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- WELCOME TO A PRESENTATION OF LINEAR AND ANGULAR VELOCITY.
THE GOALS OF THE VIDEO ARE TO CALCULATE LINEAR VELOCITY
AND ALSO TO CALCULATE ANGULAR VELOCITY.
SO SOMETIMES IT IS IMPORTANT TO KNOW
HOW FAST A POINT ON A PULLEY OR A DISK IS MOVING
OR HOW FAST A CENTRAL ANGLE IS CHANGING.
THESE MEASUREMENTS ARE MEASUREMENTS
OF LINEAR VELOCITY AND ANGULAR VELOCITY.
SO YOU PROBABLY RECALL THE FORMULA
DISTANCE = RATE x TIME.
WELL, IF WE LET THE RATE R EQUAL THE VELOCITY
WE WOULD HAVE DISTANCE = VELOCITY x TIME OR D = V x T.
NOW, IF WE SOLVE THIS EQUATION FOR VELOCITY
BY DIVIDING BOTH SIDES BY T,
WE WOULD HAVE VELOCITY = DISTANCE DIVIDED BY TIME.
AND LASTLY IF WE REPLACE THE DISTANCE WITH THE VARIABLE S
WE WOULD HAVE VELOCITY = S DIVIDED BY T.
AND WE USE S WHEN WE TALK ABOUT CIRCLES,
MAINLY BECAUSE S IS ALSO USED FOR ARC LENGTH.
LET'S GO AHEAD AND TAKE A LOOK
AT OUR TRADITIONAL QUESTION FIRST.
A RUNNER OF THE PAT TILLMAN 4.2 MILE RACE
FINISHED IN 28 MINUTES, 4 SECONDS.
WHAT WAS THE RUNNER'S AVERAGE VELOCITY?
SO WE'LL TAKE THE DISTANCE AND DIVIDE BY THE TIME.
THIS IS AN EXAMPLE OF LINEAR VELOCITY.
THE DISTANCE IS 4.2 MILES.
NOW, THE TIME IS A MIX OF MINUTES AND SECONDS.
WE NEED TO CONVERT THESE SECONDS INTO MINUTES.
4 SECONDS IS 4/60 OF A MINUTE WHICH EQUALS, AS A DECIMAL,
0.06 REPEATING.
SO WHAT WE'LL USE FOR OUR TIME IS 28.067 MINUTES.
HERE IS THE CONVERSION OF OUR SECONDS TO MINUTES.
LET'S GO AHEAD AND DIVIDE AND FIND OUR AVERAGE VELOCITY.
APPROXIMATELY 1.50 MILES PER MINUTE
OR .150 MILES PER 1 MINUTE.
NOW, THIS IS OUR CORRECT ANSWER
BUT LET'S GO AHEAD AND CONVERT THIS TO MILES PER HOURS.
SO TO CONVERT MINUTES INTO HOURS
WE KNOW THERE ARE 60 MINUTES IN 1 HOUR.
SO THIS IS LIKE MULTIPLYING BY 1
EXCEPT NOW THE UNITS OF MINUTES SIMPLIFY OUT.
IF WE MULTIPLY ACROSS THE TOP HERE
WE'RE GOING TO HAVE 8.979 MILES OVER 1 HOUR.
AND THIS IS NORMALLY WRITTEN AS 8.979 MILES PER HOUR.
ON THIS NEXT SCREEN
I JUST LISTED SOME INTERESTING LINEAR SPEEDS.
YOU MAY WANT TO PAUSE THE VIDEO AND READ THESE,
IF INTERESTED.
LET'S GO AHEAD AND TAKE A LOOK NOW AT ANGULAR VELOCITY
AND THEN WE'LL COME BACK TO LINEAR VELOCITY
AROUND A CIRCLE.
SO THE MEASURE OF HOW FAST A CENTRAL ANGLE IS CHANGING
IS CALLED ANGULAR VELOCITY,
AND TYPICALLY THEY USE THE LOWERCASE GREEK LETTER OMEGA.
SO OMEGA IS EQUAL TO THETA DIVIDED BY T,
WHERE THETA IS THE MEASURE OF THE CENTRAL ANGLE AT TIME T,
THETA MUST BE IN RADIANS
AND OMEGA IS EXPRESSED IN RADIANS PER UNIT OF TIME.
LET'S TAKE A LOOK AT A PROBLEM.
A MECHANICAL ARM ROTATES 1/3 OF A ROTATION IN 0.25 SECONDS.
SO WE KNOW OUR TIME IS 0.25 SECONDS,
BUT IT STATES THE ARM ROTATES 1/3 OF A ROTATION.
WE KNOW A COMPLETE ROTATION WOULD BE 360 DEGREES.
