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
coordinatesarerelatedb
Dynamics and Relativity: Example Sheet 2
Professor David Tong, February 2013
1. Aparticlemovesinafixedplaneanditspositionvectorattime t is x.Let(r,)
beplanepolarcoordinatesandletrand beunitvectorsinth
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
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
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 sma
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 d
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, supp
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 fa
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 happe
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
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
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
Mathematical Tripos Part IA
Lent term 2011
Dynamics and Relativity, Examples sheet 3
Dr S.T.C. Siklos
Comments and corrections: e-mail to [email protected]
1
A square hoop ABCD is made of ne smooth wire and h
Mathematical Tripos Part IA
Lent term 2011
Dynamics and Relativity, Examples sheet 2
Dr S.T.C. Siklos
Comments and corrections: e-mail to [email protected] The starred parts of questions should be left until
Mathematical Tripos Part IA
Lent term 2011
Dynamics and Relativity, Examples sheet 4
Dr S.T.C. Siklos
Comments and corrections: e-mail to [email protected] A commentary for supervisors is available.
1
A clock
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 s
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 n
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
Mathematical Tripos Part IA
Lent term 2011
Dynamics and Relativity, Examples sheet 1
Dr S.T.C. Siklos
Comments and corrections: e-mail to [email protected] Starred questions or parts of questions should be le
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
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 gravita
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)
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 ca
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 Newto
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
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