Relationship between angular velocity and Linear velocity - Derivation
Here is complete discussion about Angular velocity and linear velocity. Like relation between angular velocity and linear velocity, their derivation and formula in vector form. So before we discuss about relation between angular velocity and linear velocity. We should first learn what is the definition of Angular velocity? What is linear velocity? So, lets start with Angular velocity.
What is Angular velocity?
Angular velocity measures how fast or slow a body is moving. On the other hand we can say angular velocity means a body is moving in some angle. In other words it is defined as the rate of change of angular displacement with respect of time.
Angular displacement means a body covers some distance with some angle 𝛉. For example, if a body is rotating in circular path having radius r with some angle 𝛉, then 𝛉 can be called as angular displacement. It is denoted by 𝛉. therefore, the formula of angular displacement is 𝛉 = L/r.
So, as we discuss earlier angular velocity is rate of change of angular displacement. It is denote by ⍵. therefore, the formula of angular velocity is given by,
⍵ = rate of change of angular displacement / time
⍵ = d𝛉/dt
Example of Angular velocity
A body is moving in circular path track and covers a distance of 100 m. Body has fixed radius 5m and travels for 5 sec. then angular velocity of the body will be,
Angular displacement 𝛉 = L/r = 100/5 = 20 radian
Now, if we want to measure the how fast and slow the body is moving we have to find angular velocity.
Angular velocity ⍵ = d𝛉/dt
⍵ = 20/5 = 4 rad/sec
Hence, the body is moving with 4 rad/sec.
Now, angular velocity is vector quantity. So how we can find the direction of angular velocity. The answer is Direction of angular velocity is determined by right hand thumb rule.
What is Linear velocity?
Linear velocity is defined as the rate of change of arc arc distance (on circumference) with respect to time. For example,
Here the arc distance is 'S'. Angular velocity means a body covers some distance on circumference of the circle with some angle theta.
Linear velocity is denoted by V. The direction of linear velocity is determined by the tangent drawn perpendicular to the radius of the circle.
Basically we can say that when an object travels in circular motion. Actually it has two velocities one is Angular velocity due to angle theta and other is Linear velocity due to circumference of the circle. So, there is a unique relation between angular velocity and linear velocity.
Relation between angular velocity and linear velocity
Angular velocity is rate of change of angular displacement. On the other hand linear velocity is rate of change of arc distance. So, the relation between angular velocity and linear velocity is, linear velocity is equal to the product of radius and angular velocity. That is given by,
Linear velocity = radius ✕ angular velocity
V = r✕ω
we can also say that there is directly relationship between linear velocity and angular velocity. That means if linear velocity increases angular velocity also increases and vice-versa.
Please Note that linear velocity of a circular moving particles is always greater than that of angular velocity.
Now, we have an idea of relation between angular velocity and linear velocity. But apart from this we do some experiment to demonstrate the relation between angular velocity and linear velocity.
Experiment to show that there is relationship between angular velocity and linear velocity.
suppose a randomly shaped body is moving in circular path taking an axis of rotation 'P'. Now, lets the extended part of the body be particles and these particles are also moving through some axis.
As we know that there is relation between angular velocity and linear velocity that is linear velocity is equal to the radius multiplied by angular velocity. i.e, V = r ✕ ⍵.
Now, this randomly shaped body also have relationship between linear and angular velocities. so we conclude that there is not necessary the how shape of body is, the only point is body must be rotating through some angle in circular motion.
Derive the relation between angular velocity and linear velocity
Consider an object is rotating in a circular field and make angle θ with the center. After making an angle θ with the center, Object also covers some distance 'L' on the circumference of the circle.
Now, as we know that when an object rotate on the fixed axis. It actually have two velocities angular and linear that is angular velocity due to angle θ at the center and linear velocity due to distance 'L' on the circumference.
Now, If we want to find how fast or slow the body is moving. We have to calculate angular velocity that is,
⍵ = d𝛉/dt
⍵ = rate of change of angular displacement / time
Here, θ is the angular displacement.
Now,
Angular velocity = rate of change of angular displacement / time
V = dl/ dt -------------------- (1)
we know that,
Angular displacement (θ) = L / r
L = θ ✕ r
put the value of 'L' in equation (1) we get,
V = d θ r / dt
V = r dθ /dt
V = r ✕ ⍵
This relation is in the form of scalar quantity.
Hence, the relation between angular velocity and linear velocity formula is V = r ✕ ⍵.
relation between angular velocity and linear velocity in vector form
Linear velocity = radius ✕ angular velocity
V (vector) = r ✕ ⍵ (vector)
