Dynamics and Relativity: Example Sheet 1
Professor David Tong, January 2013
1. Inonespatialdimension,twoframesofreference S and S havecoordinates(x, t) and(x, t)respectively.The
coordinatesarerelatedbyt=tand
x = f (x, t)
Viewedfromframe S,aparticlefollows
Dynamics and Relativity: Example Sheet 2
Professor David Tong, February 2013
1. Aparticlemovesinafixedplaneanditspositionvectorattime t is x.Let(r,)
beplanepolarcoordinatesandletrand beunitvectorsinthedirectionofincreasing
randincreasingrespectively.Showt
Dynamics and Relativity: Example Sheet 4
Professor David Tong, February 2013
1. A clock C is at rest at the spatial origin of an inertial frame S. A second clock
C is at rest at the spatial origin of an inertial frame S moving with constant speed
u relati
Dynamics and Relativity: Example Sheet 3
Professor David Tong, February 2014
1. In a system of particles, the ith particle has mass mi and position vector xi with respect
to a fixed origin. The centre of mass of the system is at R. Show that L, the total
5. Systems of Particles
So far, weve only considered the motion of a single particle. If our goal is to understand
everything in the Universe, this is a little limiting. In this section, we take a small step
forwards: we will describe the dynamics of N, i
4. Central Forces
In this section we will study the three-dimensional motion of a particle in a central force
potential. Such a system obeys the equation of motion
mx =V(r)
(4.1)
where the potential depends only on r = |x|. Since both gravitational and el
6. Non-Inertial Frames
We stated, long ago, that inertial frames provide the setting for Newtonian mechanics. But
what if you, one day, find yourself in a frame that is not inertial? For example, suppose that
every 24 hours you happen to spin around an ax
7. Special Relativity
Although Newtonian mechanics gives an excellent description of Nature, it is not universally valid. When we reach extreme conditions the very small, the very heavy or the
very fast the Newtonian Universe that were used to needs repla
2. Forces
In this section, we describe a number of dierent forces that arise in Newtonian mechanics. Throughout, we will restrict attention to the motion of a single particle. (Well
look at what happens when we have more than one particle in Section 5). W
3. Interlude: Dimensional Analysis
The essence of dimensional analysis is very simple: if you are asked how hot it is outside,
the answer is never 2 oclock. Youve got to make sure that the units agree. Quantities
which come with units are said to have dim
Lent Term, 2013
Preprint typeset in JHEP style - HYPER VERSION
Dynamics and Relativity
University of Cambridge Part IA Mathematical Tripos
David Tong
Department of Applied Mathematics and Theoretical
Physics, Centre for Mathematical Sciences,
Wilberforce
Chapter 4
Rotating Frames
4.1
4.1.1
Angular velocity
The concept of angular velocity
Recall that in Chapter 3, we discussed the motion of a particle conned to a circle. In this situation,
the concept of linear velocity was replaced with angular velocity (
Mathematical Tripos Part IA
Lent term 2011
Dynamics and Relativity, Examples sheet 3
Dr S.T.C. Siklos
Comments and corrections: e-mail to stcs@cam.
1
A square hoop ABCD is made of ne smooth wire and has side length 2a. The hoop is horizontal and
rotating
Mathematical Tripos Part IA
Lent term 2011
Dynamics and Relativity, Examples sheet 2
Dr S.T.C. Siklos
Comments and corrections: e-mail to stcs@cam. The starred parts of questions should be left until you have done all the
other questions. A commentary is
Mathematical Tripos Part IA
Lent term 2011
Dynamics and Relativity, Examples sheet 4
Dr S.T.C. Siklos
Comments and corrections: e-mail to stcs@cam. A commentary for supervisors is available.
1
A clock C is at rest at the spatial origin of an inertial fram
Chapter 1
Basic concepts
1.1
1.1.1
Newtons laws of motion
Statement of the laws
The laws governing the whole of this course except for the section on Special Relativity are Newtons
three laws, so it seems right to bang them down on the table at the very b
Chapter 3
Orbits
3.1
Motion in a plane
You might wonder why there is a section on motion in two dimensions when vector methods can
be readily used to study motion in three dimensions or an arbitrary number of dimensions. The
answer is that in the case of
Chapter 2
Forces
We consider here forces acting on a single particle, which may be an idealisation of an extended
body. Force is what appears on the right hand side of Newtons second law, but one does use
Newtons laws to determine whether a force acts: th
Mathematical Tripos Part IA
Lent term 2011
Dynamics and Relativity, Examples sheet 1
Dr S.T.C. Siklos
Comments and corrections: e-mail to stcs@cam. Starred questions or parts of questions should be left until you have
done all the others. A commentary is
Mathematical Tripos Part IA
Dynamics and Relativity
Lent term 2010
Dr S.T.C. Siklos
Hand-out 8: Rolling disc
A uniform disc of mass m and radius a rolls without slipping down a line of greatest slope of
an inclined plane of angle . The plane of the disc i
Mathematical Tripos Part IA
Dynamics and Relativity
Lent term 2011
Dr S.T.C. Siklos
Hand-out 3: Projectile with linear drag
A particle of mass m is projected from the origin at velocity u. The gravitational acceleration
is denoted by g and the drag force
Mathematical Tripos Part IA
Dynamics and Relativity
Lent term 2011
Dr S.T.C. Siklos
Hand-out 1: Motion in a cubic potential
A particle of unit mass moves in a one-dimensional potential (x), where
(x) = x3 3x .
d
The force due to this potential is
(minus
Mathematical Tripos Part IA
Dynamics and Relativity
Lent term 2011
Dr S.T.C. Siklos
Hand-out 4: motion of a point charge in a uniform electromagnetic eld
We wish to solve
m = e(E + r B)
r
()
in the case when E and B are constant (in time) and uniform (sam
Mathematical Tripos Part IA
Dynamics and Relativity
Lent term 2011
Dr S.T.C. Siklos
Hand-out 5: Orbits in a non-inverse square force law.
In this example, we consider the following modication to Newtonian gravity:
f (r) =
k
a
4,
2
r
r
where k = GM as us
Mathematical Tripos Part IA
Dynamics and Relativity
Lent term 2011
Dr S.T.C. Siklos
Hand-out 7: Drum majorettes baton
We model the baton as a light rod of length with masses m1 and m2 attached to the ends. What
happens when the baton is thrown up into the
Mathematical Tripos Part IA
Dynamics and Relativity
Lent term 2011
Dr S.T.C. Siklos
Hand-out 6: Coriolis eect on a falling body
We consider the eect of the coriolis force on a particle dropped from a xed point in the
rotating frame of the Earth the top of