SO 1/3 OF 360 DEGREES WOULD BE 120 DEGREES,
BUT WE DO HAVE TO EXPRESS OUR ANGLE IN RADIANS.
I KNOW THAT PI/3 RADIANS IS 60 DEGREES.
SO 2PI/3 WOULD BE 120 DEGREES.
AND IF IT'S BEEN A WHILE
THE CONVERSION WOULD BE MULTIPLYING BY PI
OVER 180 DEGREES,
AND YOU CAN CHECK THIS BUT IT DOES COME OUT TO 2PI/3.
SO OUR ANGULAR VELOCITY IS EQUAL TO THETA,
WHICH WE SAID WAS 2PI DIVIDED BY 3 ALL OVER T,
WHICH IS 0.25,
AND, AGAIN, THIS WAS IN RADIANS.
OKAY, SO WE'LL GO AHEAD AND DIVIDE
TO FIND OUR ANGULAR VELOCITY
AND IT COMES OUT TO APPROXIMATELY 8.378,
AND THE UNITS ARE RADIANS PER SECOND.
OKAY, LET'S GO BACK TO LINEAR SPEED
AND TALK ABOUT HOW TO FIND IT AROUND A CIRCLE.
REMEMBER THE FORMULA FOR ARC LENGTH IS S = R THETA.
SO IF WE GO BACK TO OUR LINEAR VELOCITY FORMULA
WE REPLACE OUR DISTANCE S WITH R x THETA.
AND IF WE WANTED TO WE COULD REWRITE THIS
AS R x THETA DIVIDED BY T
AND THETA DIVIDED BY T IS ACTUALLY OUR ANGULAR VELOCITY
OR OMEGA.
SO THERE ARE A COUPLE WAYS HERE
TO FIND LINEAR VELOCITY AROUND A CIRCLE.
AND THIS IS SUMMARIZED ON THE NEXT SCREEN.
SO WE HAVE ONE FORMULA FOR ANGULAR VELOCITY
AND WE ACTUALLY HAVE THREE OF THEM FOR LINEAR VELOCITY
BASED UPON WHAT INFORMATION WE'RE GIVEN.
AND, AGAIN, NOTICE THAT THETA DIVIDED BY T
IS OUR ANGULAR VELOCITY
AND SO YOU CAN SEE THIS HERE CAN BE REPLACED WITH OMEGA
OR THE ANGULAR VELOCITY.
AND HERE WE HAVE THE DISTANCE BEING REPLACED
BY THE ARC LENGTH FOR THE DISTANCE AROUND A CIRCLE.
LET'S TAKE A LOOK AT A COUPLE EXAMPLES.
A TIRE WITH RADIUS 9 INCHES
IS SPINNING AT 80 REVOLUTIONS PER MINUTE.
WE FIRST WANT TO FIND THE ANGULAR SPEED OR VELOCITY
AND THEN FIND THE LINEAR SPEED OR VELOCITY.
AND DEPENDING ON WHAT TEXTBOOK YOU'RE LOOKING AT
SOME USE SPEED AND SOME USE VELOCITY.
OKAY, SO WHAT WE CAN GATHER FROM THE GIVEN PROBLEM
IS THAT OUR RADIUS IT 9 INCHES.
AND IF WE HAVE 80 REVOLUTIONS PER MINUTE
THAT IMPLIES THAT OUR THETA WOULD BE--
WELL, EACH REVOLUTION IS 2PI RADIANS,
SO WE'D HAVE 80 x 2PI.
SO OUR ANGLE IS GOING TO BE 160PI RADIANS.
SO OUR ANGULAR SPEED OR ANGULAR VELOCITY
WILL EQUAL OUR ANGLE THETA,
WHICH IS 160PI DIVIDED BY OUR TIME.
AND IT SAYS 80 REVOLUTIONS PER MINUTE.
SO OUR TIME--WE DIDN'T WRITE DOWN--IS ONE MINUTE.
SO THAT TIME ISN'T VERY EXCITING
BECAUSE THIS JUST MAKES OUR DENOMINATOR 1 MINUTE,
WHICH IS OUR ANGULAR VELOCITY.
BUT LET'S GO AHEAD AND GET A DECIMAL APPROXIMATION.
160PI DIVIDED BY 1 WOULD BE 502.655,
AGAIN, RADIANS PER MINUTE AS A DECIMAL APPROXIMATION.
NEXT TO FIND THE LINEAR SPEED IN INCHES PER MINUTE
AND MILES PER HOUR
WE'LL START WITH MINUTES BECAUSE THAT'S THE GIVEN TIME.
