Angular Speed Formula, Definition, Solved Examples

Angular Speed Formula: Angular speed is a crucial measure in understanding the rotational motion of an object. It quantifies the rate at which a body traverses a distance in terms of rotations or revolutions over a specific duration. The concept of speed enclose the speed or slowness of an object's movement.

Angular speed specifically pertains to the rotational motion of an object. The angular speed formula, denoted by ω, calculates the distance covered by the object concerning rotations or revolutions in relation to the time elapsed. Mathematically, it is expressed as:

Angular Speed = Total Distance Covered / Total Time Taken

ω= θ ​ / t

Here, the distance travelled is symbolized as θ and is measured in radians, while the time taken is quantified in seconds.

Hence, angular speed is articulated in radians per second or rad/s. For a single complete rotation, the angular speed is commonly referred to as the "angular velocity," which can be expressed as:

ω =  2π ​/  T

The relationship between Angular Speed and Linear Speed is defined by:

v=ω×R

Where:  v represents the linear speed, ω is the angular speed, and  R signifies the radius of the circular path.

Angular Speed Formula Solved Examples

Example 1: A record player is rotating at a rate of 45 revolutions per minute. Determine its angular speed.

Solution: Given: Revolutions per minute rev/min=45

Convert revolutions per minute to radians per second:

ω= 2π×revolutions / 60×seconds per minute​

ω= 2π×45/60 ​

ω= 3/4 ​ πrad/s

Example 2: A car's wheel is rotating at an angular speed of 2.5 rad/s 2.5rad/s. Calculate its linear speed if the radius of the wheel is 0.5 m 0.5m.

Solution: Given: Angular speed ω=2.5rad/s, Radius r=0.5m

v=ω×r

v=2.5×0.5

v=1.25m/s

Example 3: A bicycle wheel completes 10 revolutions in 2 seconds. Determine its angular speed.

Solution: Given: Revolutions rev=10, Time t=2 seconds

ω  =  revolutions / time ​

ω= 10 ​/ 2

ω=5 rad/s

Example 4: An electric fan blade has an angular speed of 120 rpm 120rpm. Calculate its angular speed in radians per second.

Solution: Given: Angular speed ω rpm ​ =120rpm

Convert revolutions per minute to radians per second:

ω rad/s ​ = 2π×rpm / 60​

ω rad/s ​ = 2π×120 ​ / 60

ω rad/s ​ =4πrad/s

Example 3: A bicycle wheel completes 20 revolutions in 2 seconds. Determine its angular speed.

Solution: Given: Revolutions rev=20, Time t=2 seconds

ω  =  revolutions / time ​

ω= 20 ​/ 2

ω=10 rad/s

These examples demonstrate how to calculate angular speed and linear speed using the relevant formulas.

Understanding angular speed is essential for comprehending the rotational motion of objects. Angular speed, denoted by ω, measures how quickly an object traverses a distance in rotations or revolutions over a specific time period, providing insight into the object's movement speed. The angular speed formula, ω = θ / t, expresses this relationship mathematically, where θ represents the distance in radians, and t is the time in seconds.

Angular speed is typically expressed in radians per second (rad/s), and for a full rotation, it is termed "angular velocity," calculated as ω = 2π / T, where T is the time for one complete rotation. The connection between angular speed and linear speed is given by v = ω × R, where v is the linear speed, ω is the angular speed, and R is the radius of the circular path.

Whether determining the angular speed of a rotating record player, calculating the linear speed of a car's wheel, or finding the angular speed of a bicycle wheel, these formulas provide a comprehensive understanding of rotational motion dynamics.

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