Circular motion does not persist without a force
1. An overhead view of a person swinging a rock on a rope. A force from the
string is required to make the rock's velocity vector keep changing direc- tion.
2. If the string breaks, the rock will follow Newton's first law and go straight
instead of continuing around the circle.
Circular Motion
Angular Velocity &
Centripetal Force
1
2π radians = 360° = 1 rotation
Degrees
360
Radians
revolutions
1.5
6.28
2.5
2
180
3.14
¾
57.3
2
Radians & Arc Length Lab
Objectives:
•
Students will be able to develop the relationship between
radian and arc length
•
Students will be able to demonstrate and explain why
there are 2π radians in one full revolution.
•
On a separate sheet of plain white paper, use a compass
to draw a large circle.
Measure the distance from the center point to the
outside of the circle. This is the radius.
Using scissors, cut a piece of string equal to the length
of the radius
Bend the piece of string to conform to the circle, and lay
the string along the circle you drew with your compass.
Tape the string in place.
With your pencil, make a mark on the circle at each end
of your string. Draw the “piece of pie”.
Cut a new string and repeat…
• Be sure you are placing the string end to end along
the arc of the circle.
Draw lines from the center of the circle to the markings
you have just made along the circumference of the circle,
*Each uniform, large “piece of the pie” is one radian!!!
Your circle should now look kind of like a pie cut into
sections.
Draw a smaller circle using your compass. Repeat all of
the above steps… starting from the same reference!!
Procedure:
•
•
•
•
•
•
•
•
Questions:
•
•
•
Approximate how many wires (of the length you cut), it
would take to outline the circumference of the circle?
Approximate how many slices of “pie” make up the entire
circle?
Approximately what angle lies between each slice of
“pie”?
3
ω = 2π(rad) / T
or
ω = ∆Θ / ∆t
V=ωr
Angular Velocity
1. A merry-go-round takes 6.78 seconds to make one complete turn.
What was the angular velocity?
2. If an object has an angular velocity of 0.1046 rad/sec;
a. What is its period?
b. What might the object be?
3. The angular velocity of a bicycle wheel is 50 rev/min (convert to
rads/sec). A spoke of the wheel is 50.0 cm long.
a. What is the linear velocity of a point on the tire?
b. What is the linear velocity of a point on the spoke 1/3 of the
way out from the axel?
4. A race car travels around a 1000 m radius circular track at a rate of
69.4 m/s (250km/hr).
a. What is the car’s angular speed?
b. From a point in the center of the race track, how many degrees
will the car sweep out in ten seconds?
5. Find the angular speed and linear speed of the moon in its orbit as it
makes one revolution in 27.3 days at an average distance of
384,000,000 m from the earth.
6. What is the angular velocity of…
a. The second hand of a clock (60s)?
b. The minute hand of a clock (1hr)?
c. The hour hand of a clock (12hr)?
4
ac = v2/r
ω = 2π(rad) / T
or
ω = ∆Θ / ∆t
ac = (4π2 r)/ T2
α = Δω / Δt
OR
Angular Acceleration and
Centripetal Force
Fc = m
1. The angular velocity of a bicycle wheel is 50 rev/min. Suppose the angular
velocity of the wheel increases to 100 rev/min in 10 seconds, and then comes
to rest in 8.4 seconds after that. Calculate the angular acceleration during
these two periods of time.
2. What is the angular acceleration of a phonograph turntable if it reaches its
angular speed of 33 and a third rev/min in 0.25 seconds?
3. A roulette wheel turning at 1.2 rev/second comes to rest in 18.0 seconds.
What was the deceleration of the wheel?
4. A centrifuge is accelerated from an angular velocity of 3000 rev/min to
8000 rev/min in 21.5 seconds. What is its angular acceleration?
5. You enter a room and flip the switch for a ceiling fan. It takes the ceiling
fan 12.3 seconds in order to be spinning at 2.35 rad/s. What was the angular
acceleration of the fan?
6. A merry-go-round is spinning at 1.65 rad/s. It is the end of the day so the
operator shuts off the power. It takes 35 s for the ride to come to a stop.
What was the angular acceleration of the ride?
7. How much centripetal force is required for an 850 kg race car traveling at
200 mile/hr (89.4 m/s) to go around a bend with a radius of 195 meters?
