7.1 - Circular motion

The Earth orbiting the Sun, a turning wheel, and a spinning top are all examples of circular motion. The concepts of position, velocity, and acceleration used to describe linear motion take slightly different forms when used to describe circular motion. Read the text aloud
Angular velocity
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 is expressed in degrees per second. If the angle is measured in full turns (360º), then angular velocity might be in rotations per second or cycles per second. The angular speed of motors is often given in revolutions per minute (rpm). Read the text aloud Show How fast do small electric motors turn?
(7.1) ω= Δθ Δt
ω  = angular velocity (rad/s)
Δθ  = change in angle (rad)
Δt  = change in time (s)
Angular velocity
Measuring angles in radiansFor the purpose of angular speed, a radian (rad) is a more natural unit of angle than a degree. One radian equals approximately 57.3°. 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. Read the text aloud
The sign of the angular velocity depends on direction. If counterclockwise rotation is defined to be positive then clockwise rotation is negative. Read the text aloud
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.” Read the text aloud
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 (2π rad) every 24 hours.
Relationships: ω=Δθ/Δt; one day=( 24 hr )( 60 min 1 hr )( 60 s 1 min )=86,400 s
Solution: ω=( 1 rotation )/( 1 day )=( 2π rad )/( 86,400 s )=7.27× 10 5  rad/s
Answer: 7.27×10−5 rad/s. Note that the units “radian” are not necessary, because the radian is dimensionless.
Read the text aloud
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 radians per second?
  1. 0.0042
  2. 0.026
  3. 0.39
  4. 1.0
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