Angular position and velocity

a rigid object

• an object that has a definite shape that does not change 

• the particles that make it up, all stay in fixed positions relative to each other 

• ex) the blades of a windmill or a merry-go-round

polar coordinates

• (r, θ)

• convenient to represent the position of any point p on the rigid object

• when point P rotates in a circle:

  • distance (r) from the axis is constant 

  • only angle θ changes 

arc length (s)

• as point p moves through an angle θ (measured in rad) it moves through an _____

• s = rθ

s = rθ

what is the equation to find the arc length (s)? 

2π radians

1 revolution = 360° what is it equal to in radians?

180 / π degrees

what is 1 radian in degrees?

π / 180 radians 

what is 1 degree in radians?

θ is the angular position

remembe: θ is the angular position

∆θ is the angular displacement

remember: ∆θ is the angular displacement

angular displacement 

• the angle that the object rotates through during some time interval

• the change in angular position

• ∆θ

• ∆θ = θf - θi

average angular velocity (ω)

• the angular displacement (∆θ) divided by the time interval

• is a vector (so use + or - to denote direction) 

  • if ω is + then the object spins counter-clockwise (CCW) 

  • if ω is - then the object spins clockwise (CW)  

• its magnitude is the angular speed (ω) 

• ω = ∆θ / ∆t

clockwise (CW) 

In what direction does the object spin if -ω?

counter-clockwise (CCW)

In what direction does the object spin if +ω?

smaller ω = spins slowly & turns through a small angle every second

remember: smaller ω = spins slowly & turns through a small angle every second 

larger ω = spins faster & turns through a larger angle every second

remember: larger ω = spins faster & turns through a larger angle every second

All point on the rigid object have the same angular speed but are NOT moving at the same speed

remember: All point on the rigid object have the same angular speed but are NOT moving at the same speed.

every point on the rigid object (disk) rotate through the same angle in the same time interval

remember: every point on the rigid object (disk) rotate through the same angle in the same time interval

every point on the rigid object (disk) is spinning at the same rate

remember: every point on the rigid object (disk) is spinning at the same rate 

every point on the rigid object (disk) has the same angular speed (ω).

remember: every point on the rigid object (disk) has the same angular speed (ω).

linear vs angular speed

• on a rotating rigid object (disk):

  • the point further from the center (point B) has a further distance to travel than the point closer to the center (point A)

    • point B is moving faster

    • point B has a larger speed (v)

speed (v) 

• v = rω

• if point is further from center, then larger r and therefore moving faster

v = rω

what is the equation for the objects speed (v)?

Period (T)

• the time for one revolution

the time for one revolution

what is the def. of Period (T)?

Frequency (f) 

• the number of revolutions per second 

• unit: Hertz

• 1 Hz = 1 rev/s 

• f = 1 / T

the number of revolutions per second

what is the def. of Frequency (f)? 

1 rev / s 

1 Hz = ?

f = 1 / T

What is the equation for Frequency (f)?

Angular speed (ω) relation to period and frequency

• ω = 2π / T = 2πf

• a point that does one full revolution, travels 2π radians (one full circle) in one period 

ω = 2πf

Equation for angular speed (ω) in terms of the period and frequency?