Wednesday, 1st May, 2013
9.30 am to 12.30 pm
EXAMINATION FOR THE DEGREES OF
M.A., B.Sc. AND M.Sci.
3H (JUNIOR HONOURS) MATHEMATICS
Mechanics of Rigid and Deformable Bodies
An electronic calculator may be used provided that it does not have
a facility for
7. Introduction to Lagrangian mechanics
7.1. Lagranges equations
Denition 7.1 If a mechanical system is completely specied by coordinates
q1 , q2 , . . . , qn , they are called generalized coordinates.
For a particle in three dimensions, for example, its
University of Glasgow
School of Mathematics & Statistics
Level 3, 2014/2015
MECHANICS OF RIGID
AND
DEFORMABLE BODIES
Lecture Notes Part 2 (3H)
46
5. System of particles
5.1. Summary of some formulas for a single particle
5.1.1. Rotational velocity
Let a p
6. Rigid body motion
A rigid body, denoted B, is a collection of particles continuously distributed in such a way
that the distance between any two particles (or points) is unchanged during the motion
of the body. The motion of a rigid body consists only
University of Glasgow
School of Mathematics & Statistics
Level 3, 2014/2015
MECHANICS OF RIGID
AND
DEFORMABLE BODIES
Lecture Notes Part 1 (3H and 3Q)
1
Mechanics of Rigid and Deformable Bodies
(MRDB)
This course develops a mathematical description of the
2. Central forces
2.1. The force of gravity
The sunplanet system inspired the development of the Newtonian model, which embodies
Newtons law of gravitation. This states that
Any two particles, of masses m1 and m2 , attract one another with a force proport
School of Mathematics & Statistics
3H/3Q Mechanics of Rigid and Deformable Bodies
Problem Sheet 1 Solutions
Solution 1
The velocity and acceleration are
x = 2e1 2te2 ,
The speed is
v=
x = 2e2 .
22 + (2t)2 = 2 1 + t2 .
In parametric form, the path of the p
4. Rotating frames of reference
A frame of reference which is rotating relative to an inertial frame is not itself inertial.
Examples
(a) A frame xed on a roundabout.
(b) A frame xed on a rotating planet.
(c) A frame xed in a train moving on a curved trac
University of Glasgow
School of Mathematics & Statistics
Level 3, 2014/2015
MECHANICS OF RIGID
AND
DEFORMABLE BODIES
Lecture Notes Part 3 (3H)
97
8. Continuously deformable materials in one dimension
8.1. Introduction
Classical mechanics is concerned with
3. Energy
3.1. The general case and conservative forces
According to Denition 1.6, the work done by a force F in moving a particle from x to
x + dx is
dW = F dx.
The total work done by F in moving the particle from a point A to a point B along a path
C wi
School of Mathematics & Statistics
3H/3Q Mechanics of Rigid and Deformable Bodies
Problem Sheet 2 Solutions
Solution 1
(i)
r2 = h,
Put
and m r
1
u= ,
r
and
r=
d
r=
dt
d
dt
h2
r3
= F (r).
(1)
so that = hu2
1
u
=
du
h
d
1 du
du
= h ,
u2 d
d
2
2
d u = h2
School of Mathematics & Statistics
3H Mechanics of Rigid and Deformable Bodies
Problem Sheet 6 Solutions
Solution 1
(a) From
LX =
(xi X) [mi (xi X)]
i
we obtain
0
:
dLX
[mxi X)] +
i(
(xi X)
(xi X) [mi (i X)]
x
=
dt
i
i
=
(xi X) (mi xi )
(xi X) (mi X
Exam paper 2014
The material covered in 2014 differed to some extent from
that in the syllabus covered in 2015 and not all the exam questions in 2014
are relevant in 2015
Therefore, questions A3, B3, B4 and their solutions have been deleted from
the follo
School of Mathematics & Statistics
3H Mechanics of Rigid and Deformable Bodies
Notes on the 2012 Examination and Solutions
The following notations are equivalent
r and x,
and x
r
and er ,
r
and e
In two dimensions
The vector product symbol
and
Cartes
School of Mathematics & Statistics
3H/3Q Mechanics of Rigid and Deformable Bodies
Problem Sheet 3 Solutions
Solution 1
[Using x, y, z instead of x1 , x2 , x3 .]
We have curl F = (0, 0, cy 2by) = (0, 0, 0) c = 2b. Hence F is conservative if and only
if c =
School of Mathematics & Statistics
3H/3Q Mechanics of Rigid and Deformable Bodies
Problem Sheet 5 Solutions
Solution 1
We have x = rer and, in the frame S (in which er and e are xed),
2x
= r er .
t2
x
= rer ,
t
The frame S rotates with angular velocity =
School of Mathematics & Statistics
3H Mechanics of Rigid and Deformable Bodies
Problem Sheet 7 Solutions
Solution 1
(a(t), 0)
l cos
l
l sin
.
x(t)
m
The position vector of the mass m relative to the xed origin (0, 0) is
x(t) = a(t)i + l sin i l cos j,
w
School of Mathematics & Statistics
3H Mechanics of Rigid and Deformable Bodies
Problem Sheet 8 Solutions
Solution 1
As in MRDB Lecture Notes Section 8.3.3, the equilibrium equation satised by the vertical
column is
dT
g(x) +
= 0,
dx
where T = T (x) is the
School of Mathematics & Statistics
3H Mechanics of Rigid and Deformable Bodies
Problem Sheet 9 Solutions
Solution 1
d4 y
The deformation of the beam satises the dierential equation 4 = 0, which has general
dx
solution
y(x) = a0 + a1 x + a2 x2 + a3 x3 .
Ca
9. Mechanics of beams
9.1. Introduction
Understanding the mechanics of beams is important for the construction of many mechanical devices ranging from the nanometer/micron sized cantilevers of micro-electromechanical systems (MEMS) and atomic force micros
School of Mathematics & Statistics
3H/3Q Mechanics of Rigid and Deformable Bodies
Problem Sheet 4 Solutions
Solution 1
For a particle of mass m acted on by a central force F = F (r)er , Newtons second law N2
in polar coordinates is
m[( r2 )er + (r + 2r)e
Summary of key mechanics ideas from 2C required for MRDB
Parametric curves
Constant velocity and constant acceleration
A parametric curve is a function x : I Rn
where n is 2 (plane curve) or 3 (space curve). If a point particle moves with constant velocit