Summary
Summary Physics B - Rigid Body Dynamics, 2008 Notes
- Module
- Institution
University of Cambridge - Part IB Natural Sciences Physics B: notes on thermodynamics and partial notes on dynamics (which cover some trickier points).
[Show more]
Seller
Follow
tandhiwahyono
Reviews received
Content preview
Physics Revision Notes – Thermodynamics Page 1 of 20Dynamics – Rigid Body Dynamics
Introduction
A rigid body is a many-particle system in which the distance between
particles is fixed. The location of all particles is described by 6
coordinates – 3 spatial and 3 angular.
The velocity is determined by v, the velocity of the CoM and w , the
angular velocity.
The basic two equations of angular motion are
= F
MR 0
The centre of mass moves as if it were a single particle under the
action of a force F0.
J = G0
The rate of change of angular momentum is equal to the total applied
couple.
Other basic equations:
o The velocity v of a particle at a distance r from an axis around
which a rotation at speed w is happening is
v = w ´r
o For similar reasons:
dJ
= w ´J
dt
o Angular speeds are additive. To if frame 1 is rotating with
w1 wrt 2 with respect to frame 2, which is rotating with w2 wrt 3
with respect o frame 3, then
w1 wrt 3 = w1 wrt 2 + w2 wrt 3
Relating J and w
If the body is rotating at w , the total angular momentum is given by
© Daniel Guetta, 2008
, Physics Revision Notes – Thermodynamics Page 2 of 20
J = år ´p
= å r ´ m(w ´ r )
= å m éër 2w - (w ⋅ r )r ùû
= å m éër 2w - (wx x + wyy + wz z )r ùû
In detail
æ m(y 2 + z 2 ) -å mxz ö÷÷
ççå -å mxy
çç ÷÷
J = ç -å mxy
çç
ç å m(x + z ) -å myz ÷÷÷÷ w
2 2
÷
çç - mxz
å -å myz
è å m(x 2 + y 2 )÷÷ø
I
J = Iw
[The non-diagonal elements are fairly easy to derive. The diagonal ones
should actually have x2 + y2 + z2, because one of the terms is always
knocked out by the second term in the sum]. In other words, J is
proportional to w , but not necessarily parallel to it.
The off-axes elements are rather hard to understand – they correspond
to the fact that looking at a particle at a given instant, it’s impossible
to tell exactly around which axis it’s moving.
Also, we can find the kinetic energy
T = å 12 m [(w ´ r ) ⋅ (w ´ r )]
= å 12 m [w ⋅ r ´ (w ´ r )]
T = 21 w ⋅ J
The couple is then given by
G = J = w ´ J
Note that I must be specified with its origin and with its set of axes.
Properties of I
I is a symmetric tensor. It therefore has three real eigenvalues and
three perpendicular eigenvectors.
With respect to the eigenvector basis:
© Daniel Guetta, 2008
, Physics Revision Notes – Thermodynamics Page 3 of 20
æI ö
çç 1 ⋅ ⋅ ÷÷
ç ÷
I ¢ = çç ⋅ I 2 ⋅ ÷÷÷
çç ÷
çç ⋅ ⋅ I ÷÷÷
è 3ø
J a = I a wa [No sum]
T = 12 I a wa2 [Sum]
The eigenvector axes are called the principal axes, and the Is are called
the principal moments of inertia.
An alternative way to think of this is that the principal axes are ones
around which objects are “happy” to rotate without any torque being
applied.
In w -space, surfaces of constant T form an ellipsoid, with axes of
length µ I a-1/ 2 . Also, in w -space:
grad T = I a wa = J
So J is perpendicular to surfaces of constant T at w .
We can classify the principal axes as follows:
o Spherical tops – all the I are equal, and J = I w , with I scalar.
The body is isotropic with the same I about any axis (eg: sphere,
cube).
o Symmetrical tops – I 1 = I 2 ¹ I 3 . e3 axis is unique, but e1 and e2
are any two mutually perpendicular vectors perpendicular to e3
(eg: lens, cigar).
o Asymmetrical tops – all Is different, and axes are unique.
Consider any two Is:
I 1 + I 2 = å m(y 2 + z 2 + x 2 + z 2 ) = I 3 + 2å mz 2 ³ I 3
So no I can be larger than the sum of the other two. Furthermore, if
z = 0 for every particle (ie: if we have a lamina), then
I 3 = I1 + I 2
Consider an axis at a distance a away from a principal axis and parallel
to it, and let r be the distance of each particle from the principal axis.
Then:
I = å m(r + a ) ⋅ (r + a ) = I 0 + Ma 2 + 2 ( mr )
å
⋅ a = I 0 + Ma 2
= 0 when r measured
relative to C of M
© Daniel Guetta, 2008
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller tandhiwahyono. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £2.06. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
67163 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy revision notes and other study material for 14 years now
Recently viewed by you
