Angular Acceleration: Formula Concept Unit

Angular Acceleration: Theory

Definition of Angular Acceleration: The rate of change of angular velocity with respect to time is called angular acceleration. Alternatively, the change of angular velocity per unit of time is also called angular acceleration.

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Let us consider, that the initial angular velocity of a particle is ω₁. After time t its final angular velocity is ω₂. Therefore the change in angular velocity is Δω = ω₂ − ω₁. So, by definition –

$$Angular\:acceleration (\alpha ) = \frac{Change\:in\:angular\:velocity }{time(t)}$$

$$\overrightarrow{\alpha } = \frac{\Delta \overrightarrow{\omega }}{t} = \frac{\overrightarrow{\omega _{2}}- \overrightarrow{\omega _{1}}}{t}$$

Instantaneous Angular Acceleration: It is the ratio of instantaneous change of angular velocity (Δω) to the infinitesimal change of time (Δt).Mathematically,

$$\mathrm{\alpha = \displaystyle \lim_{\Delta t \to 0} \frac{\Delta \omega }{\Delta t} = \frac{d\omega }{dt}}$$

Now,

$$\mathrm{\mathrm{\alpha = \frac{d\omega }{dt}} = \frac{\mathrm{d} }{\mathrm{d} t}(\frac{d\theta }{dt}) = \frac{\mathrm{d^{2}\theta } }{\mathrm{d} t^{2}}}$$

Where θ = angular displacement.

If the angular velocity of an object continues to increase, then the motion of the object consists of ‘angular acceleration’. When an electric fan is switched on, the speed of the fan consists of angular acceleration until it reaches maximum speed. Again, if the angular velocity of an object is constantly decreasing, its motion has ‘angular deacceleration’. In the case of angular deceleration, the value of the final angular velocity ω₂ is less than the initial angular velocity ω₁. As a result, ω₂ – ω₁ = α = a negative quantity. That is, angular deceleration is only negative angular acceleration.

What is the direction of angular acceleration?

Angular acceleration is an axial vector. The direction of angular velocity and angular acceleration may be different. The direction of positive angular acceleration is the direction of angular velocity. But if the angular acceleration is negative, that is, when the angular deceleration occurs, the direction of angular velocity and angular acceleration is opposite to each other.

Angular acceleration is a vector quantity. Its direction is along the direction of angular velocity. Here in this article, one can find the direction of angular velocity.

Alert!! In two-dimensional space, angular acceleration is a pseudoscalar. In three-dimensional space, angular acceleration is a pseudovector.

Unit and Dimension of Angular Acceleration:

The unit of angular acceleration is radian/second².

Dimension of angular acceleration =

$$\frac{dimension\: of\:angular\:velocity}{Dimension\:of\:time}$$

$$= \frac{[T^{-1}]}{[T]}= [T^{-2}]$$

SI unit of Angular Acceleration	rad/sec²
The common unit of Angular Acceleration	degree/sec²
Dimension of Angular Acceleration	[T⁻²]

Relation between Linear acceleration and angular acceleration:

Let us consider a particle rotating along a circular path of radius r. Also, its initial linear velocity is v₁ and angular velocity ω₁. After time t its linear velocity and angular velocity become v₂ and ω₂ respectively.

So, v₁ = ω₁r and v₂ = ω₂r (Since linear velocity = angular velocity × radus).

If f represents the magnitude of linear acceleration then –

$$f=\frac{v_{2} – v_{1}}{t}$$

$$or,\: f= \frac{\omega _{2}r – \omega _{1}r}{t}$$

$$or,\: f= r\times \frac{\omega _{2} – \omega _{1}}{t}$$

Again, we know that the angular acceleration α

$$\alpha= \frac{\omega _{2}- \omega _{1}}{t}$$

$$\therefore\: f = r\times \alpha $$

Linear Acceleration = Angular Acceleration × Radius of circle

The Vector relation of linear and angular acceleration is given below.

$$\therefore\: \overrightarrow{f} = \overrightarrow{r}\times \overrightarrow{\alpha}$$

Angular Acceleration: Example Practice:

Exercise-1: An electric fan is rotating at 210 rpm. By using a regulator, its motion is increased to 630 rpm in 11 seconds. What is the angular acceleration of the electric fan?

The initial velocity ω₁ = 210 × (2π/60) = 7π rad/sec.
The final velocity ω₂ = 630 × (2π/60) = 21π rad/sec.

Now the angular acceleration α = (ω₂−ω₁)/t
or, α = (21π − 7π)/11 = 14π/11
or, α = (14/11) × ((22/7) = 4 rad/sec².

Exercise-2: The angular displacement of a particle is θ = 2πt³ + πt² + 4t − 5. Calculate its angular acceleration at time t = 10 seconds.

We have angular displacement θ = 2πt³ + πt² + 4t − 5.

So the angular acceleration is given by the following equation.

$$\alpha = \frac{d^{2}\theta }{dt^{2}}$$

$$or,\: \alpha = \frac{\mathrm{d} }{\mathrm{d} t}(\frac{\mathrm{d}\theta }{\mathrm{d} t})$$

$$or,\: \alpha = \frac{\mathrm{d} }{\mathrm{d} t}(6\pi t^{2}+2\pi t+4)$$

$$or,\: \alpha = 12\pi t+2\pi $$

So, the angular acceleration at time t =10 seconds is –

$$or,\: \alpha|_{t=10} = 12\pi \times 10+2\pi $$

$$or,\: \alpha|_{t=10} = 122\pi\:rad/sec^{2}$$

Therefore, the angular acceleration of the particle at time t = 10 seconds is 122π rad/sec².

Exercise-3: A cycle moves at a constant speed of 8 m/s in a circular path of radius of 2 m. What is the angular acceleration of the object?

We know that the relationship between angular and linear acceleration is f = αr. where f = linear acceleration and α = angular acceleration and r = radius.

Now the lnear acceleration = centripetal acceleration = f = v²/r = (8)²/2 = 32 m/s² .

Therefore the angular acceleration α = f/r = 32/2= 16 rad/s²

Suggested Readings:

Angular Velocity	Angular Displacement
Speed & Velocity	Displacement: Concept

What is angular acceleration?

Angular acceleration is the time rate of change of angular velocity.

How to find the angular acceleration (α)?

It can be found by dividing the angular velocity (ω) by time (t). α = ω/t.

By which unit the angular acceleration is measured?

In SI and cgs units, the angular acceleration is measured in rad/square sec and degree/square sec respectively.

How does the angular acceleration become zero?

If the angular velocity is constant or zeroes with respect to time, then by the definition α = dω/dt gives us angular acceleration α = zero.

Why is angular acceleration so important?

The rate at which the angle of approach changes is determined by the magnitude of the angular acceleration.

Angular Acceleration Study Notes

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