8. What does centripetal mean?
9. What does centrifugal mean?
Answer centripetal or centrifugal to the following statements:
10. A false force used to describe what one feels when their frame of
reference is rotating ________________.
11. Force required for any object to travel in a circular path _____________.
12. This type of force is responsible for keeping the moon in an almost perfectly
circular orbit _________________.
13. “center fleeing” _________________.
5
x
ac
A quick Review:
Centripetal Acceleration & Centripetal Force
1. What are the two things needed in order for an object, any object, to
travel in a circular path?
a.
b.
2. Since an object moving in a circle is constantly changing direction, it is
also _______________
Some important Equations:
ac = v2/r
OR
&
ac = (4π2 r)/ T2
Fc = mac
1. It takes a 615 kg racing car 14.3 s to travel at a uniform speed around a
circular racetrack of 50.0 m radius.
a. Is the car accelerating?
b. What is the acceleration of the car?
c. What average force must the track exert on the tires to produce
this acceleration?
2. An athlete whirls a 7.0kg hammer tied to the end of a 1.3 m chain in a
horizontal circle. The hammer moves at a rate of 1.0 rev/second.
a. What I the centripetal acceleration of the hammer?
b. What is the tension of the chain?
3. Sam O’neil whirls a yo-yo in a horizontal circle. The yo-yo has a mass of 0.20
kg and is attached to a string 0.80 m long.
a. If the yo-yo makes 1 complete revolution each second, what force
does the string exert on it?
b. Sam increases the speed of the yo-yo to 2.0 rev/sec., what force
does the string now exert?
4. According to the Guinness Book of World Records, the highest tangential
speed ever attained was 2010 m/s (4500mph). The rotating rod was 15.3 cm
(0.153 m) long. Assume the speed quoted was at the end of the rod.
a. What is the centripetal acceleration at the end of the rod?
b. What is the period of rotation of the rod, T?
5. The “Enterprise” at KENNYWOOD takes 2.4 seconds to make one revolution
when it is spinning the fastest. When you are in your seat, you are 15 m from
the center.
a. What is the centripetal acceleration or the rider when the ride is
spinning the fastest?
6
Hammer throw
The modern or Olympic hammer throw is an athletic
throwing event where the object is to throw a heavy metal ball
attached to a wire and handle. The name "hammer throw" is
derived from older competitions where an actual sledge
hammer was thrown. Such competitions are still part of the
Scottish Highland Games, where the implement used is a steel
or lead weight at the end of a cane handle.
Like other throwing events, the competition is decided by who
can throw the ball the farthest. The men's hammer weighs 16
pounds (7.257 kg) and measures 3 feet 11 3⁄4 inches (121.5 cm)
in length and the women's hammer weighs 8.82 lb (4 kg) and
3 feet 11 inches (119.5 cm) in length. Competitors gain
maximum distance by swinging the hammer above their head
to set up the circular motion. Then they apply force and pick up
speed by completing one to four turns in the circle. In
competition, most throwers turn three or four times. The ball
moves in a circular path, gradually increasing in velocity with
each turn with the high point of the ball toward the sector and
the low point at the back of the circle. The thrower releases the
ball from the front of the circle. The two most important factors
for a long throw are the angle of release (45° up from the
ground) and the speed of the ball (the highest possible).
Centripetal Force Pre-Lab
1.
2.
3.
4.
5.
6.
Measure your arm length…
Determine the radius of the circle…
Determine the period (T) of the Hammer…
Calculate the angular speed (ω)…
Calculate the linear speed…
Calculate the Centripetal Force (FC)…
________________ m
________________ m
________________ s
________________ rad/s
________________ m/s
________________ N
http://www.youtube.com/watch?v=LYf8NZnh0oI
7
Centripetal Force Lab
Objective:
Name:______________
Verify the relationship between Fc, m, v , and r.
