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Angular Acceleration

Picture this - you're lucky enough to have seats right in front of the finish line at an F1 race! As the cars race down the straight towards you, you keep them in the centre of your vision. While they are far away down the straight, you only have to turn your head slowly to keep the cars central. However…

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# Angular Acceleration

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Picture this - you're lucky enough to have seats right in front of the finish line at an F1 race! As the cars race down the straight towards you, you keep them in the centre of your vision. While they are far away down the straight, you only have to turn your head slowly to keep the cars central. However as the cars get closer, you have to turn faster and faster to keep your eyes on them! You've just had to accelerate the rotation of your head, and this can be described as angular acceleration. This article defines angular acceleration, its formula and its units, and it introduces some example calculations and relates angular and linear acceleration in more detail.

## Angular acceleration definition

We use linear acceleration to describe a change in linear velocity, but how do we describe a change in the rate of rotation of a spinning object? As the rate of rotation is the angular velocity, it may not surprise you that angular accelerationis the rotational equivalent of linear acceleration. While linear acceleration describes the rate of change of linear velocity, angular acceleration is the rate of change of angular velocity $$\omega$$.

Diagram showing how the key quantities in circular motion vary as an object rotates with an angular acceleration of 0.6 rad/s for 1 second, Vaia Originals.

Similarly to angular velocity, convention states that angular acceleration which results in an increase in the rate of counter-clockwise rotation is positive, while an increase in the rate of clockwise rotation is caused by negative angular acceleration.

Angular acceleration is a pseudoscalar unit. This means that it behaves like a scalar unit as it only requires a magnitude to be fully defined, but it changes sign depending on the direction from which you are looking: a ceiling fan might go anticlockwise from below, but if you look at it from above it would go clockwise! Depending on the frame of reference, positive angular acceleration can always increase the rate of rotation in either direction. To define which direction acceleration is acting in, we choose a frame of reference, and then convention states that a positive sign indicates angular acceleration acting to increase the rate of clockwise rotation and a negative sign indicates an increase in the counter-clockwise rate of rotation.

## Angular acceleration units

The SI unit for angular velocity is radians-per-second, defining the angle an object rotates through every second. Angular acceleration defines the amount that the angular velocity changes each second, so its units are the unit for angular velocity per second: (radians-per-second)-per second. Radians-per-second-per-second is equivalent to radians-per-second-squared, as shown below:

$\alpha=\dfrac{\left(\dfrac{\text{radians}}{\text{second}}\right)}{\text{second}}=\dfrac{\text{radians}}{\text{seconds}^2}$

When studying circular motion, the standard unit for dealing with angles is the radian. A full $$360^\circ$$ rotation contains $$2\pi$$ radians, meaning:

$$360^\circ=2\pi \,\text{radians}$$, so $$1\,\text{radian}=\dfrac{360^\circ}{2\pi}=57.3^\circ$$.

To convert an angle $$\theta_{\text{degrees}}$$ into radians, this can be found as $$\theta_{\text{radians}}=\dfrac{\theta_{\text{degrees}}}{360^\circ}\times 2\pi$$.

Similarly, to convert from radians to degrees, we can use $$\theta_{\text{degrees}}=\dfrac{\theta_{\text{radians}}}{2\pi}\times 360^\circ$$.

## Angular acceleration formula

To find the angular acceleration of an object, we need to know its angular velocity at two points in time. We can then calculate the amount that the angular velocity changed each second, assuming a constant rate of angular acceleration between the two points. This gives us the angular acceleration $$\alpha$$:

$\alpha=\dfrac{\Delta\omega}{\Delta t}=\dfrac{\omega_f-\omega_i}{t_f-t_i}$

where the subscript '$$f$$' means 'final' and '$$i$$' means 'initial'.

## Example angular acceleration calculation

The diagram below shows a flywheel which is initially stationary, is accelerated for 5 seconds and then left to spin freely for 10 seconds, undergoing some friction. The angular velocity is measured at each of these points and indicated in the diagram.

Example - A flywheel is accelerated for 5 seconds until it has an angular rotation of 10 rad/s. It then spins for 10 seconds, after which its angular velocity has decreased to 9 rad/s, Vaia Originals.
1. Determine the angular acceleration while the flywheel is accelerated and while it spins freely.
2. Plot the angular velocity and acceleration against time.

To find the angular acceleration in each period, we can use the formula for angular acceleration as we know the initial and final angular velocities. We call the angular acceleration undergone in the first 5 seconds $$\alpha_1$$ and that in the next 10 seconds $$\alpha_2$$ and calculate:

$\alpha_1=\dfrac{10\,\mathrm{rad/s}-0\,\mathrm{rad/s}}{5\,\mathrm{s}-0\,\mathrm{s}}=2\,\mathrm{rad/s}^2$

$\alpha_2=\dfrac{9\,\mathrm{rad/s}-10\,\mathrm{rad/s}}{15\,\mathrm{s}-5\,\mathrm{s}}=-0.1\,\mathrm{rad/s}^2$

To plot the velocity and acceleration against time, we plot the values at each of our known time points ($$0\, \mathrm{s}$$, $$5\,\mathrm{s}$$ and $$15\,\mathrm{s}$$) and connect them with straight lines because the angular accelerations are constant in each period.

