Rotational Kinematics

• Road Map of Chapter 8• Master Analogy chart• Rotational Kinematics

– Definition of radian, relation to degrees– Angular displacement– Angular velocity– Angular acceleration– Constant angular acceleration– Examples

Comet landing – 2014 Nov 12• Mass of Comet 67P – 1013 kg.

• Radius of comet (approx.) - 1.5 km.

• Mass of  Philae lander (est.) - 100 kg.

http://en.wikipedia.org/wiki/67P/Churyumov%E2%80%93Gerasimenko

• Gravitational Force

That’s why they have to anchor it down!

news: www.bbc.com/news/

ESA: http://www.esa.int http://new.livestream.com/esa/cometlanding

twitter: #cometlanding #philae

Translation vs. Rotation

Translation Rotation

Translation vs. Rotation (cont)

Translation Rotation

Angular Displacement

• Methods for describing rotation– Degrees (90°, 180°, etc) - Independent of radius

– Revolutions (1/4, ½, etc) - Independent of radius

– Arc Length (m) - Dependent on radius!

– Arc Length at standard radius (1 m)

• Radian– Arc length on circle of radius 1 m

– At any other radius arc length

– Unitless, though I may occasionally write “rad”

– GMT analogy

Just doing 1-D kinematics on circular path of radius 1, and then scaling to circle of radius r!

Converting to Radians

• Methods for describing rotation– Degrees (90°, 180°, 360 etc) - Independent of radius

– Revolutions (1/4, ½, 1, etc) - Independent of radius

– Radians (π/2, π, 2π, etc) Independent of radius

• Degrees to Radians

• Revolutions to Radian

𝜃 𝑟𝑎𝑑

2𝜋=𝜃𝑟𝑎𝑑

360

Example 8.1

• Bicycle wheel rotates 4.5 revs

– How many radians?

– How far along 700 mm bicycle wheel?

Angular velocity

• Similar to linear velocity

• Merry-go-round– 2 horses at radius 1 m, 2 m– 1 rev/s–

–

–

Angular acceleration• Similar to linear acceleration

• From this we get tangential acceleration

• And there’s always radial acceleration

Angular acceleration – remember

• Rotating object may have tangential acceleration, if its speeding or slowing:

• But always has radial acceleration, even at constant velocity:

(for circular motion)

𝑎𝑅=𝑣2

𝑟=𝑟 𝜔2

Translation vs. RotationTranslation Rotation

Just 1-d motion along circular path

Example 8-4 – Merry go Round

• Angular velocity at 8s

• Linear velocity at 2.5 m

Example 8-4 (cont)

• Linear acceleration at 2.5 m

• Linear velocity at 8 s

t

same result

Example 8-4 (still more)

• Radial acceleration at 8s

• Total acceleration 8s and 2.5 m

Frequency and Period

• Angular velocity related to frequency

• Angular velocity related to period

• Same as before only with r = 1

Example 8.5 – Hard Drive

• Frequency

• Angular velocity

• Head speed at r = 3 cm

• # Bits read at

Example 8-6 - Centrifuge

• Acceleration

• # Revs

Example 8-6 alternative

• Use third kinematic equation

Plug in

Rolling Motion

• Wheel edge velocity matches relative velocity of room backwards

• Relative velocity of room backwards matches velocity of wheel CM forwards

• Demonstrate

Example - Bicycle

• Initial Angular velocity wheels

• Angular displacement of wheel

• Angular acceleration

• Time

Bicycle – alternative• Get linear acceleration

• Get angular acceleration