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
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
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 f
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
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)
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 mass
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 + (
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 pla
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
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
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
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
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,
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
I
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
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
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
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 vert
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 ge
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-e
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
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