Angular Acceleration - Definition, Formula, Angular Acceleration Formula

In this article, we understand the concept of angular acceleration its formula and units. It also includes the concept of angular velocity and the relation between linear and angular acceleration.

Angular velocity:

In rotational kinematics, we discuss the concept of the rotational motion of an object. Suppose an object rotates around a centre or a specific point in a particular given time t, then the velocity of the object is defined as the angular velocity. It is also termed as the rate of change of angle θ with time t. The symbol of angular velocity is omega (ω).

Therefore, angular velocity, ω=dθ / dt

Unit of angular velocity: Radian per second

Angular velocity has SI units as 1/s or s^-1

Angular velocity explains the speed at which the object rotates and gives the direction of the rotating object. The direction of angular velocity is perpendicular to the plane of rotation. The angular velocity directs away from us if the object is moving clockwise while it directs towards us if the object is moving anti-clockwise.

Figure 1 Representation of the direction of angular velocity ω with the radius of circle r

Suppose an object moves in a circle with constant speed, the motion of the object is said to be a uniform circular motion. Then the object has constant angular velocity in a uniform circular motion.

Average angular velocity: It is obtained by dividing angular displacement by the time taken for displacement.

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What is angular acceleration?/ Define angular acceleration.

Angular acceleration is defined as the rate of change of angular velocity.

Figure 2 Angular acceleration

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The angular acceleration is denoted by the symbol α. The formula for angular acceleration is,

α=dω / dt

Here, ω is the angular velocity changing with time t.

Unit of angular acceleration is measured as a change in angle per unit time squared.

SI unit of angular acceleration: Radians per squared second

Dimension of angular acceleration: [M⁰L⁰T^-2]

Angular acceleration is also known as rotational acceleration. Similar to angular velocity, the direction of angular acceleration is also perpendicular to the plane of rotation. It also directs away from the viewer if the object is moving clockwise and directs towards him if the object is moving anti-clockwise.

Angular acceleration equation

As mentioned above the angular acceleration is given by, α=dω / dt

For a finite change, α=∆ω/∆t

Then the angular acceleration equation is written as, α=ω₂-ω₁ / t₂-t₁

Here, ω₁ and ω₂ are the initial and final angular velocities; t₁ and t₂ are the initial and final time.

Angular acceleration formula with radius

According to one-dimensional kinematics, linear acceleration is represented by,

a_t=∆v / ∆t ……………………………………......................... (1)

Here, v is the velocity changing with time t.

In rotational motion, velocity v=rω…………………...... (2)

Here, r is the radius of the circle and ω is the angular velocity.

Substituting equation (2) in equation (1) then,

a_t=∆(rω) / ∆t ……………………………………………..….. (3)

We know that for a circular motion, the radius is constant. Hence,

∆(rω)=r(∆ω) ………………………………...……….. (4)

we get,

Linear acceleration, a_t=r ∆ω / ∆t ..………………………… (5)

Angular acceleration is defined as the rate of change of angular velocity.

Therefore, α=∆ω / ∆t ………………………………………. (6)

On comparing equations (5) and (6) we obtain,

a_t=rα

Or, angular acceleration α=a_t / r

Thus, angular acceleration is inversely proportional to the radius of the circle.

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Angular velocity and angular acceleration

Angular velocity is applicable in determining the rotation of a body concerning the inertial frame of reference. While to determine the rate of change of angular velocity and to depict the direction of the change, angular acceleration is used.

The rotational motion under constant angular acceleration

An object moving with a rotational motion under constant acceleration measures the angular velocity of the object with the help of equations called kinematic equations. The equations are,

ω_f=ω_i+αt

θ=ω_it+0.5αt²

ω_f²=ω_i²+2α(θ)

Angular force

If the object is rotating along a curved path, the force acting on that object is called angular force.

Angular force formula is given by, τ=Iα

Omega

Omega is defined as the symbolic representation of angular velocity. It is represented as radian per second.

We know that, ω=2πf

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