Two Dimensional & Circular Motion

Advanced Placement Physics 1Mr. Kuffer

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AP Physics 1Mr. Kuffer

A stone is thrown horizontally at a speed of +5.0 m/s from the top of a cliff 78.4 m high.

How long does it take the stone to reach the bottom of the cliff? How far from the base of the cliff does the stone strike the ground? What are the horizontal and vertical components of the velocity of the

stone just before it hits the ground?

Two Dimensional Motion

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Projectile Motion Practice & Simulation

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Motion in a Plane & Projectile Motion

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Additional Projectile Problems

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Objective: The purpose of this lab is to demonstrate an understanding of the independence of

vertical and horizontal velocities of a projectile by solving a problem of a projectile launched horizontally in a lab setting.

Materials:Ball-bearing, race-car track, stopwatch, meter stick Styrofoam cup, lab table, calculator, class notes, text

Setup / Procedure:To be explained in class. If absent, be prepared to gather notes from a lab partner.

Remember to draw any diagram(s) when needed.

Horizontal launch (to take place in the classroom) – Using vx and dy (height) of the Ball bearing, find its dx (range), vyf, and t.

Theory:The independence of vertical and horizontal motion and our motion equations

(Use Textbook) can be used to determine the position of thrown objects. If we call the horizontal displacement dx and the initial horizontal velocity vx then, at time t, (Note: vxf = vxi)

dx = vxt

The equations for an object falling with constant acceleration, g, describe the vertical motion. If dy is the vertical displacement, the initial vertical velocity of the object

is vy. At time t, the vertical displacement is

dy = vyi t + ½ gt2

Using these equations, we can analyze the motion of projectiles. (Be sure to retain the independence of the vertical and horizontal components)

Analysis Questions:

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Projectile MotionCan you get the Ball in the Cup?

Mr. Kuffer

1. The Ball bearing rolls “without friction” across the table at a CONSTANT VELOCITY. When it reaches the end of the table, it flies off and lands on the ground.

a) Draw the situation above, drawing vectors showing the Acceleration of the Ball-bearing at two positions while it is on the table and at three more when it is in the air. Draw all vectors to scale.

2. For the Ball bearing in question 1, a) Draw vectors showing the horizontal and vertical components of the

Ball bearing’s velocity at the five points.b) Using a different color, draw the total velocity vector at the five points.

3. Determine the time the ball will be in flight.4. Determine where the ball will land.5. What will the final velocity be in the …

a) X direction?b) Y direction?

6. What will the total final velocity equal?

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Projectile Problems Continued…

** Show all work on a separate sheet of paper **

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Continued…

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Additional Horizontal Projectile Practice Problems

9. A stone is thrown horizontally at a speed of 5.0 m/s from the top of a cliff 78.4 m high.

a. How long does it take the stone to reach the bottom of the cliff? b. How far from the base of the cliff does the stone hit the ground?

c. What are the horizontal and vertical components of thestones velocity just before it hits the ground?

10. How would the three answers to problem 9 change if… a. The stone were thrown with twice the horizontal speed? b. The stone were thrown with the same speed, but the cliff

were twice as high?

11. A steel ball rolls with constant velocity across a tabletop0.950 m high. It rolls off and hits the ground 0.352 m from the edge of the table. How fast was the ball rolling?

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10 A) a) 4.0 s b) 40 m c) Vx = 10 m/s

Vy = 39 m/s

10 B) a) 5.7 s b) 28 m c) Vx = 5.0 m/s

Vy = 55 m/s

C) Vx = 5.0 m/sVy = 39 m/s

B) 20 mA) 4.0 s

V = 0.8 m/s

Angular Projectile Motion

Daniel Sepulveda (now retired due to injury) punts a football at 45˚ with an initial velocity of 24 m/s.

1. What is the hang time of the ball?2. What does the punt ‘net’, assuming the return man

signals for a fair catch? 3. What was the maximum height of the punt?

