Chapter 10: Rotation

Lecture 2010/12/2009

RotationGoals for this Lecture:

Introduce the principles of rotational motion of rigid bodies

Introduce the rotational kinematic variables of Angular displacement: !Angular velocity: " Angular acceleration: #Rotational Inertia (I)Torque: $ Work and Rotational Kinetic Energy

Rigid BodiesA rigid body is defined as an object which has the following property: The distance between any two points in the body does not change. i.e. a common sense “solid”

A

B

A

B

Fixed AxisFor now, we will consider rotation around a fixed axis of rotation, i.e. the hinges of a door, or a BBQ spit central axis, but NOT the central axis of a football

Fixed axis

Fixed axis

Varying axis

RotationRotation is described by an angle ! with respect to a reference line that is perpendicular to the rotation axis. !(t) varies with t as the object rotates

Front view Side view

Rotation AngleAll points in the rigid body rotate by the same angle (!) but they travel different distances

s1r1

!s2

r2

! = =s1

r1

__ s2

r2

__

s = arc length (has dimension of length)

! is dimensionless. It is expressed in RADIANS

Rotation Eqns. of Motion

Translation

x(t) [meters]v(t) = dx/dt [m/s]

a(t) = d2x/dt2 [m/s2]

IF a(t) = constantv = v0 + at

x = x0 + v0t + 1/2 at2 v2 - v0

2 = 2a(x-x0)

Rotation

!(t) [radians]"(t) = d!/dt [rad/s]#(t) = d2!/dt2 [rad/s2]

IF #(t) = constant" = "0 + #t

! = !0 + "0t + 1/2 #t2 "2 - "0

2 = 2#(!-!0)

Position:Velocity:

Acceleration:

The new Airbus A380 engine fans extend from a central spool of radius 0.5m to a maximum radius of 1.5m. The engine has a top rotation speed of 3000 rpm. What are the linear velocities of the fan tip / base?

Linear/Angular Rotation Speed

3000 rpm = 3000 Rotation/minute * 1 min/ 60s = 50 Rotation / s

1 Rotation = 2! radians

3000 rpm = 50 Rotation/s * 2! radians / Rotation = 100! radians/s

Distance travelled: s = !r = (100! radians)*1.5m = 150! m

Speed of fan tip = 150! m/s = 471 m/s

calculating the speed of the fan base is left to you

Linear/Angular Rotation Speed

*tip speed = 1.5 times the speed of sound

Relating Linear & Angular Quantities

The point on a radius r1 moves a distance s in time t.

Position: s = !r

Velocity: v = ds/dt = d(!r)/dt = r d!/dt = "r

Acceleration: tangential: at = dv/dt = d("r)/dt = r d"/dt = #r radial: ar = v2/r = ("r)2/r = "2/r total: |a| = ar

2 + at2

s1

r1

!

at

ar

Period & FrequencyA rotating object has a rotation period given by: T = distance / speed = 2!r / v = 2!r / "r = 2!/"

and a frequency f = 1/T = "/2! --> "=2!f

s1

r1

!

Angular Velocity VectorsTranslation:

s = sxî + sy% + szk ! --> v = vxî + vy% + vzk !

Rotation: do we need vectors to define rotations? Yes! The vector direction is defined by the axis of rotation & the right hand rule

" "

Kinetic Energy of RotationThe kinetic energy of a single mass moving in a circle is : K = &mv2 = &m("r)2 = &(mr2)"

A rotating solid is made up of many small elements rotating with the same angular velocity ". The kinetic energy is given byK = &m1v1

2 + &m2v22 + &m3v3

2 + .... = &(m1r1

2)"2 + &(m2r22)"2 + &(m2r2

2)"2 + ... = "[&(miri

2)"2] = & ["(miri2)] "2 =

= & I "2 s1

r1

!

i i

Rotational Inertia

Translation

K = &Mv2

M = " mi

Rotation

K = & I "2

I = "(miri2)

= #r2dmI depends on the shape

and the mass of the object and on the axis of rotation

s1

r1

!

i

Rotation Kinetic energy2 masses (m1 and m2) are attached to the spokes of a massless wheel at radii R1 = 2m and R2 = 4m that is rotating with an angular velocity of " = 10 rad/s. What is the kinetic energy of this system?

K1 =&M1v12 = &M1("r1)2

K2 =&M2v22 = &M2("r2)2

Ktot = K1 + K2 = &M1("r1)2 + &M2("r2)2

= & [M1r12 + M2r2

2] "2

m1

m2

Rotational InertiaThe rotational inertia, I, for some common shapes