2.3 
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1 - Science of Physics
2 - Forces and Motion      Chapter 2 at a glance      2.1 - Motion      2.2 - Force, momentum, and Newton's laws      2.3 - Circular motion      2.4 - Solving harder physics problems      2.5 - Chapter review3 - Energy Transformations4 - Waves and Sound
5 - Electricity and Magnetism
6 - Light and Optics
7 - The Atom
8 - Linear Motion
9 - Vectors and Forces
10 - Forces
11 - Newton's Laws and Gravitation
12 - Circular Motion and Gravitation
13 - Torque and Static Equilibrium
14 - Momentum and Collisions
15 - Angular momentum
16 - Power and Machines
17 - Harmonic Motion
18 - Waves
19 - Sound
20 - Electricity
21 - Electric and Magnetic Fields
22 - Electromagnetism and Induction
23 - Light and Color
24 - Geometrical Optics
25 - Electromagnetic Radiation
26 - The Properties of Matter
27 - Thermodynamics and Heat Transfer
28 - The Atom
29 - The Quantum Theory
30 - TBD
31 - TBD
32 - TBD
33 - TBD
34 - TBD
35 - TBD
36 - TBD
37 - The atomic nucleus and radioactivity
38 - Nuclear reactions
39 - Relativity
40 - Particle physics and the standard model
41 - Electronics (semiconductors)
42 - Nuclear technology
43 - Lasers and photonics
44 - Metals, alloys and materials science
45 - Communications technology
46 - Buildings and structures
47 - Energy technology
48 - Biophysics
49 - Matter and Energy, Space and Time
50 - Energy Transformations
51 - Nanotechnology
52 - Website Videos
53 - ExtraStuff
55 - Modeling Motion, Friction, and Center of Mass
56 - Work and the Conservation of Energy
2.3 - Circular motion
Linear motion is described using the concepts of position, velocity, and acceleration. The Earth orbiting the Sun, a turning wheel, and a spinning top, on the other hand, are all examples of circular motion.
Angular velocity
How does physics describe rotating speeds? The rate at which a rotating object spins is called its angular velocity, which is represented by the Greek letter ω (“omega”). To understand angular velocity, consider an object with a rotational position given by its angle θ. Angular velocity describes the amount that angle changes per unit time. If the angle is measured in degrees then angular velocity would be expressed in degrees per second. If the angle is measured in full turns (360º) then angular velocity might be in rotations per second. The angular speed of motors is often given in revolutions per minute (rpm).
Angular velocity
(2.14) ω= Δθ Δt
ω  = angular velocity (rad/s)
Δθ  = change in angle (rad)
Δt  = change in time (s)
Angular velocity
What is a radian? Measuring angles in radians For the purpose of angular speed, a radian is a more natural unit of angle than a degree. One radian equals approximately 57.3 degrees. Radians are dimensionless because radians are a ratio of lengths. One radian is the angle formed by wrapping one radius of a circle around the circumference. There are 2π (about 6.28) radians in a full circle, which makes 2π rad = 360º.
Positive and negative values The sign of the angular velocity depends on direction. If counterclockwise rotation is defined to be positive then clockwise rotation is negative.
What are the units of ω? Radians are pure numbers without units in the sense that meters or seconds are units. Expressed in radians per second, angular velocity has units of 1/s or s-1. If you encounter an angular velocity expressed in units of 1/s, then interpret the value as “radians per second.”
If you are running around a perfectly circular pond, and you make 1/4 of a loop every minute, what is your angular velocity in rad/s?
  1. 0.0042 rad/s
  2. 0.25 rad/s
  3. 0.39 rad/s
  4. 1 rad/s
Show
Solved Problem 2.6: Angular velocity of Earth
The Earth rotates once every 24 hours. What is its angular velocity in radians per second?

Asked: Angular velocity ω
Given: Earth makes one full rotation every 24 hours.
Relationships: One full rotation = 2π radians
One day=24 hrs( 60 min 1 hr )( 60 s 1 min )=86,400 s
ω= Δθ Δt
Solution: ω= 2π 86,400 s =7.27× 10 −5  s −1
Answer: 7.27×10−5 s−1

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