Extra Practice Solution

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12) t = 2.76 s dx = 64.6 m dy = 9.27 m

13) t = 4.78 s dx = 64.5 m dy = 27.9 m

Ball TossWhen a juggler tosses a ball straight upward, the ball slows down until it reaches the top of its path. The ball then speeds up on its way back down. A graph of its velocity vs. time would show these changes. Is there a mathematical pattern to the changes in velocity? What is the accompanying pattern to the position vs. time graph? What would the acceleration vs. time graph look like?

In this experiment, you will use a Motion Detector to collect position, velocity, and acceleration data for a ball thrown straight upward. Analysis of the graphs of this motion will answer the questions asked above.

OBJECTIVES Collect position, velocity, and acceleration data as a ball travels straight up and

down. Analyze the position vs. time, velocity vs. time, and acceleration vs. time graphs. Determine the best fit equations for the position vs. time and velocity vs. time

graphs. Determine the mean acceleration from the acceleration vs. time graph.

MATERIALScomputer Vernier Motion DetectorVernier computer interface ballLogger Pro

PRELIMINARY QUESTIONS1. Think about the changes in motion a ball will undergo as it travels straight up and

down. Make a sketch of your prediction for the position vs. time graph. Describe in words what this graph means.

2. Make a sketch of your prediction for the velocity vs. time graph. Describe in words what this graph means.

3. Make a sketch of your prediction for the acceleration vs. time graph. Describe in words what this graph means.

PROCEDURE1. Connect the Vernier Motion Detector to the DIG/SONIC 1 channel of the interface.

2. Place the Motion Detector on the table. Cover the Motion Detector with the protective cage.

3. Open the file “06 Ball Toss” from the Physics with Computers folder.

4. In this step, you will toss the ball straight upward above the Motion Detector and let it fall back toward the Motion Detector. This step may require some practice. Hold the

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ball directly above and about 0.5 m from the Motion Detector. Click to begin data collection. You will notice a clicking sound from the Motion Detector. Wait one second, then toss the ball straight upward. Be sure to move your hands out of the way after you release it. A toss of 0.5 above the Motion Detector works well. You will get best results if you catch and hold the ball when it is about 0.5 m above the Motion Detector.

5. Examine the position vs. time graph. Repeat Step 4 if your position vs. time graph does not show an area of smoothly changing position. Check with your teacher if you are not sure whether you need to repeat the data collection.

ANALYSIS1. Sketch the three motion graphs in your lab notebook. The graphs you have recorded

are fairly complex and it is important to identify different regions of each graph. Click the Examine button, , and move the mouse across any graph to answer the following questions. Record your answers directly on the sketched graphs.

a) Identify the region when the ball was being tossed but still in your hands: Examine the velocity vs. time graph and identify this region. Label this on the

graph. Examine the acceleration vs. time graph and identify the same region. Label the

graph.

b) Identify the region where the ball is in free fall: Label the region on each graph where the ball was in free fall and moving

upward. Label the region on each graph where the ball was in free fall and moving

downward.

c) Determine the position, velocity, and acceleration at specific points. On the velocity vs. time graph, decide where the ball had its maximum velocity,

just as the ball was released. Mark the spot and record the value on the graph. On the position vs. time graph, locate the maximum height of the ball during free

fall. Mark the spot and record the value on the graph. What was the velocity of the ball at the top of its motion? What was the acceleration of the ball at the top of its motion?

2. The motion of an object in free fall is modeled by y = v0yt + ½ gt2 (equation 6), where y is the vertical position, v0y is the initial velocity in the y direction, t is time, and g is the acceleration due to gravity (9.8 m/s2). This is a quadratic equation whose graph is a parabola. Your graph of position vs. time should be parabolic. To fit a quadratic equation to your data, click and drag the mouse across the portion of the position vs. time graph that is parabolic, highlighting the free-fall portion. Click the Curve Fit button, , select Quadratic fit from the list of models and click . Examine the fit of the curve to your data and click to return to the main graph. How closely does the coefficient of the t2 term in the curve fit compare to ½ g? ( 1/2 g is 4.9… what is ‘coefficient’? How well do they compare?)

