AP Physics 1 Lesson 15.a Rotational Kinematics Graphical Analysis and Kinematic Equation Use. Name. Date. Period. Engage
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1 AP Physics 1 Lesson 15.a Rotational Kinematics Graphical Analysis and Kinematic Equation Use Name Outcomes Date Interpret graphical evidence of angular motion (uniform speed & uniform acceleration). Apply an understanding of position time graphs to novel examples. Use graphical patterns of uniform angular velocity and uniform angular acceleration to derive kinematic equations. Solve graphical analysis problems and problems involving angular kinematic equations. Period Engage At the beginning of the school year we learned how to describe the motion of an object in terms of its location (position), how far it traveled, how fast it was moving, in what direction it was moving, and how fast it changed its motion. Our objects were in linear motion. How can we do the same for objects that rotate or have angular motion? 1. Complete the table below for the motion of the three different hands on this clock and for the tire spokes of a bicycle rolling down the hill as it speeds up. You are just checking what you already know or remember, or what you can deduct from your prior knowledge. You can check your answers when you receive more information on the next page. Second Hand (sorry, I Minute Hand Hour Hand Any front Tire don t know which is Spokes first hand) In general terms, how can you describe its Angle between position? Think back second hand and how we described horizontal line to the the direction of a right vector. What is its specific, 150 degrees current position? In general terms, how 360 degrees/ can you describe how 12x60x60 sec fast it is rotating? What is its rotational speed? In general terms, how can you describe the direction of its rotation? What is the direction cw of its rotation? In general terms, how can you describe the rate at which its rotational speed is changing? How can you indicate the direction of its change in rotational speed? 1
2 Notes: Explore I Vocabulary: angular kinematics, rotation, degrees, angular velocity, revolution, angular motion, theta θ, θ is defined as the motion of a body about a fixed axis. motion about an internal axis motion about an external axis is the description of the angular motion of an object. Angular position Location on the unit circle Measured in either or radians. Uses greek letter for symbol. To convert degrees to radians : degrees x Angular displacement Defined as the change in angular position: Measured in either or radians counterclockwise from a horizontal line extending to the right from the axis of rotation Uses greek symbol. Defined as the rate of change of angular displacement. Measured in deg/sec or rad/sec. Its direction is determined as positive (+) for counterclockwise motion and negative (-) for clockwise motion. As a vector quantity the velocity vector points in the direction of your right thumb when you let your right hand fingers curve according to the rotation of the object. Right hand rule. 2
3 Angular acceleration Defined as the rate of change of. Measured in deg/sec 2 or rad/sec 2 Important: angular acceleration is not the same as centripetal acceleration. The first is due to a force tangential to the arc the object is traveling on, where the second is due to a force towards the axis of rotation. Explore II Practice with graphical analysis. 2. Sketch the appearance of the 3 angular graphs corresponding to the motion of the three hands of the clock. The lines do not have to be representing actual values. 3. Sketch the appearance of the 3 graphs corresponding to the motion of the spokes of the front tire of the bicycle as it speeds up down the hill. Notice how the shape of these graphs match the shape of the graphs you would draw for the translatory (linear) motion when the object is either moving at uniform velocity of uniform acceleration. 3
4 4. Sketch the corresponding angular position, angular velocity, and angular acceleration graphs for the 4 motions depicted above. The graphs are showing the angular velocity versus time. You can choose the location of the 0 y-value, it doesn t have to be at the bottom of the grid. A. B. C. 4
5 D. Explore III Let s explore the relationship of angular and linear motion. 4. Picture a fly has landed on the end of the second hand. How could you determine the length of the path it takes during one minute if the length of the dial is 0.30m? 5. How would you determine the linear speed of the fly as it makes its way around the center of the clock one time? The table below relates translational and angular motion. The conversion is straight forward: multiply the angular quantity by the radius of the circle of the angular motion. We often use the letter s instead of x for the displacement since we are looking at the length of the arc described by the angular motion (distance). 5
6 Explain I 6. A potter s wheel rotates. A student makes the following claim: A location on the wheel farther from the axis will have a greater angular velocity than one closer to the axis because that point covers a greater distance each revolution. a. What, if anything, is correct about this statement? b. What, if anything, is incorrect about this statement? 7. A hand on a clock moves from 12 o clock to 3 o clock. A student makes the following claim: The hand s angular displacement is 90 degrees. a. What, if anything, is correct about this statement? b. What, if anything, is incorrect about this statement? 8. Is it possible for a rotating object to have increasing angular speed and negative angular acceleration? Explain your answer. 9. Order these three cities from smallest to largest tangential velocity due to the rotation of the Earth: Washington, DC, USA; Havana, Cuba; Ottawa, Canada. 10. Which of the following rotational quantities are the same for all points on a rotating disk? Check all that apply, and explain your selections. Angular velocity Tangential velocity Angular acceleration Tangential acceleration Centripetal acceleration 11. Does the angular velocity vector of the Earth point north or south along its axis of rotation? 6
