rigid4.nb

22 November 2002

Rigid Body Acceleration

by George E. Hrabovsky, President, MAST

News from MAST

Well, we finally hashed out a set of by-laws that will be voted on in the next meeting.

Motion in a Rotating Frame

Last week we explored the velocity of a rotating rigid body. Since a rigid body can be seen as a kind of coordinate system (see "Locating the Crowd" and "Herding the Crowd") it is a natural question to ask, how do things move in a rotating frame of reference?

Let's assume that we have some position vector r. So, we have,

$FormBox[RowBox[{r, , =, , RowBox[{r _ 1 Overscript[e,^] _ 1, , +, , r _ 2 Overscript[e,^] ... ]}], (1)}]}]}], TraditionalForm]$

The velocity is the time derivative of (1),

$FormBox[RowBox[{Overscript[r, ·], , =, RowBox[{Overscript[b, ·] _ 1 Overscript[ ... ]}], (2)}]}]}], TraditionalForm]$

If we are thinking that the reference frame is rotating then the derivatives of the unit vectors does not vanish. If you think about it the rotational unit vectors can be described by Poisson's formula (from last week). So,

Overscript[Overscript[e,^], ·] _ 1 = ω × Overscript[e,^] _ 1, Overscript[ ... cript[e,^] _ 2, Overscript[Overscript[e,^], ·] _ 3 = ω × Overscript[e,^] _ 3,

where is the angular velocity. Thus (2) becomes,

Overscript[r, ·] = Overscript[r, ·] _ 1 Overscript[e,^] _ 1 + Overs ... nbsp; ]}]}], (3)}]}], TraditionalForm]

We can rearrange terms,

Overscript[r, ·] = ( Overscript[r, ·] _ 1 Overscript[e,^] _ 1 + Ove ... nbsp; ]}]}], (4)}]}], TraditionalForm]

We can rewrite this

Overscript[r, ·] = ( Overscript[r, ·] _ 1 Overscript[e,^] _ 1 + Ove ... nbsp; ]}]}], (5)}]}], TraditionalForm]

So,

The velocity is then the sum of the time derivative of the position vector and the outer product of the angular velocity and the position vector.

Acceleration in a Rotating Frame

The acceleration is the time-derivative of (6),

The derivative of a sum is the sum of the derivatives so,

Let's expand the first term into its components,

d/(d t) Overscript[r, ·] = d/(d t) ( Overscript[r, ·] _ 1 Overscript[e,^] _ 1 ... _ 3 Overscript[e,^] _ 3 + Overscript[r, ·] _ 3 Overscript[Overscript[e,^], ·] _ 3 .

From Poisson's formula we then have

d/(d t) Overscript[r, ·] = (Overscript[r, · ·] _ 1 Overscript[e,^] _ 1 ... + ω × Overscript[r, ·] + ω × ( ω × r )

Thus (8) becomes

The second term is,

d/(d t) (ω × r ) = ω × (d r)/(d t) + (d ω)/(d t) × ... ; = ω × Overscript[r, ·] + Overscript[ω, ·] × r .

So (9) becomes,

a = Overscript[v, ·] = Overscript[r, · ·] + ω &time ... p; ]}], (10)}]}]}]}], TraditionalForm]

The first term is the tangential acceleration, the second term is the centripetal acceleration, the third term is the Coriolis acceleration, and the final term vanishes for constant angular acceleration. This is called the Coriolis theorem and was first derived in 1835. Since we can view a rigid body as a coordinate frame, this describes the acceleration of a point on a rigid body.

Answer to Last Column's Theory Challenge

Since Poisson's formula

is an outer product, this implies that the angular velocity is a vector. Thus angular velocities can be added like vectors.

Theory Challenge

Can you think of a coordinate frame suggested by the rotation of a rigid body?

Books That I Like

E. A. Fox (1967), Mechanics, Harper and Row. This book is an excellent treatment of the mechanics of rigid bodies, elastic solids, and fluids. It is also, unfortunately, out of print; because of this I am following its arguments reasonably closely.

Keith R. Symon (1971), Mechanics, Addison-Wesley Publishing Company. This has a very nice section on rigid bodies and on rotating frames.

Converted by Mathematica (November 21, 2002)