s^2 5.1 g	200 m/s^2 20 g
100 m 330 ft	Freeway on-ramp	10 mm/s^2 0.0010 g	40 mm/s^2 0.0041 g	250 mm/s^2 0.025 g	1.0 m/s^2 0.10 g	4.0 m/s^2 0.41 g	25 m/s^2 2.5 g	100 m/s^2 10 g
200 m 660 ft		5.0 mm/s^2 0.00051 g	20 mm/s^2 0.0020 g	130 m/s^2 0.013 g	500 mm/s^2 0.051 g	2.0 m/s^2 0.20 g	13 m/s^2 1.3 g	50 m/s^2 5.1 g
500 m 1600 ft		2.0 mm/s^2 0.00020 g	8.0 mm/s^2 0.00082 g	50 mm/s^2 0.0051 g	200 mm/s^2 0.020 g	800 mm/s^2 0.082 g	5.0 m/s^2 0.51 g	20 m/s^2 2.0 g
1 km 3300 ft	High-speed railway	1.0 mm/s^2 0.00010 g	4.0 mm/s^2 0.00041 g	25 mm/s^2 0.0025 g	100 mm/s^2 0.010 g	400 mm/s^2 0.041 g	2.5 m/s^2 0.25 g	10 m/s^2 1.0 g

Non-uniform

In a non-uniform circular motion, an object is moving in a circular path with a varying speed. Since the speed is changing, there is tangential acceleration in addition to normal acceleration.

In a non-uniform circular motion, the net acceleration (a) is along the direction of, which is directed inside the circle but does not pass through its center (see figure). The net acceleration may be resolved into two components: tangential acceleration and normal acceleration also known as the centripetal or radial acceleration. Unlike tangential acceleration, centripetal acceleration is present in both uniform and non-uniform circular motion.

In a non-uniform circular motion, normal force does not always point in the opposite direction of weight. Here is an example with an object traveling in a straight path then looping a loop back into a straight path again.

This diagram shows the normal force pointing in other directions rather than opposite to the weight force. The normal force is actually the sum of the radial and tangential forces. The component of weight force is responsible for the tangential force here (We have neglected frictional force). The radial force (centripetal force) is due to the change in the direction of velocity as discussed earlier.

In a non-uniform circular motion, normal force and weight may point in the same direction. Both forces can point down, yet the object will remain in a circular path without falling straight down. First, let's see why normal force can point down in the first place. In the first diagram, let's say the object is a person sitting inside a plane, the two forces point down only when it 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 (as no work is performed when the motion is perpendicular to the direction of force applied. Here weight force is perpendicular to the direction of motion of the object at the top of the circle) and centripetal force points down, thus normal force will point down as well. From a logical standpoint, a person who is travelling in the 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 the object does not fall down when subjected to only downward forces is a simple one. Think about what keeps an object up after it is thrown. Once an object is thrown into the air, there is only the downward force of Earth's gravity that acts on the object. That does not mean that once an object is thrown in the air, it will fall instantly. What keeps that object up in the air is its velocity. The first of Newton's laws of motion states that an object's inertia keeps it in motion, and 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. For inhomogeneous objects, it is necessary to approach the problem as in.^[2]

One can deduce the formulae of speed, acceleration and jerk, assuming all the variables to depend on

t :$$\backslash mathbf\; =\; R\; \backslash mathbf\_R$$

$$\backslash dot\; \backslash mathbf\_R\; =\; \backslash omega\backslash mathbf\_$$

$$\backslash dot\; \backslash mathbf\_\; =\; -\backslash omega\backslash mathbf\_R$$

$$\backslash mathbf\; =\; \backslash frac\; \backslash mathbf\; =\; \backslash dot\; \backslash mathbf=\; \backslash dot\; R\; \backslash mathbf\_R\; +\; R\; \backslash omega\; \backslash mathbf\_$$

$$\backslash mathbf\; =\; \backslash frac\; \backslash mathbf\; =\; \backslash dot\; \backslash mathbf=\; \backslash ddot\; R\; \backslash mathbf\_R+\; \backslash left(\backslash dot\; R\; \backslash omega\; \backslash mathbf\_+\; \backslash dot\; R\; \backslash omega\; \backslash mathbf\_\; \backslash right)+\; R\; \backslash dot\; \backslash omega\; \backslash mathbf\_-\; R\; \backslash omega^2\; \backslash mathbf\_$$

$$\backslash mathbf\; =\; \backslash frac\; \backslash mathbf\; =\; \backslash dot\; \backslash mathbf=\; \backslash dot\backslash ddot\; R\; \backslash mathbf\_R\; +\; \backslash ddot\; R\; \backslash omega\; \backslash mathbf\_$$

