Dimensional Analysis

Summary

This video script guides viewers through various dimensional analysis problems, starting with calculating the number of seconds in a year. It then demonstrates converting miles per hour to meters per second, explaining the necessary length and time unit conversions. The script also covers converting the density of aluminum from kilograms per cubic meter to grams per milliliter, and finally, it calculates the area of a door mat in both square inches and square feet. The lesson concludes with determining the time required to read a 200-page book based on reading speed, emphasizing the importance of unit conversion in problem-solving.

Takeaways

• 📚 The lesson focuses on solving dimensional analysis problems by converting units step by step.
• ⏳ To find the number of seconds in a year, convert from years to days, then days to hours, hours to minutes, and finally minutes to seconds.
• 🗓 The average number of days in a year is 365, but for more accuracy, consider 365.25 to account for leap years.
• 🔢 The conversion process involves using multiplication of numerators and cancellation of units to isolate the desired unit, in this case, seconds.
• 🚗 To convert speed from miles per hour to meters per second, first convert miles to kilometers and then to meters, and convert hours to minutes and then to seconds.
• 🔄 Conversion factors are essential in dimensional analysis, such as 1 mile = 1.609 kilometers and 1 hour = 60 minutes.
• 🧩 The density of aluminum is given in kilograms per cubic meter, but can be converted to grams per milliliter by adjusting for mass and volume units.
• 📏 To convert from kilograms to grams, multiply by 1000, and to convert from cubic meters to milliliters, divide by 1,000,000.
• 📐 The area of a rectangle is calculated by multiplying its length by its width, resulting in square units.
• 📏 To convert square inches to square feet, divide by the square of the conversion factor from inches to feet (12 inches = 1 foot).
• ⏱ John's reading speed can be used to calculate the total time needed to read a book by converting pages to minutes and then to hours.

Q & A

How many seconds are there in a non-leap year?

In a non-leap year, there are 31,536,000 seconds. This is calculated by multiplying 1 year by 365 days, then by 24 hours, 60 minutes, and 60 seconds each.

What is the average number of seconds in a year, including leap years?

The average number of seconds in a year, accounting for leap years, is approximately 31,557,600 seconds, using the average of 365.25 days per year.

How can you convert miles per hour to meters per second?

To convert miles per hour to meters per second, first convert miles to kilometers using the conversion factor 1 mile = 1.609 kilometers, then convert kilometers to meters (1 kilometer = 1000 meters), and finally convert hours to seconds (1 hour = 60 minutes, 1 minute = 60 seconds).

What is the conversion factor from miles to kilometers?

The conversion factor from miles to kilometers is 1.609 kilometers per mile.

How many meters are there in a kilometer?

There are 1000 meters in a kilometer, as 'kilo' means a thousand.

What is the density of aluminum in grams per milliliter?

The density of aluminum is 2.7 grams per milliliter, obtained by converting the density from kilograms per cubic meter to grams per cubic centimeter and then to grams per milliliter.

How can you convert the density from kilograms per cubic meter to grams per milliliter?

To convert density from kilograms per cubic meter to grams per milliliter, divide the density in kilograms per cubic meter by 1,000,000 (since 1 cubic meter equals 1,000,000 cubic centimeters and 1 cubic centimeter equals 1 milliliter).

What is the area of a rectangular door mat with dimensions 26 inches by 18 inches in square inches?

The area of the door mat is 468 square inches, calculated by multiplying the length (26 inches) by the width (18 inches).

How many square feet is the area of the door mat with the given dimensions?

The area of the door mat is 3.25 square feet, obtained by converting the area from square inches to square feet using the conversion factor where 12 inches equal 1 foot.

If John can read 15 pages every 45 minutes, how long will it take him to read a 200-page book?

It will take John 10 hours to read a 200-page book, calculated by converting the reading rate from pages per 45 minutes to hours.

What is the conversion factor from inches to feet?

The conversion factor from inches to feet is 12 inches per foot.

Outlines

00:00

🕒 Conversion of Time Units

This paragraph introduces the concept of dimensional analysis with a focus on converting time units. It explains the process of converting a year into seconds by sequentially changing units from years to days, then to hours, minutes, and finally to seconds. The average number of days in a year is used as the conversion factor, and the importance of setting up conversion factors to cancel out all units except the desired one (seconds) is emphasized. The result of the calculation is either 31,536,000 seconds for a non-leap year or 31,557,600 seconds for a leap year or an average year.