7 12. Angular velocity and linear velocity both describe how fast something is moving. As a result they must have the same dimensions/units. a. What, if anything, is correct about this statement? b. What, if anything, is incorrect about this statement? 13. A ball rolls in a uniform circle with constant speed inside a frictionless cone (look at the picture). The weight of the ball is shown by the vector labeled mg. a. What other forces act on the ball? b. What is the direction of the net force on the ball? c. How does the magnitude of the normal force compare to mg? Explore IV The parallels between translational and rotational motion go even further. You developed a set of kinematic equations for translational motion that allowed you to explore the relationship between displacement, velocity, and acceleration. The relationships were based on using slopes or areas of the respective graphs. The relationships for angular displacement, angular velocity, and angular acceleration are identical for those of linear motion. The equations follow the same form as the translational equations, all you have to do is replace the translational variables with rotational variables, as shown in the following table. Also the instantaneous angular velocity of an uniformly accelerating object is 2 x the average angular velocity assuming it started from rest. 7
8 Explain II Sample question: A carpenter cuts a piece of wood with a high powered circular saw. The saw blade accelerates from rest with an angular acceleration of 14 rad/s 2 to a maximum speed of 15,000 rpms. a) What is the maximum speed of the saw in radians per second? Solution: b) How long does it take the saw to reach its maximum speed? c) How many complete rotations does the saw make while accelerating to its maximum speed? d) A safety mechanism will bring the saw blade to rest in 0.3 seconds should the carpenter s hand come off the saw controls. What angular acceleration does this require? How many complete revolutions will the saw blade make in this time? Now you try a few. 1. Glenn starts his day by walking around a circular track with radius 48 m for 15 minutes. First he walks in a counterclockwise direction for 1000 meters, then he walks clockwise until the 15 minutes are up. This morning, his clockwise walk is 880 meters long. When he ends his walk, what is his angular position with respect to where he starts? 8
9 2. What is the angular displacement in radians that the minute hand of a watch moves through from 3:15 A.M. to 7:30 P.M. the same day? Express your answer to the nearest whole radian. 3. A heavy vault door is shut. The angular position of the door from t = 0 to the time the door is shut is given by θ(t) = 0.125t 2, where θ is in radians. (a) The door is completely shut at θ = π/2 radians. At what time does this occur? (b) What is the angular displacement of the door between t = 0.52 s and t = 1.67 s? (c) What is the door's average angular velocity between t = 1.50 s and t = 2.50 s? 4. Your bicycle tires have a radius of 0.33 m. It takes you 850 seconds to ride 14 times counterclockwise around a circular track of radius 73 m at constant speed. (a) What is the angular velocity of the bicycle around the track? (b) What is the magnitude of the angular velocity of a tire around its axis? (That is, don't worry about whether the tire's rotation is clockwise or counterclockwise.) 5. The blades of a fan rotate clockwise at 225 rad/s at medium speed, and 355 rad/s at high speed. If it takes 4.65 seconds to get from medium to high speed, what is the average angular acceleration of the fan blades during this time? 6. The angular displacement of an object over time is given by the equation θ(t) = cos(0.350t) rad. (a) What is the object's angular velocity at time t = 2.40 seconds? 7. A cyclist starts from rest and rides in a straight line, increasing speed so that her wheels have a constant angular acceleration of 2.0 rad/s 2 around their axles. She accelerates until her wheels are rotating at 8.0 rad/s. If the radius of a tire is 0.29 meters, how far has the cyclist traveled? 9
10 8. Two cars race around a circular track. Car A accelerates at rad/s 2 around the track, and car B at rad/s 2. They start at the same place on the track and car A lets the slower-to-accelerate car B start first. Car B starts at time t = 0. When car A starts, car B has an angular velocity of 1.40 rad/s. a) At what time does car A catch up to car B? b) Sketch the angular motion graphs representing this scenario. a) b) Angular motion graphs 9. How might a magician make the Statue of Liberty disappear? Imagine that you are sitting with some spectators on a circular platform that, unknown to all of you, can rotate very slowly. It is evening, and you can see the Statue of Liberty a short distance away between two tall brightly lit columns at the rim of the platform. A large curtain can be drawn between the columns to temporarily hide the statue. The magician closes the curtain, then rotates the platform through an angle of just radians so the statue is hidden behind one of the columns when the curtain is opened. (a) If the platform rotation takes 24.0 seconds, what is the average angular speed required? (b) You are sitting 4.00 m from the center of rotation while the platform is rotating. What is the centripetal acceleration required to move you along the circular arc? (c) Calculate the centripetal acceleration as a fraction of g. 10
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