Angular Speed
Nov 03, 2022, 16:45 IST
Angular speed measures how fast the central angle of the rotating body changes over time. In this short article, we will familiarize ourselves with the angular speed.
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What is Angular Speed?
The rate of change of the angular displacement is known as angular speed, and it is expressed as follows:
W = θ / t
Where,
θ = angular displacement
t = time
w = angular speed
Angular Speed Unit
The unit of angular speed is radian per second. The same formula represents both angular speed and angular velocity. An angular velocity is a vector quantity that expresses both direction and magnitude, while angular speed describes magnitude only.
Angular speed = 2π / T
Angular Speed of Earth
Our Earth takes about 365.25 days to finish one revolution around the Sun.
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
Q1. An Earth rotates once on its axis once every 24 hours. Find the angular speed.
Ans. The angle traversed in one revolution is 2π. The time taken for this rotation is 24 hours.
Converting hours into second we get,
t = 24 hr x 60 min/hr x 60 sec/min = 86400 sec
We know that angular speed is given by the formula ω = θ /t
Putting 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
Q2. Earth takes 365 days to complete a revolution around the sun. Calculate its angular speed.
Ans. As we know, Angular speed = θ/t
where
θ = 2π / 31536000
t = 365 x 24 x 60 x 60
= 31536000 s
w = 2π / 31536000
= 1.9923 x 10-7 rad/s
Q3. The wheel of a wagon of radius 2m and is traveling with a speed of 6m per second. Calculate its angular speed.
Ans. Given Linear speed V = 6m/s,
The radius of circular path r = 2m
w = v / r
Substituting the values, we get
=6 /2 = 3 rad/s
Relationship Between Angular Speed and Linear Speed
Let the object travel in a round path of radius r and angular displacement be θ, and then we have an angle, θ = arc/radius.
Linear speed is given by the given formula:
v = s/t
Where s = linear displacement of arc
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 = linear speed
The above equation gives the relation between angular speed, linear speed, and the radius of the circular path.
Frequently Asked Question (FAQs)
Q1. What does angular speed depend on?
Ans. If an object rotates through the greater angle of the rotation in a given time, it has a higher angular speed. The units for angular speed are radians per second (rad/s).
Q2. Why is angular speed constant?
Ans. The reason for angular velocity to be constant is that the object in the circular motion covers equal angular displacement all the time.
Q3. Can angular velocity negative?
Ans. Angular velocity is generally represented by the symbol omega (ω, sometimes Ω). By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise.
Q4. Is angular speed a vector quantity?
Ans. It is a vector quantity and has both magnitude and the direction. The direction is the same as the angular displacement direction from which we defined the angular velocity.
Q5. What is the unit of angular speed?
Ans. The unit of angular speed is radian per second.
