Study Material

# Angular acceleration

We explain what angular acceleration is, how to calculate it and we give several examples

The  angular acceleration  is the change affects the angular velocity taking into consideration a time unit. It is represented by the Greek letter alpha, α. Angularsacceleration is a vector quantity; therefore, it consists of module, direction, and sense.

The unit of measurement for angularsacceleration in the International System is the radian per second squared. In this way, the angularsacceleration makes it possible to determine how the angular velocity varies over time. Angular acceleration associated with uniformly accelerated circular motions is often studied.

Thus, in a uniformly accelerated circular motion the value of the angularsacceleration is constant. On the contrary, in a uniform circular motion the value of the angular acceleration is zero. Angular acceleration is the equivalent in circular motion to tangential or linear acceleration in rectilinear motion.

In fact, its value is directly proportional to the value of the tangential acceleration. Thus, the greater the angular acceleration of the wheels of a bicycle, the greater the acceleration it experiences.

In the same way, angular acceleration is also present in a Ferris wheel, since it experiences a uniformly accelerated circular motion when it begins its movement. Of course, angular acceleration can also be found on a merry-go-round.

## How to calculate angular acceleration?

In general, the instantaneous angular acceleration is defined from the following expression:

α = dω / dt

In this formula ω is the angular velocity vector, and t is time.

The mean angular acceleration can also be calculated from the following expression:

α = ∆ω / ∆t

For the particular case of a plane motion, it happens that both the angular velocity and the angular acceleration are vectors with a direction perpendicular to the plane of motion.

On the other hand, the modulus of the angularsacceleration can be calculated from the linear acceleration by means of the following expression:

α = a / R

In this formula a is the tangential or linear acceleration; and R is the radius of gyration of the circular motion.

### Uniformly accelerated circular motion

As already mentioned above, angularsacceleration is present in uniformly accelerated circular motion. For this reason, it is interesting to know the equations that govern this movement:

ω = ω 0 + α ∙ t

θ = θ 0 + ω 0 ∙ t + 0.5 ∙ α ∙ t 2

ω 2 = ω 2 + 2 ∙ α ∙ (θ – θ 0 )

In these expressions θ is the angle traveled in circular motion, θ 0 is the initial angle, ω 0 is the initial angular velocity, and ω is the angular velocity.

### Torque and angular acceleration

In the case of linear motion, according to Newton’s second law, a force is required for a body to acquire a certain acceleration. This force is the result of multiplying the mass of the body and the acceleration it has experienced.

However, in the case of a circular motion, the force required to impart angularsacceleration is called torque. Ultimately, torque can be understood as an angular force. It is denoted by the Greek letter τ (pronounced “tau”).

Similarly, it must be taken into account that in a rotational movement, the moment of inertia I of the body plays the role of mass in linear movement. In this way, the torque of a circular motion is calculated with the following expression:

τ = I α

In this expression I is the moment of inertia of the body with respect to the axis of rotation.

## Examples of angular acceleration

### First example

Determine the instantaneous angularsacceleration of a body moving in a rotational motion, given an expression of its position in the rotation Θ (t) = 4 t 3 i. (I being the unit vector in the direction of the x axis).

Similarly, determine the value of the instantaneous angular acceleration 10 seconds after the start of the motion.

#### Solution

From the expression of the position, the expression of the angular velocity can be obtained:

ω (t) = d Θ / dt = 12 t 2 i (rad / s)

Once the instantaneous angular velocity has been calculated, the instantaneous angular acceleration can be calculated as a function of time.

α (t) = dω / dt = 24 ti (rad / s 2 )

To calculate the value of the instantaneous angular acceleration after 10 seconds, it is only necessary to substitute the value of time in the previous result.

α (10) = = 240 i (rad / s 2 )

### Second example

Determine the mean angular acceleration of a body undergoing circular motion, knowing that its initial angular velocity was 40 rad / s and that after 20 seconds it has reached the angular velocity of 120 rad / s.

#### Solution

From the following expression, the mean angular acceleration can be calculated:

α = ∆ω / ∆t

α = (ω f  – ω 0 ) / (t f – t 0 ) = (120 – 40) / 20 = 4 rad / s

### Third example

What will be the angular acceleration of a Ferris wheel that begins to move in a uniformly accelerated circular motion until, after 10 seconds, it reaches the angular velocity of 3 revolutions per minute? What will be the tangential acceleration of the circular motion in that period of time? The radius of the Ferris wheel is 20 meters.

#### Solution

First, it is necessary to transform the angular velocity from revolutions per minute to radians per second. For this, the following transformation is carried out:

ω f = 3 rpm = 3 ∙ (2 ∙ ∏) / 60 = ∏ / 10 rad / s

Once this transformation has been carried out, it is possible to calculate the angular acceleration since:

ω = ω 0 + α ∙ t

∏ / 10 = 0 + α ∙ 10

α = ∏ / 100 rad / s 2

And the tangential acceleration results from operating the following expression:

α = a / R

a = α ∙ R = 20 ∙ ∏ / 100 = ∏ / 5 m / s 2

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