### What is Angular Speed: Formula & Examples?

Ever wonder how fast the Earth orbits the Sun? How to calculate it? What is it even called? It’s called angular speed. This is the rate at which an object rotates over time. Mathematically, the definition is the change of the central angle of an object over time.

It calculates the distance or the number of revolutions an object covers during its rotation during a period of time. So, in regards to Earth orbiting around the sun, angular speed would calculate how fast the Earth is spinning around the sun over a period of time. This could be a day, a month, or a year.

The angle speed of a rotating body is determined by how fast its central angle changes over time. In this article, we will familiarise ourselves with angular speed.

**What is Angular Speed?**

Speed is a term that is used in many contexts. For instance, the speed at which you drive your car or the speed at which you pitch a ball. Similarly, speed refers to how slow or fast the object is moving. Angular speed refers to how fast an object rotates. In other words, it is described as the change in the angle of the object per unit of time.

Thus, to calculate the speed of rotation, we must know its angular speed. The angular speed formula calculates the distance the body covers in terms of revolutions or rotations to the time taken.

Furthermore, radian is quite important here. When we calculate the angular speed, we measure the angle in radians. Angles are measured in radians, where a right angle is defined as pi/2 radians. Therefore, one full revolution will contain around 6.28 radians.

**Angular Speed Unit**

Radian per second is the unit of angular speed. The formula for angular speed and angular velocity is the same. Angular velocity expresses both direction and magnitude, while angular speed only expresses magnitude.

**Angular Speed Formula**

Angular Speed (ω) is the scalar measure for rotation rate. In one complete rotation, the angle traveled is 2π and time is time period (T). The angle of rotation is given by the following formula:

Angular Speed=2π/T

From the above equation, we can conclude that ω is equivalent to 2πf, where 1/T is equivalent to f (frequency).

**Angular Speed of Earth**

Our Earth takes about 365.25 days to complete one revolution around the Sun.

Converting days into seconds, we get

T = 365.25 x 24 x 60 x 60 = 31557600 seconds

We know that angular speed = 2π/T, hence

ω = 1.99 x 10^{-7} radians /seconds.

The angular speed of Earth is 1.99 x 10^{-7} radians/seconds.

**Angular Speed Examples**

**The Earth revolves around its axis once every 24 hours. What is its angular speed?**

**Solution:**

The angle traversed in one revolution is 2π. It takes 24 hours for this rotation to complete.

Converting hours into seconds, we find:

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

The formula for angular speed is ω = θ /t.

By substituting the values in the equation, we get

ω = 2π/86400 sec

Solving, we get

ω = 0.0000726 radians/sec = 7.26 x 10^{-5} rad/sec

**Relationship between Angular Speed and Linear Speed**

Assuming that the object travels in a circle of radius r, then the angle displacement θ = arc/radius.

The following formula determines linear speed:

**v = s/t**

In this equation, s is the linear displacement of the arc, and θ = S/r.

Thus, linear speed V =(θ.r)/t = r.(θ/t)

Therefore, V = r ω

Rearranging, we get,

**ω = V/r**

V is equal to the linear speed.

The equation shows the relationship between angular speed, linear speed, and radius of the circular path.

**FAQs**

**Q: What is the Angular Speed?**

A: Angular speed is defined as the rate at which angular displacement changes.

**Q: What does angular speed tell you?**

A: Angle speed is the distance or the number of revolutions an object covers during its rotation during a period of time. The rotation rate is a scalar value that indicates how quickly an object rotates over time.

**Q: What is the symbol for Angular Speed?**

A: Angular Speed is denoted by ω.

**Q: What is Angular speed measured in?**

A: Angular speed is measured in radians/second.

**Q: Is Angular Speed scalar or vector?**

A: Angular speed is a scalar quantity.

**Q: What is the relationship between linear speed and angular speed?**

A: The relationship between angular speed and linear speed is given by the equation ω = V/r.