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This is a simulation of a ball experiencing uniform circular motion, which means it travels in a circle at constant speed. Use the sliders to adjust the speed and the radius of the path.

If you show the vectors, you will see the ball's velocity vector, in blue, and its acceleration vector, in green. The velocity vector is always tangent to the circle, and the acceleration vector always points toward the center of the circle.

If the vectors are shown, and the ball has gone about three-quarters of the way around the circle, you will also see a vector triangle. The vector triangle is showing why the ball's acceleration vector points toward the center. We're thinking about which way the acceleration vector points at the bottom of the circle. The purple vector shows the velocity just before the ball reaches the bottom point, so we can consider it to be the initial velocity. The light blue vector shows the velocity just after the ball passes through the bottom point, so we can consider it to be the final velocity. The black vector therefore represents the change in velocity at the bottom point - note that the change in velocity points up, which is toward the center of the circle for that lowest point. The acceleration is proportional to the change in velocity (it is the change in velocity divided by the time it takes the ball to move from the purple point to the light blue point), so the acceleration points in the same direction as the change in velocity - toward the center.

Simulation written by Andrew Duffy, and first posted on 8-09-2017.

This work by Andrew Duffy is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
This simulation can be found in the collection at http://physics.bu.edu/~duffy/classroom.html.
The counter has been running on this page since 8-10-2018. The number of people accessing the page since then is:

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This is a simulation of a ball experiencing uniform circular motion, which means it travels in a circle at constant speed. Use the sliders to adjust the speed and the radius of the path. If you show the vectors, you will see the ball's velocity vector, in blue, and its acceleration vector, in green.

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Circular motion plays an important role in nature and technology. So, the planets move on (approximately) circular orbits around the sun. Other examples are the rotating armature of an electric motor or the crankshaft of a gasoline engine.

This HTML5 app simulates such a circular motion and demonstrates how position, velocity, acceleration and acting force vary in time. The "Reset" button brings the rotating body in its initial position. You can start or stop and continue the simulation with the other button. If you choose the option "Slow motion", the movement will be ten times slower. You can adjust radius, period and mass by using the corresponding input fields. The radio buttons give the possibility to select one of four physical sizes.

This browser doesn't support HTML5 canvas!

The radius vector (red) joins the center of rotation (the origin of the coordinate system) to the revolving body. The velocity vector (violet) is tangential to the circle and perpendicular to the radius vector. The vector of acceleration (blue), surprisingly, is directed inside (towards the center). Here acceleration doesn't mean an increase or decrease of speed (magnitude of velocity), but a varying direction of motion. The same is true for the force (green) on the moving body. The terms centripetal acceleration and centripetal force express that these vectors are directed towards the center of the circular motion.

# data[].attributes.attachment_link.description
Circular motion plays an important role in nature and technology. So, the planets move on (approximately) circular orbits around the sun. Other examples are the rotating armature of an electric motor or the crankshaft of a gasoline engine. This HTML5 app simulates such a circular motion and demonstrates how position, velocity, acceleration and acting force vary in time.

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This animation illustrates the kinematics of an object travelling in uniform circular motion. Velocity is the rate of change of position and acceleration is the rate of change of velocity. The expression for the magnitude of acceleration can be determined by a fairly simple geometric argument.

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Uniform Circular Motion and Centripetal Force

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Centripetal force and acceleration intuition

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The direction of the force in cases of circular motion at constant speeds

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Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug's x,y position, velocity, and acceleration using vectors or graphs.

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The radian is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees (expansion at ). The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit.

The symbol rad can be used to represent the radian (for example, an angle of 1.3 radians would be written as 1.3 rad), but this is often omitted, especially in mathematical writing.

Definition

Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, , where θ is the subtended angle in radians, s is arc length, and r is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, .

As the ratio of two lengths, the radian is a " pure number" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted[citation needed ]. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant the symbol ° is used. The radian is defined as 1.

It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or , or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.

The relation can be derived using the formula for arc length. Taking the formula for arc length, or . Assuming a unit circle; the radius is therefore one. Knowing that the definition of radian is the measure of an angle that subtends an arc of a length equal to the radius of the circle, we know that . This can be further simplified to . Multiplying both sides by gives .

The concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure. Prior to the term radian becoming widespread, the unit was commonly called circular measure of an angle.

The idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi (c. 1400) used so-called diameter parts as units where one diameter part was radian and they also used sexagesimal subunits of the diameter part.

The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson (brother of Lord Kelvin) at Queen's College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, vacillated between the terms rad, radial, and radian. In 1874, after a consultation with James Thomson, Muir adopted radian. The name radian was not universally adopted for some time after this. Longmans' School Trigonometry still called the radian circular measure when published in 1890.

