Angular quantities, also known as angular magnitudes,  as its name suggests, are related to angles. Among other things they allow us to study circular motion. In this section, we are going to present you the main ones. We will see:

The International System (S.I.) unit for measuring angles is the radian (rad). A circumference is 2π radian. Contrariwise, we can also measure angles in degrees. A circle has 360 degrees. Considering the previous relationship, to convert degrees to radians, you can use the following expression:

${\theta }_{\text{degrees}}={\theta }_{\text{radians}}\frac{180}{\mathrm{\pi }}⇔{\theta }_{\text{radians}}={\theta }_{\text{degrees}}\frac{\mathrm{\pi }}{180}$

## Angular Position

The angular position φ is a fundamental angular magnitude which represents the angle that the position vector of a body forms with the positive side of the x-axis or positive x-semiaxis. Its unit in the International System (SI) is the radian (rad).

Angular position φ

The angular position corresponds to the angle that the position vector of a body form with the positive side of the x-axis or positive x-semiaxis

## Angular Displacement

The angular displacement (Δφ) represents the traveled angle. It is given by the difference between φf , the final angular position, and the initial angular position φi:

$∆\phi ={\phi }_{f}-{\phi }_{i}$

Angular displacement Δφ

The angular displacement corresponds to the angle traveled by a body moving from an initial angular position φi to a different final one φf

## Angular Velocity

It represents the angular displacement (Δφ) experienced by a body each second. Its unit in the International System (SI) is the rad/sec although sometimes you will see that you can also use revolutions per minutes, rpm (1 r.p.m. = 2π/60 rad/s). As it happened with velocity, there is an average angular velocity ωa and an instantaneous angular velocity ω (or simply angular velocity) according to the time considered, whether a time interval Δt or an instant of time dt, respectively.

ωa ω
${\omega }_{a}=\frac{∆\phi }{∆t}=\frac{{\phi }_{f}-{\phi }_{i}}{{t}_{f}-{t}_{i}}$
$\omega =\underset{∆t\to 0}{\mathrm{lim}}\frac{∆\phi }{∆t}=\frac{d\phi }{dt}$

## Angular Acceleration

It represents the variation of the angular velocity (Δω) with time. Its unit in the International System (SI) is the rad/s2. Just like with equivalent linear quantities, there is an average angular acceleration αa and an instantaneous angular acceleration α (or simply angular acceleration) according to whether an interval of time ∆t or an instant of time respectively dt is being considered.

αa α
${\alpha }_{a}=\frac{∆\omega }{∆t}=\frac{{\omega }_{f}-{\omega }_{i}}{{t}_{f}-{t}_{i}}$
$\alpha =\underset{∆t\to 0}{\mathrm{lim}}\frac{∆\omega }{∆t}=\frac{d\omega }{dt}$

## Normal or Centripetal Acceleration

Although it is not an angular magnitude properly speaking, because it is not measured in angular units, it is important to remember here the expression for normal or centripetal acceleration.

${a}_{n}=\frac{{v}^{2}}{R}={\omega }^{2}·R$

Remember that normal or centripetal acceleration is responsible for the change of direction of the velocity vector so it appears in all circular motions.

## Relationship between Angular and Linear Quantities

We can relate angular and linear quantities in circular motions using the radius R.

Linear quantity Relationship Angular quantity
s
$s=\phi ·R$
φ
v
$v=\omega ·R$
ω
at
${a}_{t}=\alpha ·R$
α
an
${a}_{n}=\frac{{v}^{2}}{R}={\omega }^{2}·R$
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## Angular Vectors Quantities

So far we have studied the angular quantities as scalars. In reality, they are vector magnitudes, but for the purposes of this level, we will consider them scalars. However, here we anticipate the relationship of linear velocity with angular velocity and tangential acceleration with angular acceleration, in vector form, which is given by the vector product:

$\stackrel{\to }{v}=\stackrel{\to }{\omega }×\stackrel{\to }{R}$

${\stackrel{\to }{a}}_{t}=\stackrel{\to }{\alpha }×\stackrel{\to }{R}$

## Solved exercises worksheet

Here you can test what you have learned in this section.

difficulty

Converts the following angles from radians to degrees or degrees to radians:

a) 230º

#### Angular quantities in tractor tires

difficulty

A farmer has a yellow tractor whose front tires have a radius of 40 cm and the rear tires have a radius of 1.20 m. When the front wheels have completed 10 revolutions, how many revolutions will the rear tires have completed?

## Formulas worksheet

Here is a full list of formulas for the section Angular Quantities. By understanding each equation, you will be able to solve any problem that you may encounter at this level.

Click on the icon   to export them to any compatible external program.

#### Linear space - angular space relationship

$s=\phi ·R$

#### Average angular velocity

${\omega }_{a}=\frac{∆\phi }{∆t}=\frac{{\phi }_{f}-{\phi }_{i}}{{t}_{f}-{t}_{i}}$

#### Angular velocity

$\omega =\underset{∆t\to 0}{\mathrm{lim}}\frac{∆\phi }{∆t}=\frac{d\phi }{dt}$

#### Average angular acceleration

${\alpha }_{a}=\frac{∆\omega }{∆t}=\frac{{\omega }_{f}-{\omega }_{i}}{{t}_{f}-{t}_{i}}$

#### Angular acceleration

$\alpha =\underset{∆t\to 0}{\mathrm{lim}}\frac{∆\omega }{∆t}=\frac{d\omega }{dt}$

#### Normal acceleration equation (u.c.m. and u.a.c.m.)

${a}_{n}=\frac{{v}^{2}}{R}={\omega }^{2}·R$

#### Tangential acceleration - angular acceleration relationship

${a}_{t}=\alpha ·R$

#### Linear velocity - angular velocity relationship

$v=\omega ·R$