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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