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3. The graph of velocity vs. time should be linear. To fit a line to this data, click and drag the mouse across the free-fall region of the motion. Click the Linear Fit button,

. How closely does the coefficient of the t term in the fit compare to the accepted value for g?

4. The graph of acceleration vs. time should appear to be more or less constant. Click and drag the mouse across the free-fall section of the motion and click the Statistics button, . How closely does the mean acceleration value compare to the values of g found in Steps 2 and 3?

5. List some reasons why your values for the ball’s acceleration may be different from the accepted value for g.

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1. Label the max dy and dx.

2. Draw velocity vectors for each point of the projectile’s trajectory.

3. What is the max height of the projectile if it is launched with an initial velocity of 4.3 m/s?

4. How long is the ball in the air?

5. What is the range of the projectile if the cart is traveling at 1.2 m/s?

HINT:“WHAT GOES UP… MUST ____________________________... BEFORE IT COMES BACK DOWN IT HAS GOT TO _____________”

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Angular Projectile Lab

Projectiles Launched at an AnglePhysics Names:Projectile Launcher LabNorth Allegheny SHMr. Kuffer Period:

Determining the Range and Apex of a Projectile

BackgroundFrom class, you know that a projectile is something that is thrown or fired but not self propelled. You also know that because of gravity pulling the projectile from its straight line path, it will ideally follow a parabolic path. Also, you know that this seemingly complicated motion can be simplified by looking at the two dimensions separately. -

PurposeThe purpose of this lab is to determine the range of a projectile launcher and the height at apex. As a test, we will fire a plastic marble projectile through a hoop at the apex and into a container at the maximum range.

ProcedureStep 1-Determining the initial velocity of the launcher

Using the supplied bracket, attach a photogate to the end of the projectile launcher, as shown below.

Using gate mode, measure the time the ball will break the photogate’s beam. Prior to loading the launcher, push the ramrod into the launcher, cocking the launcher to MEDIUM RANGE. Shoot the launcher in a safe direction, and do not catch the ball. Record the broken beam time below. The diameter of the ball is exactly 1 inch or .0254 m. Determine the initial velocity of the launcher on medium range by dividing.

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launcher

bracket

photogate

Launcher #

Step 2 - Determining range of the launcherClamp the launcher on the end of the table. Set the launcher’s angle so that when marble is launched on SHORT range, that it lands somewhere on the two adjacent tables.

Determine where your container should be placed to catch the marble AT THE SAME HEIGHT IT WAS LAUNCHED by resolving your initial velocity into x- and y-components, and then working out your horizontal and vertical mathematics.

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time =

distance = 0.0254 m

velocity= m/sec=

Marble lands somewhere on two tables

m/sec

m/sec

m/sec

(X) (Y)

Try your experiment. Did it work on the first try? What are some possible sources of error?

Continue trying and revising until you can reliably get the marble into your container.

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acceleration = 0 acceleration = g

Step 3-Determining apex of the launcherObtain one of the apex hoops. Place it on a ring stand as shown below.

Calculate mathematically how high the apex ring should be so that the marble can pass through the ring on route to the container. Show your work below.

Try your experiment. Did it work on the first try? What are some possible sources of error?

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Circular Motion

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Circular motion does not persist without a force

Angular Velocity & Centripetal Force

Vocabulary and EquationsRecall: Displacement –

Average velocity –

Average acceleration –

New Terms:

Uniform circular motion –

Axis –

Revolve –

Rotation –

Centripetal acceleration –

Period of Revolution –

Centripetal force –

Radian –

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Circular motion does not persist without a force

IMPORTANT EQUATIONSα=∆ ω

∆ t∨ac=

v2

r∨ac=

(4 π2 r )T 2

F c=m ac ω=2 π (rad )T

∨ω=∆ θ∆ t

v=ωr

Angular displacement –

Angular velocity –

Angular acceleration -

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Example Problems – Uniform Circular Motion

1. A 150 gram ball at the end of a string is revolving uniformly in a horizontal circle of radius 0.60 m. The ball makes 2.00 revolutions in one second. What is its centripetal acceleration?

