Rotational Kinematics WarmUp: Actual Unedited
Student Responses
1. Q1,Because the reel that is turning is geting
bigger.
Q2,the tangential velocity is about 55 m/s.
Q3,c
2. Q1,Its just like a bicycle and changing
gears. The changing of the gears being the winding of the tape itself. With
most of the tape being on one real its momentum quickly turns the smaller real.
Q2,It would have 360/rad per day. The earth is
complete circle, spinning once a day. An location will return to its origional
position in a day.
Q3,d
3. Q1,It is not that it winds up faster, but
that the wheel that has a radius being reduced moves quicker and therefor makes
it look as if the tape is winding faster...if you insist that it winds faster,
then it could be because the mass that is doing the pulling is greater than the
mass that is being pulled, therefor making it easier to pull and quicker as
well.
Q2,If Indianapolis sits at 50degrees north of
the equator then thata - thata initial = 57.3 and t2 - t1 = 24 hrs. Therefor:
.0035rad/sec
Q3,b
4. Q1,Becaue the amount of tape is added up to
the circle every round and it begins with a small circle. The circle gets
bigger and bigger. The bigger it is the more amount of tape it can get done
each round even though the speed of the rewinder is the same.
Q2,Since the angular velocity of the earth is
2*pi rad/day and its radius is 6.38*10^8m. Then the tangential of the object
is: V=2*pi*6.38*10^6 m/day
Q3,b
5. Q1,At the end of a tape when you are
rewinding the majority of the tape is would on the left side. due to the
cicumference of the wheel + tape it is slow at first. As the tape rewinds the
circumference of the of the left side decreases and therefore increases the
speed of the rotation.
Q2,a(tan) = rx
Where r = 6.38 x 10^6 I do not know what x is.
However, at any time, every part of a rotating rigid body has the same angular
velocity. so the velocity of an object in Indy would be the same as the
velocity of an object in Columbus, OH.
Q3,d
6. Q1,Because the radius of the role increases.
But the angular velocity stays the same. Then the arclength S=r*(theta) divided
by the same time interval shows the increase in the velocity at which the tape
roles up.
Q2,R=6.38X10^6m. I would think that the velocity
tangental would be the speed at which we move arround the earth. The problem is
that we are not on the equator so we don't have a velocity equal to
S/t=R*(theta)/t. If we assume that the earth is a sphere, we can also assume at
the equator the angle from the equator would be equal to 0 and the constant
multiple of the velocity function would be equal to one(%40 the north pole
,%A2=90, it would be zero). So in estimation we could multiply the function by
a cosine(theta) and recieve an aproximate answer. So, %A2=angle from the
equator, and %A3=(theta)=period of 1 rotation of the earth , then
r*%A3*cos(%A2)/t= (6.36e6 m)*(2*V
Q3,c
7. Q1,In the equation for centripetal
acceleration (a=v^2/r). Initially, the static side has a large radius and thus
a smaller velocity. Once the tape starts moving faster, the velocity is squared
and the radius plays a smaller part in how large the acceleration gets.
Q2,I used the equation: average angular
velocity=total angular displacement/change in time
the angular displacement was
2*pie/86,400seconds.
Then I used: tangential velocity=radius*angular
velocity
I found the estimated tangential velocity to be
4.34x10^20m/s
Q3,b
8. Q1,because the radius gets smaller and
smaller.
Q2,
Q3,d
9. Q1,The more tape that gets rewound, the
bigger the spool gets ( the bigger the circumference gets) so it can take up
alot more tape.
Q2,2*r is the cicumference, in 24hrs=1rotation.
2*r/24hrs. This is at the equator so this is an estimate of what it would be
for Indianapolis.
Q3,b
10. Q1,As the tape rewinds, the circumference of
the roll of tape increases. This, coupled with the constant speed of the motor,
means the speed of the tape increases as the tape rewinds.
Q2,(2** radius of Earth)/24hr
Q3,b
11. Q1,The angular rewinding velocity is
constant. On the other hand, the radius of the portion of the tape that has
already been rewound increases. From the equation v = ang.vel * r, we conclude
that the tangential rewinding speed
must also increase.
Q2,Assuming that Indy's latitude is
approximately 45 deg., we arrive at the following estimate:
R(Earth) = 6400 km
r(at 45 deg.) = cos45 * 6400 = 0.7 * 6400 = 4480
km
An object in Indy travels 2*r*PI in one day,
meaning v = 2 * PI * 4480 km / day = 325.8 m/s
Q3,b
12. Q1,Because as it winds, the radius of the
take-up wheel increases, thus increasing the angular velocity at its radius and
the speed at which the tape is wound.
Q2,At one revolution per day, 7.27 x 10^-5 rad/s
Q3,b
13. Q1,The tape wides up faster at the end than
at the beginning because the radius of the end side is smaller than the other
side. Therefore taking less revolutions per second to spin.
