Dimensional+Analysis

=Dimensional Analysis=

Introduction
Dimensional Analysis is a basic and essential component of AP Physics. Most Dimensional Analysis is preliminary work. You must be able to perform the operations quickly in order to begin the problem proper, although many actual AP question do not require evaluation.

Basics
//Units// are the measurement of a //quantity//. For example, the //quantity// velocity is measured in the //units// of distance per time, i.e. meters per second (m/s), miles per hour (mi/h), or any other combination of a distance over time. The fundamental units, which are the foundation of physics, are distance, mass, and time. The fundamental units compose most of the equations you will encounter. The most commonly used standard of measurement is the SI (System International) standard. You will need to know most of the SI units, their abbreviations, and their corresponding quantities.

Foming New Equations
Here we will take two different equations and use dimensional analysis to derive a third. You will need to be able to manipulate equations frequently throughout the year.

Solve Equation A for Δx, making Equation B. A) v avg = Δx/Δt (v o + v f )/2 = Δx/Δt B) [(v o + v f )/2] × Δt = Δx

Solve Equation C for Δt, making Equation D. C) v f = v o + a avg × Δt v f – v o = a avg × Δt D) (v f – v o )/a avg = Δt

Substitute Equation D into Equation B, making Equation E. E) Δx = (v o + v f )/2 × (v f – v o )/a avg

Simplify Equation E, making Equation F. Δx × 2 × a avg = (v f – v o ) × (v f – v o ) Δx × 2 × a avg = v o v f – v o 2 + v f 2 – v o v f Δx × 2 × a avg = – v o 2 + v f 2 Δx × 2 × a avg + v o 2 = v f 2 F) v ﻿2 = v o 2 + 2aΔx

Finding Units of a Quantity
Here we will use units of an equation to find units of a quantity and constant. This demonstrates an important rule: the units of one side of an equation __must__ match those ofthe opposite.

m = kg We can derive the units of acceleration (a) with the equations vavg = Δv/Δt and aavg = Δv/Δt. Replace the variables with units and solve.
 * 1.** Given F = ma, find the units of the quantity Force.

First: v avg = Δx/Δt ? = m/s = v

Then: a avg = Δv/Δt ? = (m/s)/s = m/s 2 = a

Thus in F = ma, m ×a = kg × m/s 2, making F = kg × m/s 2 kg × m/s 2 = Newton (N) The unit for Force is the Newton.

First solve the equation for k. F s = –kΔx F s /Δx = –k
 * 2.** Given F s = –kΔx, find the spring constant k.

Then supplant the variables with the units. x = m F = N = kg × m/s 2 –[(kg × m/s 2 )/m]= ? –kg/s 2 =? The units for k are –kg/s 2

Converting Units
The third use for dimension alanalysis is to convert units to different measurement standards. You will need to know a few conversion factors. Some conversions are common sense like converting minutes to seconds, others however, you will have to memorize, such as converting miles to meters or vice versa. Most answers to problems that involve conversion must be given in SI units. Here is an example problem that involves dimensional analysis.

v avg = Δx/Δt v avg =100mi/2h v avg = 50mi/h v avg = 50mi/h × 1m/0.621mi × 60min/1h × 60s/1min = 289855m/s Make sure to writeyour answer appropriately: 290km/s
 * 1.** Find the average velocity of a plane that flew 100 miles in 2.00 hours in SI units.