Physics 105A
Analytical Mechanics
If you don’t know that
bodies at rest tend to stay
at rest, you don’t know the
First thing about
mechanics, dude!
28 June 2016
Manuel Calderón de la Barca Sánchez
Philosophia Naturalis Principia
Mathematica
28 June 2016
MCBS
Newton’s Laws
 1st Law: Inertia.
A body moves with constant velocity unless acted on by a
force.
 2nd Law: “F=ma”
The time rate of change of the momentum of a body equals the
force acting on the body.
 3rd Law: Action-Reaction
For every force of action on one body, there is an equal and
opposite reaction force on another body.
28 June 2016
MCBS
Implications: Inertial mass
 Inertia: resistance to motion.
 Apply same F to two bodies,
If a change in speed, you
want; a force must you apply.
If you keep a constant force,
and a larger acceleration,
find you, then smaller, must
the mass be! Yes!
measure their accelerations, a1, a2.
If a1 < a2, body 2 has less resistance to motion.
F = m1a1 = m2 a2 : a1 = m2 : a2>a1, m1 > m2.
a2
28 June 2016
m1
MCBS
The kilogram standard
The kilogram is the last physical quantity of worldwide use that is
defined not by a universal property of nature, but by a specific
physical object!
 The kilogram standards are diverging by a few tens of micrograms
over the centuries, and we don’t know why.
 Last year, the GCPM discussed redefining the kg using Planck’s
constant, but it is still not standard.

28 June 2016
MCBS
mg vs. mi
 2nd Law:
SF = mi a,

Universal Gravitation:
Fg = Gm1gM2g/r122

Coulombs Law:
Fc = Kq1eQ2e/r122


28 June 2016
MCBS
We know q1e and mi are not the
same.
How come m1g is the same thing as
mi?
General Relativity: Matter tells
space how to bend, space
shows matter how to move.
 1st Law:
Notes about the three laws
Defines what is meant by “zero force”.
Defines an “inertial frame”: one in which the first law holds.
Holds for all free particles in an inertial frame.
 2nd Law:
Allows us to be quantitative!
Implies the first law.
– Would not be true if F was prop to v, or to third derivative of position.
Talks about a quantity, F, that is independent of the body or how it
moves. Remarkable that it tells us something about the body’s intrinsic
properties, and how it moves through space!
 3rd Law: Implies
28 June 2016
(or is equivalent to) momentum conservation.
MCBS
Using Newton’s Laws
 Given a force F, apply F=ma to find acceleration.
 Knowing a = d2x/dt2 and two initial conditions, find x(t) and
v(t).
 Two kinds of problems:
You are given a physical setup, find all forces.
You are given a force, F(t), or F(x) or F(v), solve differential
equation
F = ma = mx
28 June 2016
MCBS
A plane and two masses
Mass M1 is held on a plane with inclination angle q, mass M2
hangs over the side.
The masses are connected by a massless string which passes
through a massless, frictionless pulley.
Coefficient of kinetic friction between M1 and the plane is m.
M1 is released from rest.
Assuming that M2 is sufficiently large such that M1 gets pulled
up the plane, what is the acceleration of the masses? What is
the tension in the string?
q
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m
MCBS
Platform and Pulley


A person stands on a platform and pulley
system.
The masses of the platform, person and pulley
are M, m and m, respectively.
Assume the mass of the pulley is all in its center.
The rope is massless.
 Let the person pull up so both the person and
platform have an acceleration a upward.
 Assume that the platform is constrained to stay
level (perhaps via rails).


Find
the tension in the rope,
the normal force between the person and the platform,
and
the tension in the rod connecting the pulley and the
platform.
MCBS
28 June 2016
Masses and pulleys: Atwood’s machines

Atwood’s machine:
Any system consisting of:
masses,
strings, (massless, for now)
pulleys (massless, for now)
m1
 Example at right:
m2
Massless strings and pulleys.
Find acceleration of masses, and Tension of string.
 Key to Atwood’s machine problems:
Write down the F = ma
Relate the accelerations by noting that length of string doesn’t change
28 June 2016 –
“Conservation of string”
MCBS
Homework: Atwood’s Machines
Atwood’s 1
 Consider the Atwood’s machine in the
figure. It consists of three pulleys, a
short piece of string connecting one
mass to the bottom pulley, and a
continuous long piece of string that
wraps twice around the bottom side of
the bottom pulley and one around the
top side of the top two pulleys. Assume
all string segments are essentially
vertical. Find the accelerations of the
masses.

28 June 2016
MCBS
m
2m