A plot of the angular velocity (yellow) and the angular acceleration (blue) of the flywheel from 0 to 15 seconds, Vaia Originals.

## Relating angular acceleration to linear acceleration

In circular motion, the angular displacement $$\theta$$ is the equivalent of the displacement $$s$$ in the study of linear motion. The kinematic equations for velocity, acceleration and displacement have angular equivalents as well.

 Quantity Linear equation Angular equation Velocity $$v=\dfrac{\Delta x}{\Delta t}$$ $$\omega =\dfrac{\Delta \theta}{\Delta t}$$ Acceleration $$a=\dfrac{\Delta v}{\Delta t}$$ $$\alpha=\dfrac{\Delta \omega}{\Delta t}$$ Displacement $$s=v_i(t_f-t_i)+\frac{1}{2}a(t_f-t_i)^2$$ $$\theta=\omega_i(t_f-t_i)+\frac{1}{2}\alpha(t_f-t_i)^2$$ $$v_f^2-v_i^2=2as$$ $$\omega_f^2-\omega_i^2=2\alpha\theta$$

A fan is stationary at an angular displacement of 90 degrees ($$\frac{\pi}{2}\,\mathrm{rad}$$). When the fan is switched on at $$t=0\,\mathrm{s}$$, it begins to rotate with an angular acceleration of $$2\pi\,\mathrm{rad/s}^2$$. Find the angular velocity and angular displacement of the fan at $$t=3\,\mathrm{s}$$.

To find the angular velocity of the fan, we can rearrange the angular kinematic equation for acceleration:

$\alpha=\dfrac{\omega_f-\omega_i}{t_f-t_i}$

so

$\omega_f=\alpha(t_f-t_i)+\omega_i$

Therefore, the angular velocity $$\omega_f$$ of the fan after accelerating is

$\omega_f=2\pi\,\mathrm{\tfrac{rad}{s^2}}\times (3\,\mathrm{s}-0\,\mathrm{s})+0\,\mathrm{\tfrac{rad}{s}}=6\pi,\mathrm{\tfrac{rad}{s}}$

To find the angular displacement of the fan after accelerating for 3 seconds, we can use the equation for displacement:

\begin{align}\theta &=\omega_i(t_f-t_i)+\frac{1}{2}\alpha(t_f-t_i)^2=\\&=0\,\mathrm{\tfrac{rad}{s}}\times (3\,\mathrm{s}-0\,\mathrm{s})+\frac{1}{2}\times 2\pi\,\mathrm{\tfrac{rad}{s^2}}\times (3\,\mathrm{s}-0\,\mathrm{s})^2=\\&=9\pi\,\mathrm{rad}\end{align}

This gives us the amount of displacement that occurred over the time period, so to find the current displacement we need to add the initial angular displacement of $$\frac{\pi}{2}\,\mathrm{rad}$$. Therefore, the angular displacement of the fan at 3 seconds is $$9.5\pi\,\mathrm{rad}$$. However, as there are only $$2\pi\,\mathrm{rad}$$ in a full rotation, this displacement can be simplified to $$1.5\,\mathrm{rad}$$, equivalent to an angle of $$270^\circ$$.

## Angular Acceleration - Key takeaways

• Angular acceleration α is the rotational equivalent of linear acceleration. While linear acceleration describes the rate of change of linear velocity, angular acceleration is the rate of change of angular velocity $$\omega$$.
• Angular acceleration is defined in SI units of radians-per-second squared ($$\mathrm{rad/s^2}$$).
• To find an object's angular acceleration, we divide the change in angular velocity by the change in time. This gives an average change in angular velocity per second, which is the angular acceleration.
• Convention states that a positive angular acceleration increases the counter-clockwise rate of rotation, while negative angular acceleration acts to increase clockwise rotation.
• When studying circular motion, the standard unit for dealing with angles is the radian. A full $$360^\circ$$ rotation contains $$2\pi$$ radians, meaning one radian equals $$57.3$$ degrees.

The angular acceleration is the rate of change of angular velocity. To find the average angular acceleration between two points in time (t1 & t2), we need to know the angular velocity at those two points (ω1 & ω2). We can then use the equation below to find the average angular acceleration:

angular acceleration α = (ω- ω1) / (t- t1

As angular acceleration defines the rate that the angular velocity is increasing or decreasing at, we need to know the angular velocity (ω1 & ω2) at two points in time (t1 & t2) . The average angular acceleration between these times can then be found using the below equation:

angular acceleration α = (ω- ω1) / (t- t1

Angular acceleration α is the rotational equivalent of linear acceleration. While linear acceleration describes the rate of change of linear velocity, angular acceleration is the rate of change of angular velocity ω. It is defined in SI units of radians-per-second squared.

