The concepts angular velocity and angular acceleration are the most familiar concept to explain how fast a body can change its position and how rapidly it travels along the circular path.

**When you rotate a ball in the circular orbit, it rotates at a certain angle with a certain velocity—this velocity results in acceleration. Let us discuss angular velocity vs angular acceleration in this post.**

**Angular velocity:**

We have discussed the angular velocity in the previous article.

**The differential change of displacement of an object rotating along the circular orbit at an angle ‘θ’, with the time is called the angular velocity.**

The formula for angular velocity is,

**Angular acceleration:**

The concept of angular acceleration is similar to linear acceleration.

**The rate of change of velocity of an object rotating at angle ‘θ’ in a circular orbit with time is called angular acceleration.**

It is denoted by the Greek letter ‘**α**’.

If a body is moving in a circular path with velocity ω_{i }initially and it changes its velocity to ω_{f}, then the acceleration of the moving body is given by

But ∆ω = ω_{f} -ω_{0}

Angular acceleration is given by the difference between angular velocities of initial and final velocity.

**Angular Velocity Vs Angular Acceleration**

The comparison between the angular velocity and angular acceleration is given in the below table, which may help you to understand.

Angular Velocity | Angular Acceleration |

The differentiation of angular displacement with time gives the angular velocity. | The second order differentiation of angular displacement time gives the angular acceleration. |

The unit of angular velocity is radians/second. | The unit of angular acceleration is radians/second^{2} . |

The dimensional formula of angular velocity is [M^{0}L^{0}T^{-1}] | The dimensional formula of angular displacement is [M^{0}L^{0}T^{-2}] |

It has magnitude, but the direction changes with the coordinate axes; hence it is a pseudo vector quantity. | It has magnitude and a specific direction, which remains constant through out the action; hence it is a vector quantity. |

Radius of the circular orbit does not exhibit any effect on the angular velocity. | The radius of the circular orbit exhibits an inverse effect on the angular acceleration. |

**Facts related to angular velocity and angular acceleration:**

- In a two-dimensional space, the angular acceleration can change its sign or can be inverted with the coordinates. It is called pseudo-scalar quantity.

- When the velocity of the rotating object increases, the acceleration is positive.

**When you switch on a fan, it starts rotating from zero and keeps on increasing whenever you turn the nob to get more air. In that case, acceleration is positive.**

- When the velocity decreases while rotating, the angular acceleration will become negative.

**Whenever** **you turn the nob of a fan in an anti-clockwise direction to lower the speed, you can observe negative acceleration.**

- In case of increase in the angular velocity, the angular acceleration and the velocity will be in the same direction.

- The angular acceleration will act opposite to the angular velocity whenever there is a decrease in the velocity.

- In a uniform circular orbit, the velocity vector exhibit constant magnitude.

- Angular acceleration will become zero in a uniform circular orbit.

- Angular acceleration decreases in the rotational path of maximum radius.

**Solved Problems.**

**A car is moving in a circular path. Initially, the angular velocity of the car is 26km/hr, and after 34 min, it increases its speed to 49km/hr. Calculate the angular acceleration of the car.**

Solution:

The initial velocity ω_{i} = 26km/hr

The change in velocity ω_{f} = 49km/hr

Time is 34 min = 0.56 hr

The angular acceleration is

α = 2.15 rad/sec^{2}.

**T****he wheel of a cycle is rotating with an angular acceleration of 12rad/sec**^{2}** in 3seconds. Calculate the angular velocity.**

^{2}

Solution:

The Angular acceleration of the wheel is 12 rad/sec2

Time taken to accelerate is 3 seconds

The angular acceleration is given by

Then the velocity can be written in terms of angular acceleration as

∆ω = α.∆t

∆ω = 12 × 3

∆ω = 36 rad/sec.

**A disc of a radius of 12cm is rotating in a circular path with an angle of 35°. The time taken to complete rotation is 12 seconds. Calculate the angular velocity and hence find out the angular acceleration if it increases its velocity to 4 rad/sec for the same 12 seconds****.**

Solution:

The angle of rotation = ∆θ = 35°

Time taken for one complete rotation ∆t = 12 seconds

Angular velocity is given by the formula

ω = 2.91 rad/sec.

The angular acceleration is given by

The velocity is changed to 4 rad/sec for the same time interval so that the angular acceleration is given by;

α = 0.090 rad/sec^{2}.

**A tire is rotating with an acceleration of 65 rad/sec. ****I****ts change in velocity is given by 92 rad/sec**^{2}**. ****C****alculate the time taken by the tire to gain the given acceleration?**

^{2}

Solution:

The angular acceleration = 65rad/sec^{2}^{}

The angular velocity = 92 rad/sec

The angular acceleration is

Rearranging the above equation, we can calculate the time as,

∆t = 1.41sec.

**Frequently Asked Questions**

**Does the angular velocity depend on the mass of the rotating object?**

Yes, the angular velocity inversely depends on the mass.

**When a freely rotating body of a certain mass is supposed to exert some velocity, if the mass is more, then the velocity decreases.**

**Doe****s the radius affect the**** ****angular acceleration?**

Suppose angular acceleration is maximum, the radius of the rotational path matters.

**Greater the radius of the orbit, the object’s attraction towards the center becomes less. This results in the decrease of the velocity and hence the acceleration.**

**Why does the angular acceleration is zero in a uniform circular orbit?**

Angular acceleration refers to a change in the Angular velocity; either the magnitude has to be changed or the speed.

**In a uniform circular orbit, the velocity remains constant throughout. ****The neither magnitude nor the radius changes****. This shows that there will be no acceleration produced.**

**Does the tangential acceleration and angular acceleration are same?**

There are two types of acceleration;

- Linear acceleration

- Angular acceleration

**When a body is accelerating in the circular orbit, it is said to be angular acceleration. This angular acceleration is further divided as**

**Tangential****acceleration**

**Radial acceleration**

**So tangential acceleration is derived from angular acceleration. It is not the same as angular acceleration.**

**Can the angular acceleration be negative?**

** **The negative angular acceleration depends on the coordinate axis in which the angular velocity is acting.

**When an object is supposed to travel in a circular path with a certain velocity, the change in the velocity gives rise to angular acceleration. If the change in velocity decreases the amount, then the angular acceleration will be negative.**