# 3. Rotational Dynamics Flashcards

Rotational Reference Frames

-R denotes a rotational reference frame, e.g. a room rotating around a central axis -I indicates an inertial reference frame

Rotation Vector

-the rotation vector is Ω_ -this is generally set to Ω_ = Ωez^ -where Ω is the rate of rotation, and Ω>0 indicates anticlockwise rotation

Period of Rotation

τ = 2π/Ω

Centrifugal Force Description

-proportional to distance from the axis of rotation -e.g. when you are in a car and it goes around a bend, your momentum is in the tangential direction so you experience a radial force

Coriolis Force Description

-associated with motion -if a person in a rotational reference frame throws a ball, the ball deflects to follow the path of rotation

Rotaional Reference Frame

Position

to describe position x_(t), we choose a basis {ej_} for j=1,2,3

-in R, choose {ej~_(t)} fixed in the room:

ej~_ = P[Ωt] ei_

-so that:

x_(t) = xj~(t) ej~_(t)

Inertial Reference

Position

- to describe position x_(t), we choose a basis {ej_} for j=1,2,3
- in I, {ej_} is fixed in time:

x_(t) = xj(t) ej_

Relationship Between Position in Inertial and Rotational Reference Frames

-the relationship between the two sets of coordinates is: xj(t) ej_ = xj~(t) ej~(t)

Rotation Matrix

-P[Ωt] is a 3x3 matrix describing rotation around the z-axis by Ωt -with entries cos(Ωt), -sin(Ωt), 0 in the first row, sin(Ωt), cos(Ωt), 0 in the second row and 0,0,1 in the third row

Rotational Basis Vectors

e1~_ = cosθ e1_ - sinθ e2_ e2~_ = sinθ e1_ + cosθe2_ e3~_ = e3_

Inertial Basis Vectors

e1_ = cosθ e1~_ + sinθ e2~_ e2_ = -sinθ e1~_ + cosθ e2~_ e3_ = e3~_

Inertial Reference Frame

Velocity

dx_/dt |I = dxj/dt ej_

Rotational Reference Frame

Velocity

dx_/dt |R = dxj~/dt ej~_

-the apparent velocity

Relationship Between Velocity in Inertial and Rotational Reference Frames

dx_/dt|I = dx_/dt|R + Ω_ x x_

Relationship Between Acceleration in Inertial and Rotational Reference Frames

d²x_/dt²|I = d²x_\dt²|R + 2Ω_xdx_/dt|R + Ω_x(Ω_xx_)

Equation of Motion in the Inertial Reference Frame

m d²x_/dt²|I = F_

Equation of Motion in the Rotational Reference Frame

m d²x_/dt²|R = F_ - 2m Ω_ x dx_/dt|R - mΩ_ x (Ω_ x x_)

-the secon term on the RHS represents the Coriolis force and the third term the Centrifugal force

Ficticious Forces in the Rotational Reference Frame

-in moving to the rotational reference frame, ficticious forces arise which keep track of the fact that we are in a rotating reference frame

Properties of the Centrifugal Force

-if Ω_ = Ω ez^ and x_ = r er^ + z ez^

Fu_ = mΩ²r er^

- this is always positive, as expercted since the centrifugal force points outwards
- a ‘radial gravity’
- a conservative force

Fu_ = -∇(1/2 mΩ²r²)

Coriolis Force Properties

Fc_ . x_ = 0

- so Fc_ is perpendicular to the direction of motion
- no work done
- deflecction is energetically free

Coriolis Dominated Dynamics

|Fu_|/|Fc_| ~ Ωr/U << 1

- Fu_ negligible and Fc_ dominant
- Fu_ can also be balanced out in the governing equation:

–by gravity in a planetary context

–by the reaction force from the outer wall in a tank

Coriolis Dominated

Equation

Fu_ = 0

=>

mx’‘_ = -2mΩ_ x x’_ + F_

Coriolis Dominated

Solutions

F_ = (0,F)

=>

x = a/4Ω² [2Ωt - sin(2Ωt)]

y = a/4Ω² [1 - cos(2Ωt)]