Unit symbol

The International Bureau of Weights and Measures and International Organization for Standardization specify rad as the symbol for the radian. Alternative symbols used 100 years ago are c (the superscript letter c, for "circular measure"), the letter r, or a superscript R, but these variants are infrequently used as they may be mistaken for a degree symbol (°) or a radius (r). So, for example, a value of 1.2 radians would most commonly be written as 1.2 rad; other notations include 1.2 r, 1.2 rad, 1.2 c, or 1.2 R.

Conversions

Conversion between radians and degrees

As stated, one radian is equal to 180/π degrees. Thus, to convert from radians to degrees, multiply by 180/π.

For example:

Conversely, to convert from degrees to radians, multiply by π/180.

For example:

Radians can be converted to turns (complete revolutions) by dividing the number of radians by 2π.

Radian to degree conversion derivation

The length of circumference of a circle is given by , where is the radius of the circle.

So the following equivalent relation is true:

[Since a sweep is needed to draw a full circle]

By the definition of radian, a full circle represents:

Combining both the above relations:

Conversion between radians and gradians

radians equals one turn, which is by definition 400 gradians (400 gons or 400 g). So, to convert from radians to gradians multiply by , and to convert from gradians to radians multiply by . For example,

Advantages of measuring in radians

In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians. This is because radians have a mathematical "naturalness" that leads to a more elegant formulation of a number of important results.

Most notably, results in analysis involving trigonometric functions are simple and elegant when the functions' arguments are expressed in radians. For example, the use of radians leads to the simple limit formula

which is the basis of many other identities in mathematics, including

Because of these and other properties, the trigonometric functions appear in solutions to mathematical problems that are not obviously related to the functions' geometrical meanings (for example, the solutions to the differential equation , the evaluation of the integral , and so on). In all such cases it is found that the arguments to the functions are most naturally written in the form that corresponds, in geometrical contexts, to the radian measurement of angles.

The trigonometric functions also have simple and elegant series expansions when radians are used; for example, the following Taylor series for sin x :

If x were expressed in degrees then the series would contain messy factors involving powers of π/180: if x is the number of degrees, the number of radians is , so

Mathematically important relationships between the sine and cosine functions and the exponential function (see, for example, Euler's formula) are, again, elegant when the functions' arguments are in radians and messy otherwise.

Dimensional analysis

Although the radian is a unit of measure, it is a dimensionless quantity. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is equal to the ratio of the length of the enclosed arc to the length of the circle's radius. Since the units of measurement cancel, this ratio is dimensionless.

Although polar and spherical coordinates use radians to describe coordinates in two and three dimensions, the unit is derived from the radius coordinate, so the angle measure is still dimensionless.

Use in physics

The radian is widely used in physics when angular measurements are required. For example, angular velocity is typically measured in radians per second (rad/s). One revolution per second is equal to 2π radians per second.

Similarly, angular acceleration is often measured in radians per second per second (rad/s ²).

For the purpose of dimensional analysis, the units of angular velocity and angular acceleration are s −1 and s −2 respectively.

Likewise, the phase difference of two waves can also be measured in radians. For example, if the phase difference of two waves is ( k⋅2π) radians, where k is an integer, they are considered in phase, whilst if the phase difference of two waves is (), where k is an integer, they are considered in antiphase.

SI multiples

Metric prefixes have limited use with radians, and none in mathematics. A milliradian (mrad) is a thousandth of a radian and a microradian (μrad) is a millionth of a radian, i.e. .

There are 2 × 1000 milliradians (≈ 6283.185 mrad) in a circle. So a trigonometric milliradian is just under of a circle. This "real" trigonometric unit of angular measurement of a circle is in use by telescopic sight manufacturers using (stadiametric) rangefinding in reticles. The divergence of laser beams is also usually measured in milliradians.

An approximation of the trigonometric milliradian (0.001 rad) is used by NATO and other military organizations in gunnery and targeting. Each angular mil represents of a circle and is % or 1.875% smaller than the trigonometric milliradian. For the small angles typically found in targeting work, the convenience of using the number 6400 in calculation outweighs the small mathematical errors it introduces. In the past, other gunnery systems have used different approximations to ; for example Sweden used the streck and the USSR used . Being based on the milliradian, the NATO mil subtends roughly 1 m at a range of 1000 m (at such small angles, the curvature is negligible).

Smaller units like microradians (μrad) and nanoradians (nrad) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. More common is arc second, which is rad (around 4.8481 microradians). Similarly, the prefixes smaller than milli- are potentially useful in measuring extremely small angles.

Notes and references

External links

# data[].attributes.attachment_link.description
The radian is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees (expansion at ).

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Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug's x,y position, velocity, and acceleration using vectors or graphs.