You already knew the concept of speed and velocity. But the concept of angular speed and velocity are the physical quantities that need to be understood.
When an object is supposed to move along the circular path, which makes a certain angle is called angular motion. The concept of angular speed and angular velocity are quantities derived from the angular motion of the object. Let us study these concepts in detail.
Angular Speed vs Angular Velocity
Before going to discuss the comparison between angular speed and angular velocity, let us study the meaning of angular speed and angular velocity and the formula that is used for the calculation.
What is angular speed?
Assume that you are rotating a ball in the circular orbit, then the angular speed can be define as below.
Angular speed is the measure of how rapidly a body changes its angle with time while rotating in the circular orbit.
Formula to calculate the angular speed.
To measure the angular speed of the rotating object, we have to calculate the number of revolutions that the body covers per unit time. The angle of rotation should be taken in terms of radians.
For a right angle, we define the radian as π/2, that makes while moving, so, for a complete rotation, it has 2π radians.
The angular speed is denoted by the symbol ω; it is given by the equation,
Where; θ is the angle of rotation and t is the time taken for one rotation.
What is angular velocity?
When an object is rotating in a circular orbit with some speed, then the angular velocity can be define as same as how the linear velocity can be defined.
Formula to calculate angular velocity.
To calculate the angular velocity, we have to know the direction in which the body is rotating .
Let us assume that the object is rotating in the anti-clock wise direction; then the angular velocity is given as;
Where; dθ is the change in angular displacement
dt is change in time.
Comparison between angular speed and angular velocity:
By knowing the differences between angular speed and angular velocity, one can understand the concept easily.
|Angular speed||Angular velocity|
|Angular speed is a scalar measure of the rotating object.||Angular velocity is a vector measure of the rotating object.|
|Angular speed specifies only the magnitude.||Angular velocity specifies both magnitude and the direction.|
|Unit of angular speed is radians/second.||Unit of angular velocity is also radians/second.|
|It does not have any proper direction to rotate.||It rotates in a particular direction along the axes, either in clockwise or counter-clockwise.|
|Speed varies along the circular path as the angle changes.||Velocity remains constant even though the angles keeps on changes.|
|Angular speed gives the absolute value for velocity vector, so that it should be positive or zero.||Angular velocity may become negative whenever it rotates along the negative axis.|
Let us consider a body rotating in a uniform circular orbit with a radius ‘r’. The body is moving from one position to another by making an angle ‘θ’ with time ‘t’.
Angular velocity is given by;
The speed at which the body is displaced from one position to another is given by,
s is the displacement which is nothing but the arc length of the circle; given by,
Now, substituting the values
Which is the magnitude of the angular velocity
Speed = |ω| r
The above equation implies that the angular speed is the magnitude of angular velocity and the radius of the path through which the object is traveling.
Some solved problems.
A ball is rotating in a circular path at a certain speed. It rotates π radians per every 6 seconds. Calculate the speed of rotation.
The speed is given
The rotation per second is 1/6 , the speed is given as
i.e,. ω = 6π rad/sec.
A tire is rotating in a circular orbit of a radius 12cm. The angle of rotation is 9 radians per every 3 seconds. Find out the angular speed?
The angular speed is given by;
For a complete rotation, the tire revolution is 360°. Hence the revolution is of 2π radians.
ω = 6π rad/sec.
A disc of a diameter 25m is rotating with a speed of 16m/s. Calculate the angular velocity of the tire.
Given: The diameter of the tire = 25m
Radius is given by
ω = 1.28 units/sec.
Calculate the speed of the Earth, which takes 365 days to revolve around the sun.
Earth takes 365 days = t = 365 × 24 × 60 × 60
t = 31536000 sec.
Since the earth is revolving in circular orbit, for one complete revolution, it takes 2π radians.
Angular speed is
ω = 1.99 × 10-4 units/sec.
Frequently Asked Questions.
What is meant by pseudo vector?
When a physical quantity has both magnitude and direction, then the quantity is said to be a Vector.
A Pseudo vectors also have both the magnitude as well as direction. But it changes its orientation when the coordinate axes change.
How does the angular velocity depend on the direction?
The angular velocity acts along direction of the rotational axis.
If the velocity is acting towards the axis of rotation, the object is subjected to rotate in anti-clockwise direction. If the velocity is acting against the axis of rotation, the object is subjected to rotate in clockwise direction.
How does the angular velocity remain constant in circular motion?
Velocity of a body remains the same even though the direction may change.
When a body is subjected to circular motion, the direction of the body may keep on changing. Since it is a vector quantity, the magnitude balances the change in the position, and the angular velocity remains constant.
How does the centripetal force impact the angular velocity?
The centripetal force acts perpendicular to the velocity along the circular path.
The friction force contributes to the centripetal force, which is equal to the angular velocity. The larger the centripetal force smaller will be radius, but the velocity remains the same.
Does the radius change the angular speed of a body?
Radius does not cause any type of change in the angular speed.
Angular speed is the same at every point of the circular path, but not the linear speed. It is because the moving body travels at the same angle at the same time at every point of uniform circular path.
When does the angular velocity become negative?
If the object is rotating in a clockwise direction, then the velocity becomes negative.
The sign of the velocity vector depends on the coordinate system. Velocity becomes negative only when the object is moving from left to right of the coordinate axis.