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Honors Physics 1Class 01Fall 2013

Logical problem solving

Estimation

Significant digits

Dimensional analysis

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Logical and “supralogical” problem solving

In most Physics classes you will be presented with well defined questions and just enough information to solve them. This is just to get the essential ideas across and is not representative of what you’ll face outside.

In research problem solving you will be faced with ill-defined questions and incomplete information with lots of information you don’t need. This is “the real world”.

I will try to present you with some variety. “Clear” problems to get the idea and “ill-defined” problems to get you thinking.

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Logical approach:

Read the problem thoroughly. (Diagram the sentence if necessary.)

Decide what you want to find out.

Identify the essential physics underlying the problem. (Make sketches, diagrams, plots, charts. Pay special attention to symmetries.)

Gather information. List what you know.

Identify relationships that could help the information you want from the information you have.

Make some estimates or predictions so you can check your arithmetic as you go along.

Logical and “supralogical” problem solving

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A supralogical approachSometimes you cannot see a clear path from what you

know to what you want.• Start out “playing” with what you know to see if

something useful comes from the manipulation.• Start working the problem in a certain direction until

you see whether this approach will lead to a path or not.

• Use brainstorming techniques to come up with new approaches.

• Analyze your thinking after the fact to see if you can add a new tool to your set.

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Tools

Math you have certainly had: Vectors, Algebra, Trig, Geometry

Math you may not have yet had: Differential equations, Integral equations– Newton’s Laws

Physical numbers

Conservation laws– Energy– Linear Momentum– Angular momentum

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Significant digits

Some rules:

The leftmost non-zero digit is the most significant digit.

If there is no decimal place the rightmost non-zero digit is the least significant digit.

If there is a decimal point, then the right most digit is the least significant digit, even if it is zero.

When reporting results, don’t report more significant digits than is consistent with the information that went into the calculation. (More on uncertainty propagation later.)

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Uncertainty estimation

The uncertainty in an input number is either as it is stated or +/-1 in the least significant digit.

To fully compute uncertainty in a result you should add or subtract the uncertainty in each input number to maximize and minimize the result.

There are simple approaches to estimate uncertainty in a result based on calculus (Taylor expansion).

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Estimation (“Fermi”) problems

• How many kids laughed so hard yesterday that a beverage came out their noses?

• How many liters of water do all Rensselaer students flush each day?

• How far in meters does an automobile tire go for one monolayer of atoms to be worn from the tire?

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Nose-Beverage estimate

6 x 109 people

Average years as a “kid” ~ 10

Average days as a kid ~4000

Average number of time beverage event occurs/kid ~ 2?

How many are spurting milk right now? (5E5*10sec/(80000 sec/day)~50)

956 10 1 10 .

# events 2 5 104000 60 .

all day kid yrs events

days all yrs kid

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Dimensional analysis

You can frequently get to the essence of a problem or figure out how physical parameters are related just by matching up the units or dimensions.

Basic dimensions - MKSAMass (M) - kgLength (L) - mTime (T) - sCurrent (I) - ATemperature - K

Useful parameters:Velocity (LT-1)Energy (ML2T-2) – JMomentum (MLT-1)Power (Energy/T) - WCharge (IT) - CFrequency (T-1) - Hz

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Activity

• Based on dimensional analysis alone, find a relationship between the kinetic energy and velocity of an object.

• Based on dimensional analysis alone, find a relationship between the momentum and kinetic energy of an object.