Angular Speed Formula

You must have heard about speed, but do you know we are talking about a specific type of speed over here. If you haven’t heard of angular speed, you’re at the right place. This article will help you understand how to calculate the angular speed formula of an object. Moreover, you will learn about the difference in the angular speed and angular velocity.

Definition

Speed is a term that is used in various contexts. For instance, how fast you are driving your car or how fast you pitch a ball. Similarly, speed is basically referring to how slow or fast the object is moving. Thus, angular speed is how quickly an object rotates. In other words, it is described as the change in the angle of the object per unit of time.

Therefore, if we want to calculate the speed of the rotational motion, we will require the angular speed of it. Angular speed formula calculates the distance the body covers in terms of revolutions or rotations to the time taken.

Furthermore, radian is quite an important thing here. Whenever we calculate the angular speed, the angle we measure is in radians. Radians are a way of measuring angles where we define the right angle as pi/2 radians. Therefore, one full revolution will contain around 6.28 radians.

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Formula

We see that angular speed is the rate at which an object changes it angles which we measure in radians in a given time. Angular speed has a magnitude (a value) only.

ω = θ /t

Over here:

ω refers to the angular speed in radians/sec

θ is the angle in radians (2π radians = 360 degrees)

t refers to the time, sec

It is important to note that angular speed and angular velocity make use of the same formula. However, the difference between the two is that angular speed is scalar quantity whereas angular velocity is a vector quantity.

Solved Questions on Angular Speed 

Question- Calculate the angular speed of the earth which rotates on its axis every 24 hours.

Answer- Over here we see that angle traversed, 1 rotation refers to θ = 2π. Further, the time it takes to rotate is 24 hours, so t = 24 hr. First of all, we need to convert the hours into second, we will do that by:

t = 24 hr × 60 min/hr × 60 sec/min = 86400 sec

Further, upon applying the angular speed formula and placing the figures accordingly, we will get:

ω = θ /t

ω = 2π/86400 sec

ω = 0.0000726 radians/sec = 7.26 × 10^-5 rad/sec

Therefore, the angular speed of the earth is 7.27 × 10^-5 rad/sec

Question- There is a carnival happening where children are flocking to the Ferris wheel in groups. Further, we see that there is a signboard which states that the angular speed of the Ferris wheel is 0.13 rad/sec.  Calculate the number of revolutions completed by the wheel within the time duration of 12 minutes.

Answer- After looking at the figures, we see that we have our angular speed, as, ω = 0.13 rad/sec. Further, time is t = 12 min. Now, we will first convert the minutes in seconds which means

t = 12 min × 60 sec/min = 720 sec.

Now we will use the equation ω = θ /t  and solve for θ .

ω = θ /t

ω t = θ

(0.13 rad/sec)(720sec) = θ

θ = 93.6 rad

θ = 93.6/ 2π revolutions

θ = 14.9 or ~15 revolutions

Therefore, the Ferris Wheel will complete 15 revolutions within 12 minutes.