Phy 105A: Analytical Mechanics
28 June 2016
Manuel Calderón de la Barca Sánchez
Problem Solving: Analytical Thinking
 Reading: necessary, but not sufficient
 Solving problems: active part of learning.
 True test of understanding:
ability to solve problems
ability to explain solution to someone else
find at least two ways to solve a problem
 I’ll throw lots of problems at you!
 All are different…
Are there strategies that are useful in general?
28 June 2016
MCBS
Strategies
 Draw a diagram
 Known knowns and known unknowns
 Algebra is your friend.
Working with symbols is faster than
working with numbers.
 Use Dimensional analysis:
How fast are
you going?
Can tell you if you’re wrong.
 Use Limiting cases.
 Check orders of magnitude
Estimating things to nearest power of 10.
28 June 2016
Estimate the number of hairs on your head.
Estimate your angular momentum about the
center of the earth.
MCBS
I’m going at
5 kg/hour
Dimensional Analysis: Pendulum
 A mass m hangs from a massless rod of length ℓ and
swings under a gravitational field g.
What can we say about the frequency?
What can we say about the total energy?
q
28 June 2016
MCBS
g
ℓ
m
Example: Ball dropped with air drag
 A ball is dropped from rest at height h.
Assume the drag
force takes the form F=-mva. You are told that the ball’s
position and velocity is given by
g
v(t) = - (1- e )
a
-a t
gæ 1
-at ö
y(t) = h - çt - (1- e ) ÷
ø
aè a
How can you trust these expressions? Could they be valid?
Check limiting cases, “small t ” or “ large t ”, what do you
expect should happen?
28 June 2016
MCBS
Additional Practice Examples
 Spring (Units/Dimensional Analysis):
A spring with spring constant k has a mass m on its end. The
spring force is F=-kx, where x is the displacement from the
equilibrium position. What can we say about the frequency of
the oscillations?
 Circular Pendulum (Limiting cases):
A mass hangs from a string of length ℓ. Conditions have been
set so that the mass swings around in a horizontal circle, with
the string making a constant angle q with the vertical. We will
g
w=
find later that
cosq
– Check limits for q at 0 and at a right angle.
28 June 2016
MCBS