WE'LL GO AHEAD AND USE THIS FIRST FORMULA
SINCE WE ALREADY FOUND THE ANGULAR VELOCITY.
SO OUR LINEAR VELOCITY IS EQUAL TO THE RADIUS 9 INCHES
TIMES OUR ANGULAR VELOCITY, WHICH IS HERE,
AND OUR UNITS HERE ARE RADIANS PER MINUTE.
IF WE MULTIPLY THIS TOGETHER
WE WILL OBTAIN 4523.893 INCHES PER 1 MINUTE.
AND I'M WRITING IT LIKE THIS
BECAUSE I KNOW I HAVE TO CONVERT THIS
INTO MILES PER HOUR.
TO CONVERT THE MINUTES INTO HOURS
WE'LL MULTIPLY BY 60 MINUTES/1 HOUR.
THIS WILL CONVERT THE MINUTES TO HOURS.
AND NEXT WE NEED TO CONVERT INCHES INTO MILES.
IT'S GOING TO TAKE TWO CONVERSIONS HERE.
THERE ARE 12 INCHES IN 1 FOOT.
THIS WILL TAKE CARE OF THE INCHES.
NOW WE NEED TO CONVERT FEET INTO MILES.
AND THERE ARE 5,280 FEET IN 1 MILE.
AND NOW THE UNIT OF FEET IS GONE.
NOW THE FEET ARE GONE.
SO NOW WE JUST NEED TO MULTIPLY ACROSS THE TOP,
MULTIPLY ACROSS THE BOTTOM AND THEN DIVIDE.
SO HERE IT IS IN MILES PER HOUR,
BUT NOW WE JUST NEED TO DIVIDE THIS
TO GET THIS MILES PER 1 HOUR.
AND I'M RUNNING OUT OF SPACE HERE SO I'LL WRITE THAT HERE.
4.284 MILES.
OKAY. LET'S TAKE A LOOK AT ONE MORE EXAMPLE.
RAQUEL AND JUAN ARE ON A MERRY-GO-ROUND.
RAQUEL IS 2 FEET FROM THE CENTER
AND JUAN IS 5 FEET FROM THE CENTER.
IF IT TAKES 6 SECONDS TO MAKE 2 REVOLUTIONS,
DETERMINE THE ANGULAR AND LINEAR VELOCITY.
FOR RAQUEL THE RADIUS WOULD BE 2 FEET,
AND IT'S 5 FEET FOR JUAN,
SO I'LL SAY R SUB J = 5 FEET.
NEXT THE TIME IS 6 SECONDS AND WE NEED TO FIND THETA.
IF IT MAKES 2 REVOLUTIONS,
WELL, 1 REVOLUTION WOULD BE 2PI RADIANS,
SO 2 REVOLUTIONS WOULD BE 4PI RADIAN.
I THINK THAT'S ALL THAT WE NEED TO BEGIN.
LET'S GO AHEAD AND REFERENCE OUR FORMULAS HERE.
THE ANGULAR VELOCITY--
NOTICE IT DOES NOT REQUIRE THE RADIUS,
SO THE ANGULAR VELOCITY IS GOING TO BE THE SAME
FOR BOTH JUAN AND RAQUEL.
THE ANGULAR VELOCITY IS EQUAL TO THETA
WHICH IS 4PI DIVIDED BY 6 SECONDS.
SO THE ANGULAR VELOCITY WILL BE 4PI RADIANS
DIVIDED BY 6 SECONDS,
WHICH IS APPROXIMATELY 2.094 RADIANS PER SECOND.
LINEAR VELOCITY--NOTICE THAT IT DOES REQUIRE THE RADIUS.
THEREFORE, WE'RE GOING TO HAVE A LINEAR VELOCITY FOR RAQUEL
AND ALSO ONE FOR JUAN.
SO LET'S FIND THE LINEAR VELOCITY FOR RAQUEL FIRST
AND THAT'LL BE EQUAL TO THE RADIUS x THE ANGULAR VELOCITY.
SO THE RADIUS FOR RAQUEL WAS 2 FEET x THE ANGULAR VELOCITY,
AND IF WE MULTIPLY THIS WE WILL HAVE APPROXIMATELY 4.189.
AND FOR JUAN THE ONLY THING THAT'S GOING TO CHANGE
IS NOW THE RADIUS IS NOW 5 FEET,
AND THIS PRODUCT IS 10.47.
SO WE CAN SEE
THAT EVEN THOUGH THEY'RE ON THE SAME MERRY-GO-ROUND
SINCE JUAN IS SITTING FURTHER OUT
FROM THE CENTER OF THE CIRCLE,
HE'S ACTUALLY GOING MORE THAN TWICE THE SPEED OF RAQUEL.
OKAY. I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU.
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