## Angular Speed Formulas - Rotational Speed Definition & Problems

Previously, we have learned the topics of angular speed, so here, we will discuss the formula of angular speed. The formula for the angular speed is used to calculate the distance one particular body covers in terms of rotations or the revolutions to the time taken.

Speed is all about how slow or how fast an object moves. We have already heard about speed but do we know what we are talking about a specific type that is of speed over here. So if we haven’t heard of the angular speed that is, we are at the right place. Moreover, we will learn about the difference between the angular speed and angular velocity.

The angular speed which we will learn is the speed of the object in rotational motion. The Angular Speed Formula generally computes the distance that is covered by the body in terms of revolutions or rotations to the time taken. It is said to be represented by letter or symbol ω and is given as:

Angular speed =\[\frac{Total Distance Covered}{Total Time Taken}\]

### Circular or Rotational Motion

What exactly do we mean by circular motion or rotation? Rotational motion is the circular motion of an object about an axis of rotation when an object moves in a circular path. For example, circular motions include a race car speeding around a circular curve or a toy which is attached to a string swinging in a circle or the circular loop-the-loop on a roller coaster. Other examples are a wheel turning on its axle, the spin of a tornado on its path of destruction or that of a figure skater spinning during a performance at the Olympics. We can notice that sometimes the objects will be spinning while in a circular motion like the planet Earth which is spinning on its axis while revolving around the Sun.

When we are solving problems that involve rotational motion, we use variables that are similar to linear variables such as distance, velocity, acceleration, and force. but we should notice that it takes into account the curvature of rotation of the motion. Here we define generally the angle of rotation which is the angular equivalence of distance and the angular velocity which is the angular equivalence of linear velocity.

### How to Find Angular Speed

The speed which is known as the angular speed is the measure of how fast the central angle of a rotating body changes with respect to time. Angular speed formula is the relationship between angular speed and the speed which is linear along with a few angular speed problems.

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

Therefore if we want to calculate the speed of the rotational motion then we will be requiring the angular speed of it. The angular speed formula calculates the distance the body covers in terms of revolutions or rotations to the time taken.

Furthermore, the radian is quite an important thing here. To calculate the angular speed, the angle we measure is in radians. Radians are said to be 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.

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

Symbol ω = \[\frac{\Theta }{t}\]

Where:

Symbol ω refers to the angular speed in radians/sec.

The symbol 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 as well as the angular velocity make use of the same formula. However, the difference between the two is that angular speed is a scalar quantity whereas the angular velocity is a vector quantity.

### Formula of Angular Speed

The linear speed at which we already have a point on an object which is rotating depends on its distance from the centre of rotation. The angular speed which we know is the angle that an object generally moves through in a certain amount of time. The angular speed has units of radians per second rad/s. There are 2π radians in a full circle. At a distance r from the centre of the rotation is a point on the object which has a linear speed that is equal to the angular speed multiplied by the distance r. The unit of linear speed is meters per second, m/s.

The speed which is the linear speed = angular speed x radius of the rotation

v = ωr

v = linear speed (m/s)

ω = angular speed (radians/s)

r = radius of the rotation (m)

### Conversion of Degree to Radian and Radian to Degree.

The angle of rotation is often measured in the unit called the radian. (Radians are dimensionless units and are defined as the ratio of two distances, radius and arc length.)

One complete revolution around the circumference of a circular object covers 2π radians or 360^{o} in degree. Radian and degree are interconvertible and we can convert them using this relationship

1 revolution = 2π radians = 360o

1 radian = 360o/ 2π = 57.3o (approx.)

### Examples of Questions Solved with the Help of Angular Velocity (or speed) Formula

Example 1:- Earth takes a whole year (365 days) to complete a single revolution around the sun. Calculate the angular speed with which the earth is rotating around the sun.

Solution: The plane earth requires a full year to make a single rotation around the sun. So, the time taken in second will be

t = 265 × 24 hr × 60 min/hr × 60 sec/min = 31,53,60,00 seconds.