05:00

🚗 Speed Conversion from Miles per Hour to Meters per Second

The second paragraph delves into converting speed from miles per hour to meters per second. It breaks down the problem into two parts: converting miles to meters and hours to seconds. The conversion factors used include 1 mile being equal to 1.609 kilometers and 1 kilometer being equal to 1000 meters. Time conversion involves changing hours to minutes and then to seconds. The process involves setting up the conversion factors to cancel out unwanted units, leaving meters per second as the final unit. The mathematical calculation results in a speed of 20.1 meters per second.

10:01

📏 Density Conversion from Kilograms per Cubic Meter to Grams per Milliliter

This paragraph discusses the conversion of density from kilograms per cubic meter to grams per milliliter. It involves converting mass from kilograms to grams and volume from cubic meters to milliliters. The process requires understanding the cubic relationship when converting meters to centimeters and then to cubic centimeters. The conversion factors include 1 kilogram being equal to 1000 grams and 1 cubic meter being equal to 1,000,000 cubic centimeters. The final calculation shows the density of aluminum as 2.7 grams per milliliter, highlighting a common conversion between physics and chemistry.

15:06

📐 Area Calculation and Conversion for a Rectangular Door Mat

The fourth paragraph focuses on calculating the area of a rectangular door mat with given dimensions in inches and converting it to both square inches and square feet. The area formula (length times width) is applied first to find the area in square inches. Then, a conversion from square inches to square feet is performed using the fact that there are 12 inches in a foot. The final results are presented as 468 square inches and 3.25 square feet, demonstrating the process of unit conversion in area calculations.

⏳ Time Calculation for Reading a Book

The final paragraph addresses the problem of calculating the time it takes to read a 200-page book based on the rate of reading 15 pages every 45 minutes. The process involves converting pages to minutes and then minutes to hours using the given reading rate and the conversion factor between minutes and hours (60 minutes in an hour). The calculation results in a total of 10 hours needed to read the entire book, illustrating the application of unit conversion in time calculations.

Keywords

💡Dimensional Analysis

Dimensional analysis is a problem-solving technique used in physics and other sciences to convert quantities from one unit to another. It involves using conversion factors to cancel out the unwanted units and obtain the desired unit. In the video, dimensional analysis is the main theme, as it is used to solve various problems such as converting years to seconds, miles per hour to meters per second, and kilograms per cubic meter to grams per milliliter.

💡Conversion Factor

A conversion factor is a ratio that expresses how many of one unit are equal to another. It is used in dimensional analysis to change units from one system to another. The script explains how to use conversion factors to convert between days and years, hours and minutes, and miles to meters, among others.

💡Leap Year

A leap year is a year that is exactly divisible by 4, except for years that are divisible by 100 but not by 400. It has 366 days instead of the usual 365 to compensate for the fractional difference between the year's length and the calendar. In the script, the concept of a leap year is mentioned when discussing the average number of days in a year for the purpose of converting years to seconds.

💡Miles per Hour (mph)

Miles per hour is a unit of speed expressing how many miles an object can travel in one hour. The script uses mph as an example to demonstrate how to convert this unit to meters per second, which is another unit of speed commonly used in scientific contexts.

💡Meters per Second (m/s)

Meters per second is the SI unit of speed, indicating how many meters an object travels in one second. The video script explains the process of converting from miles per hour to meters per second, which involves multiple steps of unit conversion.

💡Density

Density is defined as mass per unit volume and is a characteristic property of a substance. In the script, the density of aluminum is given in kilograms per cubic meter, and the task is to convert this to grams per milliliter, illustrating the concept of density in different units.

💡Kilograms per Cubic Meter

Kilograms per cubic meter is a unit of density commonly used in physics to express the mass of a substance contained in a unit volume. The script uses this unit to show how to convert the density of aluminum from a physical science context to a chemical science context.

💡Grams per Milliliter

Grams per milliliter is a unit of density used in chemistry, indicating the mass of a substance per unit volume in milliliters. The video script demonstrates the conversion from kilograms per cubic meter to grams per milliliter, highlighting the difference in density units between physics and chemistry.

💡Area

Area is a measure of the extent of a two-dimensional surface or shape and is expressed in square units. The script discusses calculating the area of a rectangular door mat in square inches and then converting it to square feet, showing the concept of area in different units.

💡Square Inches and Square Feet

Square inches and square feet are units of area measurement. The script uses these units to illustrate the conversion process from a smaller unit (square inches) to a larger unit (square feet), which is a common practice in various fields.

💡Reading Rate

Reading rate refers to the speed at which a person reads, typically measured in pages per unit of time. The script provides an example of calculating how long it will take to read a 200-page book based on a reading rate of 15 pages every 45 minutes, demonstrating the application of reading rate in time management.