Q2,The magnitude of the tangential velocity of
an object in Indianapolis, due to the rotation of the earth is equal to one
another.
Q3,a
14. Q1,From v=rw, when v is constant as r is
decreased w is increased. With a tape, as the radius decreases from film being
pulled off, w increases.
Q2,Using the equation v=rw, the radius of the
earth can be r and 6.28 radians is theta over 24 hours is w. The tangential
velocity is 464m/s.
Q3,b
15. Q1,Because at the end there is more tape at
the other side so the radius of the other side is bigger.
Q2,W=6.48/86,400s=7.5*10^-5 Rad/s
Q3,b
16. Q1,The motor winds at a constant speed in
terms of revolutions per unit time. The circumference of the receiving tape wheel
increases as it accumulates tape. Thus, it can spool more tape per revolution,
and thus, more tape per unit time.
Q2,2x pi x R per day
R = 6.38 E 6 m
v = 4 E 7 m/day
where R = radius of the earth
unit analysis yields 463.7 m/s
Q3,b
17. Q1,As you get closer to the end of the tape,
the radius gets larger.
Q2,
Q3,b
18. Q1,The tape reel on the left is turned when
rewinding the video tape. A smaller mass of tape acummulates on this reel when
the tape is at the end than when the tape is at the beggining. Thus when the
same force turns this lesser accumulated mass, it causes a greater acceleration
and causes the reel to pick up velocity faster. Q2,The cicumference at this
latitude is approx.=4*10^7m. The earth turns around its axis in 86400s.
Therefore the approx. tangential velocity=4E7m/86400s=463m/s.
Q3,d
19. Q1,Since acceleration is equal to the angle
times the radius, the side that has the smaller amount of tape , has to go
around faster and faster to unwind the tape.
Q2,The speed of a particle is directly
proportional to the body's angular velocity: rw, where 'w' is measured in
radians per seconds. 'r' is the constant distance from the axis of rotation,
which in this case is the radius of the earth.
Q3,b
20. Q1,because the radius of the circle of tapre
decreases, which is related to how fast it will turn. *&*! @H( @(&
BHSVS%^&*^(& N*&(V^&$^%$%^&$*^V^V& &*(&*(&
&%^^#$#
Q2,
Q3,d
21. Q1,It is because the mass being rotated is
less at the end than at the beginning. Therefore less force is needed to rotate
the tape.
Q2,By using the time it takes the earth to
rotate in day its radius I was able to find that the velocity is 8000m/s
Q3,b
22. Q1,It moves faster because it is traveling a
greater distance in the same amount of time as the inner part of the tape.
Q2,We could use the equation v=r(omega), but we
don't know the radius of the earth. Since not a word has been taught about
this, I am not sure how to manipulate the formulas to come up with an answer.
Q3,b
23. Q1,Because near the end, more of the tape is
on the spool that is providing the rotational force, meaning that the radius
(and the mass) of the other spool is less than what it started with.
Q2,Earth is approx. 8000 miles in diameter, and
Indianapolis is approx. 40 deg. north of the equator, so the distance of indy
to the rotational axis of the earth should be cos(40)*cos(40)*4000 miles. The
distance traveled in 24 hours would be: 2*pi*cos(40)*cos(40)*4000 miles, making
the tangetial velocity 2*pi*cos(40)*cos(40)*4000 miles/day, or 274.7 meters/sec
Q3,b
24. Q1,The angular velocity is given by: w=(theta(2)-theta(1))/(t(2)-t(1)).
The value of theta is equal to s divided by r.
So that means that theta is inversely proportional to the radius. So as the
radius get smaller that means that theta gets larger. From the equation above a
larger theta results in a larger angular velocity.
Q2,I am assuming that the earth rotates 360
degrees in a time period of 24 hours. So the tangential velocity is given by
the change in theta over the change in time. I am going to assume that theta is
140 degrees. So 140/360=.389. This means that it has traveled .389 of its whole
rotation. The tangential velocity is then equal to (140degrees x 2pi
rad/360degrees)/(.389x24hours)=about .262rad/hr.
Q3,b
25. Q1,The video tape rewinds faster because the
radius of the tape is smaller at the end than at the beginning; therefore
causing the tape to have a higher tangential velocity at the end in comparison
to the beginning.
Q2,The velocity is approximately 2xpi
radians/day, which will turn out to be a net of zero.
Q3,b
26. Q1,Because the VCR needs time to stop the
rewind process.
Q2,108 km/day^2
Q3,b
27. Q1,The linear velocity is greater when the
raius is greater. V = r*W
Q2,The earths rotation is constant (I hope) no
acceleration so A tan = r(0) = 0
Q3,b
28. Q1,As you are rewinding, the tape winds
faster at the end. This is because the spool keeps the same angular velocity,
but has a larger radius. This means that it has a larger circumfrence so it
takes more tape to cover it, but it still takes the same time for the spool to
rotate.
