Angular Speed: Definition, Unit, Formula and Solved Examples

Angular speed can be defined as the rate at which an object rotates.

• Speed is defined as the rate at which an object moves from one location to another.
• Angular speed is the rate at which the central angle of a rotating body changes with respect to time.
• It is a scalar quantity.
• The SI unit of angular speed is radian per second (rad/s).

The formula for angular speed is given by

\(\omega=\frac{\theta}{t}\)

Where

• \(\omega\) is the angular speed
• \(\theta\) is angular displacement
• t is the time

Key Terms: Angular speed, Angular velocity, Torque, Angular displacement, Rotational motion, Circular motion, Linear velocity, Linear displacement

What is Angular Speed?

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Angular speed implies how quickly an object rotates.

• In other words, we can say that it is described as the change in the angle of the object per unit of time.
• So, if we want to compute the speed of a rotational motion, we'll need to know its angular speed.
• The angular speed formula is used to calculate the distance traveled by a body in terms of rotation and revolutions per unit of time.

The radian is a very essential concept that is mentioned below. It is in radians that we can determine the angular speed from the angle we measure.

• Radians are a method of measuring angles in which the right angle is defined as π/2 radians.
• As a result, a whole revolution contains around 6.28 radians.
• We can use speed to describe how quickly or slowly an object moves.
• The rotational speed of an object is defined as its angular speed.

The Angular Speed formula calculates the distance traversed by the body in terms of revolutions or rotations per unit of time.

• The distance traveled is shown as a symbol and is measured in radians.
• The amount of time taken is expressed in seconds.
• As a result, the angular speed can be expressed in radians per second (rad/s).

The following is the definition of angular speed for a single full rotation:

ω = 2π/t

The measure of how quickly the angle of the center of a spinning body varies in relation to time is known as angular speed.

Angular Speed Units

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The radian per second is the unit of angular speed.

• Both angular speed and angular velocity are calculated using the same formula.
• Angular velocity is a vector quantity that describes both magnitude and direction.
• Whereas angular speed just describes magnitude.

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Angular Speed Formula

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The scalar measure of the rotation rate is Angular Speed (ω). The angular distance traversed in one complete rotation is 2 and the period is period (T).

The following formula calculates angular speed:

Angular Speed = 2π/T

We can deduce from the above equation that ω is equal to 2πf and that 1/T is equal to f. (frequency).

Angular Speed of Earth

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One cycle around the Sun takes our planet 365.25 days to complete.

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.

Solved Examples

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Ques. Every 24 hours, the Earth rotates once on its axis. What is the angular speed of this object?

Ans. In one revolution, the angle traveled is 2. This rotation takes 24 hours to complete.

Converting hours into the second we get,

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

We already know that the formula for angular speed is = /t.

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

Ques. A carnival is taking place, and youngsters are rushing to the Ferris wheel in large crowds. We can also observe that there is a sign stating that the Ferris wheel's angular speed is 0.13 rad/sec. Calculate the number of revolutions the wheel has completed in the time limit of 12 minutes.

Ans. Looking at the figures, we can see that our angular speed is equal to 0.13 rad/sec. In addition, the time is t = 12 minutes. We'll start by converting the minutes to seconds, which gives

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.

Relation between Angular and Linear Speed

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The formula for linear speed is as follows:

v = s/t

Where 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

Where V is equivalent to the linear speed.

The equation expresses the relationship between the circular path's angular speed, linear speed, and radius.

Things to Remember

• The rate of change of angular displacement is defined as angular speed.
• Radian/second is the unit of measurement for angular speed.
• The angular speed of an object is a scalar number.
• The equation ω = V/r describes the link between angular and linear speed.
• The symbol for Angular Speed is ω.