The earth completes make a single full rotation, and one rotations covers 360o, so the can be written as = 2π.

And after substituting these values in the formula of angular speed,

Angular speed (ω) = \[\frac{Total Distance Covered}{Total Time Taken}\]= \[\frac{\Theta }{t}\]

= \[\frac{2\pi }{31,53,60,00sec}\] = 1.9923 × 10-7 rad/sec.

Thus the angular velocity of earth while moving around the circle will be 1.9923 × 10-7 rad/sec.

Example 2:- The wheels of a huge truck of radius 1m is travelling with the speed of 5m per second. Calculate its angular speed.

Solution: The travelling speed of the truck can be considered as the Linear speed is = 5m/s,

The radius of the wheel of a truck (or the circular path) id r = 1 m (given)

The formula used to find the angular velocity ω = r

Substituting the values, we get

ω = \[\frac{5}{1}\] = 5 rad/sec.

Hence the angular speed of the wheels of that truck will be equal to 5 rad/sec.

## FAQs on Angular Speed Formula

**1. How do you Calculate Angular Speed?**

In a uniform circular motion, the angular velocity denoted by “w” is a vector quantity and is equal to the angular displacement that is Δθ, a vector quantity which is further divided by the change in time Δt. The formula for calculating angular speed will be written as, = \[\frac{\Delta \Theta }{\Delta t}\], note that the same formula is used to calculate both Angular speed and Angular velocity, the only difference will be that the velocity is a vector quantity, while speed is scalar in nature. The speed is equal to the arc length travelled, denoted by S divided by the change in time that is Δt which is also equal to |w|R.

**2. How do you Calculate Linear and Angular Speed?**

If v represents the linear speed of a rotating object, and the radius of the circle or arc traced by the moving object is denoted by symbol r (radius) and ω its angular velocity. Then, v = rω. This equation proves that Angular velocity shares a directly proportional relation with the linear speed (velocity).

If we take a unit circle, the radius of a unit circle will have to be 1 unit. So in this case the linear speed will be equal to the angular speed

= rω = 1×ω = ω,

With this equation, we can define linear speed, as the angular speed of the object when it travels in a unit circle or arc.

**3. How do you Calculate Angular Speed from RPM?**

The revolutions per minute can be converted to angular velocity in degrees as well by units of per second by multiplying the rpm by 6. We can say that since one revolution is 360 degrees and there are 60 seconds per minute. If we see that the rpm is 1 rpm, the angular velocity in degrees which is per second would be 6 degrees per second as we see that since 6 multiplied by 1 is 6.

Angular speed, in degrees per second can also be converted into RPM, revolutions per minute by multiplying the angular velocity by ⅙.

**4. What is Angular Speed in Physics?**

Angular speed is the speed of the object in rotational motion. The angular speed formula computes the distance which is covered by the body in terms of revolutions or we can say rotations to the time taken. It is represented by the symbol |ω|. It is a scalar quantity and can be calculated by this formula

Angular speed |ω| =\[\frac{Total Distance Covered}{Total Time Taken}\]= \[\frac{\Theta }{t}\], where represents the distance travelled by the object moving in the circular motion and is measured in radians. While the t is the time required to cover that certain amount of distance.

**5. Calculate the angular speed of the earth, when it rotates on its axis every day?**

The planet earth takes a full 360o rotation on its axis every day (and we who live on this planet don’t even feel the rotation). Thein this situation will be equal to 2π. i.e. = 2π.

Now, for the time, the earth consumes a whole day for this rotation and a day includes 24 hours. The 24 hours will be needed to be converted into seconds, in order to find the angular speed of the earth.

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

Applying these values to the formula of angular velocity, we get

ω =\[\frac{\Theta }{t}\] (formula to find angular velocity)

ω = 2π/86400 sec = 0.0000727 radians/sec = 7.27 × 10-5 rad/sec.

Hence the angular velocity for the earth while it is moving on its own axis in a day is 7.27 × 10-5 rad/sec.