The thrust of a rocket is an excellent example of how to use the law of conservation of
momentum for an isolated system. Consider an isolated (deep space) rocket of mass M, at time t,
moving with velocity v, as shown in the figure on the left. An instant
Random walks.
Consider flipping a coin, for each head you take a step to the right, tail to the left. After N steps
how far will you be away from your starting point? This problem may not seem like a physics
question, but already in class we have come acr
Energy Dissipation in an Oscillating System
We have learned that oscillations are a natural response of a system to being disturbed from a
stable equilibrium. If the amplitude of the oscillation is small then harmonic motion results, for
larger amplitude
Driven Oscillations and Resonance
In the last section we learned about damping and how a damping force can remove energy from
the system, but what about the opposite effect? Can a force do positive work and feed energy
into the system? A constant force wi
Driven Oscillations and Resonance
We have learned that oscillations are a natural response of a system to being disturbed from a
stable equilibrium. If the amplitude of the oscillation is small then harmonic motion results, for
larger amplitude anharminic
Keplerian Orbits
1
force law their trajectories can be
r2
reduced to conic sections. This is called Keplers problem after Johannes Kepler who studied
planetary motion. If one of the particles is so massive that it can be approximated as fixed, then
the ot
The Lorentz factor gamma
Einsteins theory of relativity supersedes Newtonian mechanics. However, that does not mean
that all mechanics problems need to be solved relativistically. Newtons second law still holds:
dP
F , however the definition of the momen
Kinematics and dynamics, an analytic approach
Given a (t ) find v (t ) and r (t )
a (t )
t
t0
dv (t )
dv (t ) a (t )dt
dt
t
t
t0
t0
dv (t ) a (t )dt v (t ) v (t0 ) a (t )dt
t
(1) v (t ) a (t ) dt v (t0 )
t0
therefore given a (t ) and the initial velocit
Electron conduction in a metal
We have just solved approximately a problem, uniformly accelerating electron, that you could
have done exactly while in high school. The power of the computational techniques becomes
apparent in a slightly more complicated p
Kinematics and dynamics, a computational approach
We begin the discussion of numerical approaches to mechanics with the definition for the velocity
r
r (t t ) r (t )
lim
or r (t t ) r (t ) v (t )t for small t. This definition
t 0 t
t 0
t
comes from math
Molecular Dynamics
Physics Nobel Prize winner Richard Feynman once said that "If we were to name the most powerful
assumption of all, which leads one on and on in an attempt to understand life, it is that all things are made
of atoms, and that everything
Center of momentum frame
Newtons laws of motion hold in any inertial frame of reference. However, it is often simplest to
examine an isolated system of particles from the frame of reference where the total momentum is
zero. Since the system is isolated, t
Small oscillations
The theory of small oscillations is an extremely important topic in mechanics. Consider a system
that has a potential energy diagram as below:
U
C
B
c
C
A
x
There are three points of stable equilibrium, A, B, and C. The general coordina
Force and dynamics with a spring, analytic approach
It may strike you as strange that the first force we will discuss will be that of a spring. It is not
one of the four Universal forces and we dont use springs every day. Or do we? The basic
motion of a s
Gamma
Einsteins theory of relativity supersedes Newtonian mechanics. However, that does not mean
that all mechanics problems need to be solved relativistically. Newtons second law still holds:
dP
= F , however the definition of the momentum for a particle
Many of the foundations of classical mechanics were first discovered when philosophers (early
scientists and mathematicians) tried to explain the motion of planets and stars. Newton is most
famous for unifying the motion of heavenly bodies with those on e
Sound in solids Using the spring model for solids immediately leads to the question, What
happens if I disturb one of the atoms from its equilibrium position? You know the answer, it
will experience a force that acts to return it to equilibrium. However i
Contact Forces So far we have not dealt with contact forces, i.e. normal, tension, etc. Consider
this very simple situation to illustrate the key issues. Two blocks in contact are in deep space,
i.e. gravity plays no role; a force is applied as shown in t
Air resistance or drag is a very common type of friction experienced in many situations, a leaf
falling from a tree, riding your bicycle or a jet flying through the air. It is often impossible to
ignore the effects of air drag, even though many introducto
Kinematics and dynamics, an analytic approach
Given a (t ) find v (t ) and r (t )
a (t )
t
t0
dv (t )
dv (t ) a (t )dt
dt
t
t
t0
t0
dv (t ) a (t )dt v (t ) v (t0 ) a (t )dt
t
(1) v (t ) a (t ) dt v (t0 )
t0
therefore given a (t ) and the initial velocit
Work
Work is the change in energy of a system (neglecting heat transfer). To examine what could
2
L ML
cause work, lets look at the dimensions of energy: E M 2 L F L so
T T
dimensionally energy is equal to a force times a length. Consider a constant fo
Energy
So far we have learned just one fundamental principle of physics, the second Newtons law.
dp
Mathematically, it is expressed
F this implies that P constant for an isolated
total
dt
system. We have also learned two fundamental forces:
m1m2
qq
r and
Relativistic Dynamics
Many students are interested in how Einsteins theory of relativity affects our understanding of
mechanics. The answer is: profoundly. To thoroughly understand relativity, first special relativity and
then general, you must suspend yo