Relation Between Linear Speed(V) And Angular Speed(Ω)

Linear Speed (v) And Angular Speed (⍵) are concepts that determine the speeds of objects based on their style or direction of motion. The concept of Linear Speed vs Angular Speed can be seen through the lens of translational motion vs rotational motion.

While Linear Speed measures the distance travelled per unit time by an object moving in a straight line, Angular Speed measures the time in which the change of angle takes place for an object in rotational motion. 

Linear Speed is denoted by the symbol ‘v’ and Angular Speed by the symbol ‘⍵’. 

Note: It must be noted that both Angular Speed and Angular Velocity have the same formulas, that is, the measure of angular displacement per unit time, and can therefore be calculated and denoted through the same symbols. 

Linear Velocity vs Angular Velocity

Linear Velocity can be defined as the speed of an object moving in the forward direction. Here the angle of movement remains linear; that is, there is no angular change throughout the movement of the object from its initial to the final position. Linear velocity can also be defined as displacement per time unit. 

Therefore, the SI unit of Linear Velocity is meters per second or m/s.

On the other hand,Angular Velocity can be defined as the speed of spinning movement of an object on a circular/angular surface. Here the motion is rotational and, therefore, follows an angular spin instead of a linear movement. There could be an angle change (as will be the case in most scenarios until the angle shift is 360 degrees) while the object moves from its initial to the final position. Angular velocity is a vector quantity.

Relationship between Linear Speed and Angular Speed

Linear speed is independent of an angular change and thus has no angular shift. However, angular speed is dependent on linear speed. 

To better understand this, one must be aware of the relationship that these quantities share, and that which translational motion and rotational motion share with each other. 

Translational motion determines the movement of an object in a linear/straight-line path. The object here moves without an angular change, in the same plane where the point of origin lies. Rotational motion, on the other hand, determines the movement of an object in a circular or an angular path, where the initial position of an object and the final position of an object share two distinct entities: a linear displacement of the object from point A to point B; and a change in angle from the origin stretched to point A and point B, respectively. 

Therefore, looking at Angular Speed through the lens of the above-mentioned concept of rotational motion,  depends on two things. These are the object’s linear speed measured from the radius of rotation; the angular displacement of the object.

Formulas—Linear Speed vs Angular Speed 

As per the established relationship between Linear speed and Angular speed or Linear speed vs Angular speed, the relationship between their symbolic representations and formulas also take up the angular displacement and radius of rotation into perspective while calculating angular speed. 

Linear Speed = distance traveled / time taken

OR

Linear Speed (v) = Radius of rotation (r) x Angular Speed (ω)

and 

Angular Speed (ω)= Linear Speed (v) / Radius of rotation (r)

OR

Angular Speed (ω) = final angle – initial angle / time = angular displacement (θ) / time (t)

In the final equation of Angular speed (ω=θ/t), the unit for angular speed will be radians/sec (rad/s).

Linear speed is the necessary value that helps determine the speed of a moving object. In the case of angular speed, linear speed combined with angular displacement determines how fast or slow an object is when in rotational motion. Therefore, in both cases, linear speed remains an important aspect of the values to be determined. 

Conclusion

In this lesson, we looked at the relationship between linear speed (v) and angular speed,, and some linear speed vs angular speed questions. 

After completing this lesson, you should be able to determine the definitions of translational/linear motion, rotational motion, linear speed, angular speed, linear velocity, angular velocity. Not only that but you should also be able to prove the derivation through understanding the formula. Finally, as discussed in the header: ‘Formulas—Linear Speed vs Angular Speed; you should also be able to discuss the formulas and formula-relationship between linear speed and angular speed.