# 2Π Radians = 360 = 1 Rotation

Circular Motion

Angular Velocity &

Centripetal Force

2π radians = 360° = 1 rotation

360
1.5
6.28
2.5
2
180
3.14
¾
57.3

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.

Procedure:

•  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!!

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”?

Circular Motion with a Ladybug Web Lab

Courtesy of Mr. Neff

To start this lab, visit the Circular Motion with a Ladybug Web Lab link on my website. If this link does not work, type in this URL: http://phet.colorado.edu/en/simulation/rotation and click .

1.  Spin the plate by grabbing it and dragging it. Give it a period of 5-6 seconds.

1. The green vector shows the linear velocity.
What direction does it always point?

Based on what you just answered, what descriptor is generally placed on this velocity?

1. The pink vector is the linear acceleration.
What direction does it always point?

Based on what you just answered, what descriptor is generally placed on this acceleration?

2.  Now stop the plate. Place the ladybug one unit away from the center, and place the beetle on the same radius as the ladybug but three units from the center. Spin the plate about the same speed as before.

1. How does the beetle’s velocity compare to the ladybug’s? Why?
1. How does the beetle’s acceleration compare to the ladybug’s? Why?

3.  Now, click on the Rotation tab at the top of the screen.

Show the θ, ω, v graphs. Show the X-Velocity and Y-Velocity, but not the speed.

1. You see something called the Angular Velocity (middle graph). Play with the simulator and try to explain what Angular Velocity (ω) is.
1. How does Angular Velocity (ω) differ from Velocity (bottom graph) (should be Linear Velocity)?

4.  Though Linear Velocity and Angular Velocity are different, they are closely related. Show the speed (should be called “the” Resultant Velocity), and turn off the X & Y-Velocities. Spin the plate at a few speeds, and observe the relationship between the values for Angular Velocity, ω (from the middle graph) and the Linear “Speed”, v (from the bottom graph). What is that relationship?

5.  Speaking of Linear Velocity (“speed”), there is a relationship between it and the centripetal acceleration. Toggle back and forth between having the “v” graph on the bottom and having the “a” graph on the bottom for a few different speeds.

Below, use your numbers to verify that ac = v2 / r ,

where ac = centripetal acceleration
v = tangential velocity

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;

1. What is its period?
2. What might the object be?

3.  The angular velocity of a bicycle wheel is 50 rev/min. A spoke of the wheel is 50.0 cm long.

1. What is the linear velocity of a point on the tire?
2. 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 250 km/hr.

1. What is the car’s angular speed?
2. 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 km from the earth.

6.  What is the angular velocity of…

1. The second hand of a clock?
2. The minute hand of a clock?
3. The hour hand of a clock?

Angular Acceleration and Centripetal Force

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 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” ______.

Centripetal Acceleration & Centripetal Force

A quick Review:

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:

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 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?

## 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 (7.257kg) and measures 3feet 11 3⁄4 inches (121.5cm) in length and the women's hammer weighs 8.82lb (4kg) and 3feet 11inches (119.5cm) 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.  Measure your arm length… ______m

2.  Determine the radius of the circle… ______m

3.  Determine the period (T) of the Hammer… ______s

4.  Calculate the angular speed (ω)… ______rad/s

5.  Calculate the linear speed… ______m/s

6.  Calculate the Centripetal Force (FC)… ______N

Centripetal Force Lab

Objective: Name:______

Verify the relationship between Fc, m, v , and r.

Fc = m (v2 / r)

Trial # Mass of Stopper Mass of Washers Fc = m (9.8m/s2)

(kg) (kg) (N)

1
2
3
4
5
6
7
8
9
10
11
12

(m) Swings (t/20) Τ (ν = ω r)

1.00
1.00
1.00
1.00
0.6
0.6
0.6
0.6
1.2
0.8
0.5
0.3

GRAPHS: Each person will be required to create three (3) graphs from your data. The graphs include:

#1 For trials 1-4 ν vs. Fc (vary Fc)

#2 For trials 5-8 ν vs. m (vary m)

#3 For trials 9-12 ν vs. r (vary r)

TIPS AND REMINDERS:

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

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?

______

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.