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Review of Undergraduate Physics

(GRE Physics Preparation)PH 5090

(1)Measurement & Dimension Analysis

Amir Shahmoradi

Summer 2010



SI base unitsle Système I nternational d'unités

The rest of units are defined as set combinations of these seven.FYI:

The most well determined physical constant:

Rydberg constant, up to a relative accuracy of ~8×10−.

The least well determined physical constant:

Newtonian constant of gravitation, up to a relative accuracy of ~10−.

Physical Quantity Unit name Symbol

Length meter m

Mass kilogram kg

Time interval second s

Electric current ampere A

Temperature kelvin K

Amount of substance mole mol

luminous intensity candela cd



SI base unitsle Système I nternational d'unités

Unit Definition

meter The length of the path traveled by light in vacuum during a time interval of 1/299 792 458

of a second.

kilogram The international standard of mass is Platinum-Iridium cylinder kept at the International

Bureau of Weights and Measures.

second The duration of 9 192 631 770 periods of radiation corresponding to the transition

between the two hyperfine levels of the ground state of the Cesium 133 atom.

ampere The current that if maintained in two straight parallel conductors of infinite length, of

negligible circular cross-section, and placed 1 meter apart in vacuum, would produce a

force between the conductors equal to 2 × 1 0 newton per meter of length.

kelvin The fraction 1/273.16 of the temperature of the triple point of water.

mole Amount of substance that contains as many elementary entities as there are in 0.012 of

Carbon 12. The elementary entity can be anything like atoms, molecules, ions, electrons.

candela The luminous intensity, in a given direction, of a source that emits monochromatic

radiation of frequency 540 × 10 (green color) and that has a radiant intensity in that

direction of 1/683 watt per steradian.



SI prefixesle Système I nternational d'unités

factor prefix symbol factor prefix symbol

10 exa E 10− atto a

10 peta P 10− femto f

10 tera T 10− pico p

10 giga G 10− nano n

10 mega M 10− micro µ

10 kilo k 10− milli m

10 hecto h 10− centi c

10 deca da 10− deci d



Recognized non-SI units

physical quantity name symbol SI value

area barn b 10−

energy electron volt eV 1.6 × 10−

length angstrom 10−

length fermi fm 10−

length micron µm 10−

pressure bar bar 10 −

time day d 86400

mass tonne t 10

volume litre l, L 10−

Example:

Cross-section of the Uranium nucleus: ~1



Some Approximate Lengths

Measurement Length

Distance to Andromeda galaxy 2 × 1 0

Distance to nearest star (Proxima Centauri) 4 × 1 0

Distance to Pluto 6 × 1 0

Radius of Earth 6 × 1 0

Height of Mt. Everest 9 × 1 0

Thickness of book page 100

Length of a typical virus 100

Radius of hydrogen atom 0.5

Radius of proton 1

Planck length 2 × 1 0−



Some Approximate Time Intervals

Measurement Length

Lifetime of proton (predicted) 1 × 1 0

Age of the universe 5 × 1 0

Human life expectancy 2 × 1 0

Length of a year ~3.15 × 10 ≈ × 10

Length of a day 86400

Mean lifetime of free neutron 886 ≈ 15

Average time between human heartbeats 0.8

Muon lifetime 2

Lifetime of the most unstable particle 1 × 1 0−

The Planck time 1 × 1 0−



Some Approximate Masses

Object Mass

Known universe 1 × 1 0

Our galaxy 2 × 1 0

Sun 2 × 1 0

Earth 6 × 1 0

Moon 7 × 1 0

Elephant 5 × 1 0

Planck mass 2 × 1 0−

Speck of dust 7 × 1 0−

Proton 2 × 1 0−

Electron 9 × 1 0−



Why is it important to know order of

magnitude approximations?Sample question from one of the ETS GRE Physics exams:



Problem Solving Strategies1. Dimensional Analysis

The SI dimensional basis:

quantity dimension

length (m) L

time (s) T

mass (kg) M

electric current (A) I

temperature (K) Θ

amount of substance (mol) N

luminous intensity (cd) J

Basic idea:

1. Find the dimension of the quantity to be obtained in the question.2. Combine the quantities given in the question to build a new entity that has

the dimensions of the requested quantity.




Q1: What is the dimension of acceleration, force, weight, pressure, energy,Hamiltonian, Lagrangian, action, angular speed, angular momentum, capacitance,gravitational constant, electric displacement, electric field, magnetic dipole moment,magnetization?




Q2: Based on the three physical constants, ℏ,,, can you derive the Planck mass,

length & time?




Q3: A mass hangs from a massless string of length . The acceleration due to gravity is .

What can we say about the frequency of oscillations?

What can we say about the energy of the system at any moment during the oscillation?




Q4: A spring with spring-constant has a mass on its end. The force is() = , where is the displacement from equilibrium.

What can we say about the frequency of oscillations?What can we say about the energy of the system




Q5: A satellite travels in an orbit just above the earth’s surface. What can we

say about its speed?



Problem Solving Strategies2. Approximations, limiting cases

Q: The time-averaged potential of a system of electric charges is given by

Φ =

4ε

−

1 +

α

2,

What can you say about the system and the charge distribution?

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