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CHAPTER 1
Understanding Units In Physics almost every number comes with a unit. The units of a number explain to everyone paying attention exactly what standard the observer has used in the measurement. The units are part of the number and must be retained. Base Units and Derived Units •To create a system of measurement, a small set of base units must be defined. Only these need to be defined in terms of outside standards. We will be using the MKS base system. Not surprising the three base units are:
- Meter (length)
- Kilogram (mass)
- Second (time) •By contrast, derived units are defined as combinations of the base units, needing no other outside standard. Examples include:
- meters per second, m/s (speed and velocity)
€ kg⋅m s (momentum)
€ kg m⋅s^2 , a.k.a the Newton (force) Watch your units! Many times you must make sure they are in the same base (SI vs. British, for example). Also, you must watch your prefixes (convert centimeters to meters, for example).
Converting Units Whether you are working with base or derived units, you must be comfortable converting between different standards of measurement. The key to converting units is understanding that you are just multiplying by a well chosen one. What does this mean? Example 1: How many minutes are in 2.3 hours?
Example 3: Chapter 1, # A bottle of wine known as a magnum contains a volume of 1. liters. A bottle known as a jeroboam contains 0.792 U.S. gallons. How many magnums are there in one jeroboam? (From the back of the text, 1 U.S. gallon = 3.785 liters.)
Chapter 1 5 Dimensional Analysis Each unit measures a certain “dimension.” Following the book’s example, we will let certain letters represent different dimensions: L = dimensions of length (meters, feet…) M = dimensions of mass (grams, slugs…) T = dimensions of time (seconds…) The game here is to take an equation and replace variables with the letters above. If an equation is dimensionally correct, both sides of the equation will give the same “dimension” after some algebra is done. During addition or subtraction dimensions are like units. €
L + L =
€
L/T+ L/T =
€
L + M =
During multiplication or division dimensions act like numbers. €
L
L
€
ML + MT =
€
MTL
LT+ LM
Example 5: Which of these equations is dimensionally correct? In this example x denotes position (dimensions of length), v denotes speed (dimensions of length divided by time), a denotes acceleration (dimensions of length divided by time squared) and t denotes time. €
x
final
= x
inital
+ v
inital
t + a^2 t
€
x
final
= x
inital
+ v
inital
t + at
€
x
final
= x
inital
+ v
inital
t + a^2 t
€
x
final
= x
inital
+ v
inital
t + at
- neither equation
- both equations