Q2,Indianapolis is at approx. 39deg N, so it has
a radius of about 5004 km from the earth's axis. The angular velocity of the
earth is 2*pi rad/day. This leads to Indy having a tangential velocity of 31446
km/day.
Q3,b
29. Q1,The radius has more time to increase at
the end than at the begining causing the speed to increase. v=wr
Q2,v=rw
w=(dw)/(dt) ; w=2pi/(86400s)
r=6.38*10^6m
v=pi/86400s*(6.38*10^6m)=231 m/s
Q3,b
Comments,I found question number 2 difficult to
grasp. I had a problem thinking about the tagential velocity of Indianapolis
being the same as the equator. But I guess that thought behind the word
estimate. Thanks
30. Q1,the moment of inertia is lowers as less
tape is on the take up wheel the wheel then has less resistance to changes in
velocity and speeds up.
Q2,Indianapolis is aproximately 37 degrees north
of the equator, multiplied by the radius of the earth ( 6380000 meters), so the
magnitude would equal 2.3 x 10^8 m/s.
Q3,b
31. Q1,becuase the tape is on a circular disk.
the more tape that is wound around the disk the slower it goes becuase it has
to turn farther to make a complete rotation on the outside portion of the disk.
when it is near the end. it has a smaller circumerance and therefore can move
faster.
Q2,the tangential velocity is the rate of speed
at which you are being pulled off the away for the circulare motion that you
are in. it is the radius of the earth times the angulare speed in rad/s. the
earth goes 2(pi) radian in 24 hours. therefore the rad/s measure is which is
pi/43200. the radius of the earth is 6.38E6. the answer is 147.68pi m/s
Q3,d
32. Q1,The average angular velocity is defined
as the change in the angle divided by the change in time, so I would assume
that as the angle increases the velocity aslo increases. As the tape goes from
one side of the cassette to another the angle increases.
Q2,The velocity is equal to rw, and estimating
the longitude of indianapolis to be around 40 degrees. I estimated that the magnitude
would be around 300 m/s.
Q3,d
Comments,I think that the second problem was
difficult. I think that we should spend some time in class to explain the
answer, and explain why it is the answer.
33. Q1,The angular velocity is the same, but
when there is more tape on the reel, any arbitrary angle will cover more length
of tape.
Q2,This will be a very rough estimate becuse I
don't have the latitude of Indianapolis handy. I estimate the latitude of Indy
to be 40 dergess. The radius of the Earth is 6380000 meters. Using trig
functions and properties of circles you can find the circumference of the
circle Indianapolis follws on its rotation around the axis ( 30708213 meters).
It travels this distance in one day. This is 355.4 meters per second.
Q3,b
34. Q1,The tape has the momentum pushing it to
go faster and faster so the tape gains speed toward the end.
Q2,Since the earth rotates it will always change
the tangent position to Indy. The velocity is the speed of the earth rotation.
Q3,a
35. Q1,because the radius of the front is bigger
tehn the radius of the rear.
Q2,atan=(6.38x10^6)(.70rad)
arad=(6.38x10^6)(465m/s^2)
magnitude=(atan^2+arad^2)^1/2
magnitude=2.9x10^9m/s
Q3,d
36. Q1,The spool's circumferance is larger so
one rotation has more distance of the tape. Also more distance at the same
rmp's will be faster.
Q2,roughly 18,000 miles/day
Q3,b
37. Q1,There is less weight on the other spool
that it has to pull against to unwind it
Q2,You would have to know the circumference of
the ring at that latitude, then take that divided by 24 hrs.
Q3,b
38. Q1,Because the radius fron where the tape is
being removed from is constantly decreasing so the linear acceleration of the
tape is increasing
Q2,Less than 463 m/s because that is the
tangential velocity at the equator. The distance from Indy to the axis of
rotation is less than the radius of the Earth. Therefore the tangential
velocity is less
Q3,b
39. Q1,The spindle which receives the rewinding
tape is turned at a constant speed by the VCR. As more tape is accumulated on
the spindle, the distance from the center of the spindle to the outside of the
rewound tape keeps getting greater. As that distance gets greater, the
tangential velocity of the already rewound tape gets larger and larger. This
pulls the tape onto the spindle at a higher and higher speed.
Q2,If the earth's diameter is 8000 miles and
Indy is at 40 degrees N latitude, then the length of a chord parallel to the
equator passing through Indy would be cos 40 x 8000 or 6128 miles. 6128 x pi
would give the distance the Indy would travel in a day. This is about 19,253
miles. Divide that by 24 hrs. and you find that Indy has a tangential velocity
of about 802 miles per hour.
Q3,b