An object’s moment of inertia I defines how resistant it is to angular acceleration around a specific axis. To accelerate an object's rotation, it's necessary to apply an external torque τ (a twisting force). The relationship between an object’s moment of inertia, the applied torque and resulting angular acceleration α is given by the equation below:

α =τ /I

The angular acceleration defines the rate of change of angular velocity, meaning it is the time derivative of an object’s angular velocity. The equation to find angular acceleration between two points is shown below:

Angular acceleration α = (ω- ω1) / (t- t1

This can also be written in derivative form as:

angular acceleration α=dω/dt

## Angular Acceleration Quiz - Teste dein Wissen

Question

Linear acceleration represents the rate of change of linear velocity. What does angular acceleration represent?

The rate of change of velocity of a body moving along a circular path.

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Question

If a stationary object undergoes an angular acceleration of $$5\,\mathrm{\tfrac{rad}{s^2}}$$, which direction will it start rotating?

Clockwise.

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Question

An object rotates clockwise at a rate of $$4\,\mathrm{\tfrac{rad}{s}}$$. If it undergoes a positive angular acceleration of $$2\,\mathrm{\tfrac{rad}{s^2}}$$, what will its angular velocity be after 1 second?

$$-2\,\mathrm{\tfrac{rad}{s}}$$. The angular speed decreases, as the object is initially rotating clockwise but a positive angular acceleration acts in a counter-clockwise direction.

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Question

What is the SI unit for angular acceleration?

Degrees-per-second.

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Question

How many radians are there in one 360-degree rotation?

There are $$2\pi$$ radiants in a full rotation.

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Question

How many degrees is one radian?

$$57.3$$ degrees.

There are $$2\pi$$ radians in $$360$$ degrees, therefore:

$\frac{360^\circ}{2\pi}=57.3^\circ$

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Question

Convert $$100$$ degrees into radians.

$$\theta_{\mathrm{radians}}=\dfrac{\theta_{\mathrm{degrees}}}{360^\circ}\times 2\pi$$

$$\theta_{\mathrm{radians}}=\dfrac{100^\circ}{360^\circ}\times 2\pi=1.75\,\mathrm{rad}$$.

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Question

$$\theta_{\mathrm{degrees}}=\dfrac{\theta_\mathrm{radians}}{2\pi}\times 360^\circ$$

$$\theta_{\mathrm{degrees}}=\dfrac{1.1\,\mathrm{rad}}{2\pi\,\mathrm{rad}}\times 360^\circ=63^\circ$$.

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Question

How do we find the angular acceleration of a rotating object?

To find the angular acceleration of an object, we need to know its angular velocity at two points in time, allowing us to find the amount that the angular velocity changed each second. This gives us the angular acceleration $$\alpha$$:

$$\alpha=\frac{\Delta\omega}{\Delta t}=\frac{\omega_f-\omega_i}{t_f-t_i}$$.

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Question

What is the equation to find angular velocity?

To find the angular velocity $$\omega$$ of a rotating object, we need to determine its rate of change of angular displacement $$\theta$$ by recording the position at two points in time:

$$\omega=\frac{\Delta\omega}{\Delta t}=\frac{\theta_f-\theta_i}{t_f-t_i}$$.

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Question

An object is initially stationary, and after 5 seconds it is rotating at $$10\,\mathrm{rad/s}$$. What was its angular acceleration?

Assuming constant acceleration, it is:

$$\dfrac{10\,\mathrm{rad/s}-0\,\mathrm{rad/s}}{5\,\mathrm{s}-0,\mathrm{s}}=\frac{10\,\mathrm{rad/s}}{5\,\mathrm{s}}=2\,\mathrm{rad/s^2}$$.

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Question

A flywheel is initially spinning at $$1500\,\mathrm{rad/s}$$, then its energy is transferred to a generator over 8 seconds until it is stationary. What angular acceleration did the generator impart?

$$\dfrac{0\,\mathrm{\tfrac{rad}{s}}-1500\,\mathrm{\tfrac{rad}{s}}}{8\,\mathrm{s}-0\,\mathrm{s}}=\dfrac{-1500\,\mathrm{\tfrac{rad}{s}}}{8\,\mathrm{s}}=-187.5\,\mathrm{\tfrac{rad}{s^2}}$$

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Question

True or false - Angular acceleration is a measurement of how many degrees an object rotates each second.

True.

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Question

True or false - a positive angular acceleration means an increasing rate of clockwise rotation.

False - counter-clockwise angular rotation is positive, so a positive angular acceleration decreases the rate of clockwise rotation.

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