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in this tutorial we're going to go over

physics problems associated with angular

speed and angular velocity so let's

start with this one the frequency of a

spinning will is 30 Hertz if the

diameter of the world is 50 centimeters

what is the angular speed of the wheel

in radians per second so what equation

do we need to calculate the anguish

speed the angular speed is represented

by this equation Omega it's equal to 2

pi F you can also use that equation to

calculate the angular velocity so we

have the frequency the frequency is 30

Hertz and the frequency represents the

number of cycles that occur per unit

time so in this case the cycles are

rotations or revolutions so if the

frequency is 30 Hertz that means that

the will makes 30 revolutions every

second now when you think of 2 pi what

do you think of one revolution around a

circle is equal to 360 degrees and 360

degrees is 2 pi radians because 180 is

equal to pi so one revolution around a

circle is 2 pi radians so omega is equal

to 2 pi radians per revolution

multiplied by the frequency which is 30

revolutions per second so notice that

the unit revolutions cancel I want to

set it up in such a way so you could see

how frequency can give you the angular

speed in radians per second so in this

example it's 2pi times 30 so Omega is

going to be 60 PI radians per second now

if you want to get the decimal value of

that this is going to be a hundred and

eighty eight point five radians per

second so that's the angular speed in

this problem

let's move on to Part B what is the

period the period is the time divided by

the number of cycles that occurs so if

we have 30 cycles that occur in one

second then the period is one second per

30 cycles in this case 30 revolutions so

period is simply 1 over F is the

reciprocal of the frequency so 1 divided

by 30 is point zero 3 3 seconds so what

that means is that the will makes one

revolution in point zero through three

seconds

so the period tells you the time it

takes to make one revolution or one

complete cycle around a circle whereas

the frequency tells you the number of

cycles that can be made in one second so

in one second the will is going to spin

30 times that's the frequency and the

period tells you that the will will spin

once every point zero three three

seconds so hopefully this clarifies the

definition between period and frequency

so frequency tells you how many cycles

will occur in one second and the period

tells you how long it takes just to make

one cycle number two what is the linear

speed of a will that is rotating that

twenty-five radians per second and we're

given the radius of the will it's thirty

centimeters

so how can we find the linear speed of

the will given its rotational speed of

25 Raiders per second so what should we

do the equation that we need is this

equation V is equal to Omega times R so

all it is is just Omega which is 25

multiplied by the radius of the will but

we'll need that in meters so to convert

30 centimeters into meters divided by

130 divided by 100 is point 3 0 so 25

times point 3 is 7.5 so the linear speed

is 7.5 meters per second now let's make

sense of the units so Omega was in

radians per second and the radius I like

to think of it as the left per Radian so

the radius is point 3 meters but it's

really 0.3 meters per 8 Ian and so this

will give you 7.5 meters per second

now for those of you who want to know

why I like to think of the radius as

being the left per Radian here's why

let's say if we draw an arc so this is

going to be s the arc length and this is

the angle theta and this is the radius

of the circle the arc length is equal to

the angle in radians times the radius of

the circle so if you solve for R the

radius is the arc length divided by the

angle so let's say if we have an arc

length of 30 meters and let's say the

angle is 2 radians so it's going to be

30 meters per 2 radians so the radius

will be 15 meters per Radian so what

does this mean this is the arc left when

the angle is 1 Radian so when the angle

was 2 radians the arc length was 30

meters now if you reduce the angle to 1

Radian the arc length will be equal to

the radius which is 15 years every what

- for some reason but let's correct that

so that's why I like to think of the

radius as being the arc left when the

angle is 1 Radian so I think of it as

meters per Radian so all of this leads

me to the conclusion that 1 Radian is

equal to the length of the radius so

when the angle is 1 Radian the arc

length of the circle is equal to the

left of the radius of the circle which

in this example is 50 meters so this is

useful for conversions let's say if I

want to convert linear speed from

radians per second to rotations per

minute or revolutions per minute so now

in the conversion factor between radians

and meters is very useful and is equal

to the radius of the circle so let's say

if the radius of the circle is 8 meters

I could come up with this conversion one

Radian represents an arc length of eight

meters if the rate is the same years so

the radius would be the eight meters per

Radian let's work on number three a disc

spins at a rate of 5,000 radians every

10 minutes what is the angular velocity

of the disc in radians per second so to

calculate the angle of velocity we can

use this equation it's the angular

displacement divided by the change in

time the angular displacement is the

amount that the disc spins in radians so

it spins a distance of 5,000 radians and

we have the time which is 10 minutes but

we'll need that in seconds now one

minute is equal to 60 seconds which

means that 10 minutes is 60 times 10 or

600 seconds so the distance 5,000

radians every 600 seconds

so therefore the angular velocity is 8.3

3 radians per second

so that's it for Part A now let's move

on to Part B what is the linear velocity

of the disc in meters per second if the

diameter of the disc is 20 centimeters

so we're given the diameter of the discs

is 20 centimeters now we need to use

that to find the radius the radius of

the disc hast be half of the diameter so

it's 10 centimeters

so we can say the radius is 10

centimeters per Radian so that means the

length of one Radian or one Radian has

an arc length of 10 centimeters but we

need to convert centimeters into meters

so we got to divide that by 100 so

that's point one meters per Radian so

now let's calculate the linear velocity

so V is equal to Omega times R Omega is

8 point 33 radians per second and the

radius is point one meters per Radian so

the arc length is point one meters when

the angle is one Radian so these two

will cancel and then we'll have the

units in meters per second so eight

point 33 times point one is simply point

eight three three meters per second so

this is the answer to Part B now let's

move on to Part C what is the angular

velocity of the will in rotations per

minute which is equivalent to

revolutions per minute so how can we

find that answer

so all we got to do is convert the

angular velocity from radians per second

to rotations per minute or simply

revolutions per minute so how can we go

about doing that well let's start with

what we have 8.33 radians per second now

we can convert second since minutes 60

seconds is equal to one minute so the

unit's seconds cancel next we can

convert radians to revolutions so it's

two pi radians per revolution I need to

put radians on the bottom so one

revolution is equal to two pi radians

and so now we have the unit revolutions

per minute which is basically rpms so

it's 8.33 times 60 divided by 2 pi and

so this is going to be seventy nine

point five rotations per minute that's

the answer so here's the last question

for this video a wheel with a radius of

1.4 meters spins at an angular speed of

45 rotations per minute or revolutions

per minute what is the linear speed of

the will in miles per hour go ahead and

try this problem so let's convert it

from revolutions per minute to miles per

hour so we have 45 revolutions per

minute now we need to convert minutes to

hours 60 minutes is equivalent to one

hour so that's the first thing we could

do now how do we convert revolutions to

Mouse well first we need to convert

revolutions to radians one revolution

around a circle is 2 pi radians so we

can cancel the unit revolutions now how

do we go from radians to meters the

radius will help us to do so the radius

is 1.4 meters so it's really 1.4 metres

per Radian so what this means is that if

we have an angle of one Radian the arc

length is going to be 1.4 meters so the

radius connects the angle with the R

cliff so an angle of one Radian equates

to an arc length of 1.4 meters so we can

cancel the unit radians so now we have

meters let's convert meters to

kilometers a thousand meters is equal to

one kilometer and one kilometer

is point six to one for miles now you

can look up these conversions online or

hopefully you know them already so

meters cancel and also kilometers cancel

so what we have left over is miles per

hour so it's 45 times 60 times 2 pi

times 1.4 divided by a thousand

multiplied by 0.6 to 1/4 so the will is

moving at a linear speed of fourteen

point seven six miles per hour

so that's the answer