Highlights

Introduction to dimensional analysis problems

Conversion of years to seconds using days, hours, minutes, and seconds

Average days in a year is 365, with leap years having 366

Conversion factor from years to days is 365.25 for an average year

Conversion from days to hours, knowing there are 24 hours in a day

Conversion from hours to minutes and then to seconds, each with 60 units

Setting up conversion fractions to cancel out all units except seconds

Multiplication of all numerators to get the answer in seconds

Result of 31 million 536 thousand seconds in a year with 365 days

Using 365.25 days gives a slightly different result of 31,557,600 seconds

Conversion of speed from miles per hour to meters per second

Two-part conversion: miles to kilometers and hours to seconds

Conversion factors: 1 mile = 1.609 kilometers and 1 km = 1000 meters

Conversion of time units: 1 hour = 60 minutes and 1 minute = 60 seconds

Final calculation of speed in meters per second results in 20.1 m/s

Density conversion from kilograms per cubic meter to grams per milliliter

Conversion of mass from kilograms to grams and volume from cubic meters to milliliters

Cubic conversion requires raising the conversion factor to the third power

Final density of aluminum is 2.7 grams per milliliter

Area calculation of a rectangular door mat in square inches and square feet

Conversion from square inches to square feet by dividing by 144

Calculation of reading time for a 200-page book based on pages per minute

John can read the entire 200-page book in 10 hours

Transcripts

in this lesson we're going to work on

some

dimensional analysis problems

so let's begin with this one how many

seconds are there in a year

so what we're going to do is we're going

to convert

from

years

to days

and then days

to hours

hours to minutes and minutes to seconds

so let's start with what we're given

we're given one year and we want to

convert that into seconds

so what is the conversion factor that's

going to take us from

years to days

we know on average just 365 days per

year

except if you're dealing with a leap

year which there's 366.

well technically if you average it is

365.25

but for problems like this if you put

365 days

you'll be okay

now let's convert

days to hours

there's 24 hours in a day

so we could cross out the unit days and

now let's convert hours to minutes

there's 60 minutes in an hour

and there's 60

seconds in a minute

so notice how we set up the conversion

uh fractions

we do in such a way that every unit will

cancel

except the desired unit which is seconds

that's the only unit that we should have

at the end of this problem

to get the answer we need to multiply by

all of the numbers

on the numerators of the fractions

so it's going to be 1 times 365

times 24 times 60 times 60.

the answer is going to be 31 million

536 000

seconds

so that's how many seconds there are

in a year that's defined as 365 days

now if you choose 365.25

your answer will change slightly it

would be

31 557

600 seconds

let's move on to the next problem

a car is traveling at 45 miles per hour

how fast is it going in meters per

second

so how can we convert

from

miles per hour

to meters per second

so this is a two-part problem first we

need to convert

the units of length miles to meters

what we could do is convert miles to

kilometers

and then kilometers to meters

next we need to convert the units of

time from hours to seconds

which we already know how to do we can

convert from hours to minutes and then

minutes to seconds

now it's helpful to write the conversion

factors that you're going to use

one mile is equal to 1.609

kilometers

and one kilometer

think of the word kilo

a kilo is a thousand so one kilometer is

a thousand meters

and we know that one hour is equal to 60

minutes

and one minute is equal to 60 seconds

so those are the conversion factors that

we're going to use in this problem

now let's go ahead and get started

so we have 45 miles

per hour

and let's convert miles to kilometers

so we have miles on top we want to put

the same unit on the bottom

so here's our conversion factor

i'm going to put this part on the bottom

of the second fraction

and then the other part

on the top

of the second fraction

so the units miles will cancel

now let's use our next conversion factor

to convert from kilometers

to meters

so since we have kilometers on top in

the second fraction

i'm going to put kilometers on the

bottom of the third fraction

and then the other side of the equation

is going to go on top

so now we can cross our kilometers

so we have our desired unit meters so we

can leave that alone now let's focus on

units of time

let's convert hours to

but first we'll convert hours to

minutes

so let's use this conversion factor

notice we have hours on the bottom so we

need to put it on top of the four

fraction

so we're going to put this

here

the other part

of the conversion factor is going to go

on the bottom

so now we can cross out hours

finally we could use the last conversion

factor to go from minutes to seconds

and now we can cross out the unit

minutes

so we're left with meters on top

and seconds on the bottom

so we have the speed in meters per

second

so now we need to do the math everywhere

you see a one you could ignore

we're going to multiply by the numbers

on top

and then divide by the numbers on the

bottom

let's begin it's going to be 45 times

1.609

times a thousand

divided by 60 and then divide that

result by 60.