Fc = m (v2 / r)
Trial #
Mass of Stopper
(kg)
Mass of Washers
(kg)
Fc = m (9.8m/s2)
(N)
1
2
3
4
5
6
7
8
9
10
11
12
Radius
(m)
Time for 20
Swings
Τ
(t/20)
(s)
(s)
ω = 2π(rad)
Τ
(rad/s)
ν
(ν = ω r)
(m/s)
1.00
1.00
1.00
1.00
0.6
0.6
0.6
0.6
1.2
0.8
0.5
0.3
8
GRAPHS: Each person will be required to create three (3) graphs
from your data. The graphs include:
#1
#2
#3
For trials 1-4
For trials 5-8
For trials 9-12
ν vs. Fc (vary Fc)
ν vs. m (vary m)
ν vs. r (vary r)
TIPS AND REMINDERS:
*** USE YOUR FLAG TO KEEP THE RADIUS CONSTANT.
THE FLAG SHOULD BE ½ INCH BELOW THE TUBE BOTTOM AT ALL
TIMES.
*** WHEN VARYING THE Fc, THE NUMBER OF WASHERS SHOULD
CHANGE BY 3 OR 4 EACH TRIAL.
*** BE CAREFUL!!!!
***YOU DO NOT HAVE TO CUT THE STRING. JUST MOVE THE
FLAG UP AND DOWN TO VARY THE RADIUS
9
Simulated Gravity:
1. Most of the energy of train systems is used in starting and stopping. The
design of the rotating train platform saves energy, because passengers can
board or leave a train while the train is still moving. Study the sketch and
convince yourself that this is true. The small circular platform in the middle
is stationary, and is connected to a stationary stairway.
a. If there is to be no relative motion between the train and the edge of the
platform, how fast must the train move compared to the rim speed of the
rotating platform?
_________________________________________________________
b. Why is the stairway located at the center of the platform?
_________________________________________________________
2. The design below shows a train that makes round trips in a continuous loop
from Station A to Station B.
a. How is the size of the round platform and train speed related to the
amount of time that passengers have for boarding?
_____________________________________________________
b. Why would a rotating platform be impractical for high speed trains?
______________________________________________________________
____________________________________________________
10
3. Here are some people standing on a giant, rotating platform in a fun house.
In the view shown, the platform is not rotating and the people stand at rest.
When the platform rotates, the person in the middle stands as before. The person
at the edge must lean inward as shown. Make a sketch of the missing people to show
how they must lean in comparison.
4. The left-hand sketch below shows a stationary container of water and some
floating toy ducks. The right-hand sketch is the same container rotating
about a central axis at constant speed. Note the curved surface of the
water. The duck in the center floats as before. Make a sketch to show the
orientation of the other two ducks with respect to the water surface.
5. Consider an automobile tire half filled with water. In the cross-sectional
views below, the left- hand sketch shows the water surface when the tire is
not rotating. The right-hand sketch shows the water surface when the tire
rotates about its central axis.
Now suppose the tire is rotating while in orbit in outer space. Draw the shape of
the water surface in the cross-sectional view below.
In your mind, scale up the rotating tire model to a rotating space habitat
orbiting in space. If the space habitat were half filled with water, could
inhabitants float on the surface as they do here on earth? Discuss this with
your classmates.
11
Concepts of Physics Study Guide Chapter 9
The following study guide is a list of objectives that you should have an
understanding of for the test on Chapter 9. It is not meant to be a series of
questions and answers that would appear on the test. You should review each
of these items and your class notes. It would also be beneficial for you to
reread the chapter and review the questions at the end of the chapter.
1. You should be able to define each of the vocabulary terms found on
page 133.
2. You should have an understanding of all the concept summary
statements on page 133.
3. You should be able to explain the difference between rotations and
revolutions.
4. You should be able to differentiate between linear speed, tangential
speed and rotational speed.
5. You should be able to explain Figure 9.3 on page 124.
6. You should be able to explain centripetal force and how it originates.
7. You should be able to give examples of centripetal forces.
8. You should be able to explain centrifugal force and how it originates.
9. You should be able to compare and contrast centripetal and
centrifugal forces.
10. You should be able to explain how centrifugal force is related to
simulated gravity.
11. You should be able to explain Figure 9.10 on page 129.
12. You should know be able to explain how the human body adapts to
simulated gravity.
13. You should be able to explain how distance and RPM are factors in
simulated gravity.
14. You should be able to answer any problem on pages 133-135, the
review work sheets or any other work sheets handed out.
12