2. The Moon’s nearly circular orbit about the Earth has a radius of about 384,000 km and a period, T, of 27.3 days. Determine the acceleration of the Moon toward the Earth.

3. Estimate the force a person must exert on a string attached to a 0.150 kg ball to make the ball revolve in a horizontal circle of radius 0.6 m, as in Example 2. The ball makes 2.00 revolutions per second. (you are unable to solve for this problem… for now… we will talk!)

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Practice Problems – Uniform Circular Motion

1. What is the direction of the force that acts on the clothes in the spin cycle of a washing machine? What exerts the force?

2. Describe all the forces acting on a child riding a horse on a merry-go-round. Which of these forces provides the centripetal acceleration of the child?

3. Will the acceleration of a car be the same when it travels around a sharp curve at 60 km/h as when it travels around a gentle curve at the same speed? Explain.

4. What does centripetal mean?

5. What does centrifugal mean?

Answer centripetal or centrifugal to the following statements:

6. A false force used to describe what one feels when their frame of reference is rotating ________________.

7. Force required for any object to travel in a circular path _____________.

8. This type of force is responsible for keeping the moon in an almost perfectly circular orbit _________________.

9. “center fleeing” _________________.

10. “center seeking” ________________.

11.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?

12.A runner moving at a speed of 8.8 m/s rounds a bend with a radius of 25 m. What is the centripetal acceleration of the runner, and what exerts the centripetal force on the runner?

13.An airplane traveling at 201 m/s makes a turn. What is the smallest radius of the circular path (in km) the pilot can make and keep the centripetal acceleration under 5.0 m/s2?

14.A 16 gram ball at the end of a 1.4 m string is swung in a horizontal circle. It revolves once every 1.09 second. What is the magnitude of the string’s tension?

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2π radians = 360° = 1 rotation

Degrees Radians revolutions360

1.5

6.28

2.5

2

180

3.14

¾

57.3

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Radians & Arc Length LabObjectives:

• 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”?

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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)?

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.

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ω = 2π(rad) / Tor

ω = ∆Θ / ∆t

ω = 2π(rad) / Tor

ω = ∆Θ / ∆t

V = ω r

Fc = m x ac

ac = v2/r OR ac = (4π2 r)/ T2

α = Δω / Δt

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” _________________.

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 _______________

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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 (Idowu) 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. Kimble 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?

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

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ac = v2/r OR ac = (4π2 r)/ T2

& Fc = mac

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. 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/s5. Calculate the linear speed… ________________ m/s6. Calculate the Centripetal Force (FC)… ________________ N

http://www.youtube.com/watch?v=LYf8NZnh0oI

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)

123

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456789101112

Radius Time for 20 Τ ω = 2π(rad) ν (m) Swings (t/20) Τ (ν = ω r)

(s) (s) (rad/s) (m/s)1.001.001.001.000.60.60.60.61.20.80.50.3

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:

*** 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.

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*** 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.

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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.

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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.

AP PhysicsUnit 2 Multi-Dimensional Motion Outline

DueTopic Book

Section(s) Assignment Video(s)Date Class

Number

9/17 21 Equations of Kinematics in Two Dimensions

3.1-3.2 P:12-13, 15, 18, 20-21, 23-24, 27-28,

2.1, 2.2, 2.3

9/18 23Projectile Motion 3.3 FOC:1, 3-4, 6

P: 30-31, 34, 36-37, 42-43, 49-51

2.4, 2.5

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9/19 24 Relative Motion 3.4 FOC: 14-16P:52-56, 58, 61, 63-64

2.6, 2.7

26 Rotational Motion and Angular Displacement

8.1 FOC: 1P: 1-2, 6-7, 11-13, 17, 19

2.8

27Angular Velocity and Angular AccelerationEquations of Rotational Kinematics

8.2

8.3

FOC: 6P:20, 22, 26-28, 30-31

2.9

2.10-2.12

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Angular and Tangential Variables

Centripetal and Tangential Accelerations

8.4

8.5

FOC: 10-11P:34, 36-37, 42, 44FOC: 13-14P:45-46, 49, 52

2.13, 2.14

2.15-2.17

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