+ \left(2\ddot R \omega \mathbf_+ 2\dot R \dot\omega \mathbf_- 2\dot R \omega^2\mathbf_R \right)

+ \dot R \dot\omega \mathbf_+ R \ddot\omega \mathbf_- R \dot\omega \omega\mathbf_

- \dot R \omega^2 \mathbf_- R 2\dot\omega\omega \mathbf_- R \omega^3 \mathbf_

$$\backslash mathbf=\; \backslash left(\backslash dot\backslash ddot\; R\; -\; 3\backslash dot\; R\; \backslash omega^2\; -\; 3R\; \backslash dot\backslash omega\; \backslash omega\; \backslash right)\; \backslash mathbf\_R\; +\; \backslash left(3\; \backslash ddot\; R\; \backslash omega\; +\; 3\backslash dot\; R\; \backslash dot\backslash omega\; +\; R\; \backslash ddot\; \backslash omega\; -\; R\; \backslash omega^3\; \backslash right)\; \backslash mathbf\_$$

Further transformations may involve

curvature = c = (1)/(R) , \omega = (v)/(R) = vc and corresponding derivatives:

\end

Applications

Solving applications dealing with non-uniform circular motion involves force analysis. With a uniform circular motion, the only force acting upon an object traveling in a circle is the centripetal force. In a non-uniform circular motion, there are additional forces acting on the object due to a non-zero tangential acceleration. Although there are additional forces acting upon the object, the sum of all the forces acting on the object will have to be equal to the centripetal force.

&= ma_r \\ &= \frac \\ &= F_c\end

Radial acceleration is used when calculating the total force. Tangential acceleration is not used in calculating total force because it is not responsible for keeping the object in a circular path. The only acceleration responsible for keeping an object moving in a circle is the radial acceleration. Since the sum of all forces is the centripetal force, drawing centripetal force into a free body diagram is not necessary and usually not recommended.

Using

F _{net} = F _{c} , we can draw free body diagrams to list all the forces acting on an object and then set it equal to F _{c} . Afterward, we can solve for whatever is unknown (this can be mass, velocity, radius of curvature, coefficient of friction, normal force, etc.). For example, the visual above showing an object at the top of a semicircle would be expressed as F _{c} = n + mg .

In a uniform circular motion, the total acceleration of an object in a circular path is equal to the radial acceleration. Due to the presence of tangential acceleration in a non uniform circular motion, that does not hold true any more. To find the total acceleration of an object in a non uniform circular, find the vector sum of the tangential acceleration and the radial acceleration.$$\backslash sqrt\; =\; a$$

Radial acceleration is still equal to $\backslash frac$. Tangential acceleration is simply the derivative of the speed at any given point: $a\_t\; =\; \backslash frac$. This root sum of squares of separate radial and tangential accelerations is only correct for circular motion; for general motion within a plane with polar coordinates

(r, \theta) , the Coriolis term $a\_c\; =\; 2\; \backslash left(\backslash frac\backslash right)\backslash left(\backslash frac\backslash right)$ should be added to a _{t} , whereas radial acceleration then becomes $a\_r\; =\; \backslash frac\; +\; \backslash frac$.

See also

• Angular momentum
• Equations of motion for circular motion
  • Fictitious force
• Geostationary orbit
• Geosynchronous orbit
• Pendulum (mechanics)
• Reactive centrifugal force
• Reciprocating motion
  • Sling (weapon)

External links

• Physclips: Mechanics with animations and video clips from the University of New South Wales
• Circular Motion – a chapter from an online textbook, Mechanics, by Benjamin Crowell (2019)
• Circular Motion Lecture – a video lecture on CM
• https://www.amazon.com/Unique-Perceptions-Physics-Commentaries-solutions/dp/1542776058 – an online textbook with different analysis for circular motion

Notes and References

1. Book: Knudsen . Jens M. . Elements of Newtonian mechanics: including nonlinear dynamics . Hjorth . Poul G. . Springer . 2000 . 3-540-67652-X . 3 . 96.
2. Gomez . R W . Hernandez-Gomez . J J . Marquina . V . 25 July 2012 . A jumping cylinder on an inclined plane . Eur. J. Phys. . IOP . 33 . 5 . 1359–1365 . 1204.0600 . 2012EJPh...33.1359G . 10.1088/0143-0807/33/5/1359 . 55442794 . 25 April 2016.