so you should get

20.1

meters per

second so that's how you can convert

from

miles per hour to meters per second

now let's move on to number three

the density of aluminum metal is 2700

kilograms per cubic meter what is the

density in grams per milliliter

go ahead and try that one

so we're given the density in kilograms

per cubic meter

and we want to convert it

to

the density in

grams per milliliter

so we need to convert the mass from

kilograms to grams we could do that in a

single step

one kilogram

is a thousand grams

and then we need to convert the volume

portion of density from cubic meters to

milliliters

so what we can do is convert

cubic meters

to cubic centimeters

by the way one meter

is a hundred centimeters

and then we can convert cubic

centimeters to milliliters one

milliliter is equivalent to one cubic

centimeter

so let's begin

we're given 2 700 kilograms per cubic

meter

that's the density of aluminum metal

we could use this one to go from

kilograms to grams

so one kilogram

is equivalent to a thousand grams

so now we can cross out the unit

kilograms

now let's use the next one let's go from

meters to centimeters

we know that

one meter is equal to 100 centimeters

since we have meters on the bottom i

decided to put meters on top in a third

fraction

but now this is the part you need to pay

special attention to

notice that we don't just have meters we

have cubic meters

that's a meter times a meter times a

meter

what we need to do

is

raise this to the third power

so this becomes cubic meters

if we raise it to the third power

one meter raised to the third power

that's one meter times one meter times

one meter

that becomes

one meter cube

now a hundred centimeters

raised to the third power

that's a hundred centimeters times a

hundred centimeters times a hundred

centimeters

so that is one million

cubic centimeters

or you can write it as

one times ten to the sixth

so basically what you do is

you can take the

cube of this equation

and you get one cubic meter

is equal to a million

cubic centimeters

so now we can cross out the unit

cubic meters

if you're wondering why this screen

looks a little different i have to do

some video editing

now let's finish this problem

so right now we can cross out the unit

cubic meters

we have grams on top

but we need to get milliliters on the

bottom

so we could use our final

conversion factor

one cubic centimeter

is equal to one milliliter

so now we could cancel

the unit cubic centimeters

so what we have

is the unit grams on top

and the unit milliliters on the bottom

so now we just got to do the math

so we're going to multiply 2700 by a

thousand

and take that result

divided by a million

or one times ten to the sixth

the final answer is going to be 2.7

grams per milliliter

anytime you need to convert from

kilograms per cubic meter

to grams per milliliter simply divide by

a thousand

because in physics

the density of objects are typically

reported

in kilograms per cubic meter but in

chemistry

you'll find that the density is usually

reported in grams per milliliter so this

is a common conversion that you may

encounter when

going between physics and chemistry

number four a rectangular door map has a

length of 26 inches and a width of 18

inches what is the area of the door mat

in square inches and square feet

so let's draw a picture let's say this

is the door mat

the length is 26 inches

the width is 18 inches

to calculate the area

the area of a rectangle is the length

times the width

it's l times w

so in this example we need to multiply

26 inches

by

18 inches

this is going to be 468. now we're

multiplying inches to the first power

times inches to the first power

so when you multiply by a common base

you need to add the exponents one plus

one is two so we get the area in square

inches

so that's the answer to the first part

of the question

now we need to get the area in square

feet so we're going to convert it

from square inches to square feet

so how many inches in a foot

we know that there's 12 inches

in one foot

so we have inches on the top left we're

going to put it

on

the bottom right

now notice that it's inches squared

so we need to square

this conversion factor so we can get

square feet

so the answer is going to be 468 divided

by 12 squared 12 squared is 144

so you should get 3.25

square feet

as your final answer

so that's how you can convert from

square inches to square feet

number five

john can read 15 pages of a certain book

every 45 minutes

how many hours will it take him to read

the entire 200 page book

so think about what we're given

we're given 200 pages

and from that

we need to find out

hours how many hours

it will take them to read 200 pages

so we're converting pages

to hours

how can we do this

what conversion factors do do we have to

go from pages to hours

well the first sentence connects pages

to minutes

so we know that he can read 15 pages

every 45 minutes

and we know the conversion from minutes

to hours

one hour is 60 minutes

so we have everything that we need

what we need to do is convert from pages

to minutes

and then minutes to hours

so let's start with what we're given

which is 200 pages

and let's use this to convert from pages

to minutes

so in 45 minutes he can read

15 pages

so we can cancel out

the unit pages

and then we'll use this conversion

factor to go from minutes to hours

there's 60 minutes

in one hour

so we can cross out minutes

and now we can get the answer

so we're going to multiply by the

numbers on top and divide by the numbers

on the bottom so it's 200 times 45

that's 9000 divided by 15

that gives us 600 divided by 60

it gives us 10.

so it's going to take him 10 hours

to read 200 pages

of this book