Circular motion. Announcements:

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1 Circular motion Announcements: Clicker scores through Wednesday are now posted on DL. Scoring is points for a wrong answer, 3 points for a right answer. 13 clicker questions so far, so max is 39 points. If you have a 0, you should investigate. Possible problems: Not using clicker correctly Attending the wrong lecture Clicker not registered Web page: 1

2 Some projectile motion warnings Following are true only when Δy 0!!!!! (when the launch and landing points are at the same height) The time of flight is twice the time to reach the maximum height. The initial and final vertical velocities have the same magnitude and opposite directions. Maximum range is obtained with an angle of 45

3 Clicker question Q. A rifle is accurately aimed at a rabid monkey hanging from the branch of a tree. The instant the gun is fired, the monkey releases the branch and starts falling. The monkey is well within the range of the rifle. The initial speed of the bullet is v o. What happens? gun Set frequency to BA A: The bullet finds its target, regardless of the value of v o. (Assuming v o is large enough to reach the air below the monkey.) B: The bullet hits the monkey only if v o is very large so the monkey does not fall very far. C: The bullet misses. v 0 3

4 Solution to monkey & hunter problem We want to find out if the bullet and monkey ever occupy the same place at the same time. At what time does the bullet reach the x-position of the monkey? v! 0 θ 0 From x x 0 + v x0 Δt and v x0 v 0 cosθ 0 : t x x 0 v 0 cosθ 0 d v 0 cosθ 0 At that time, what are the y-positions of bullet and monkey? Monkey starts at y 0 d tanθ 0 with v y0 0 so y monkey d tanθ 0 1 gt Bullet has v y0 v 0 sinθ 0 at y 0 0 so y bullet v 0 sinθ 0 t 1 gt Bullet hits monkey if y bullet y monkey ; that is, if v sinθ t d tanθ d But v 0 sinθ 0 t v 0 sinθ 0 d tanθ 0 so it always hits! v 4 0 cosθ 0 d

5 Clicker question 1 Set frequency to BA Q. A person stands at the edge of a cliff and throws one ball up and another ball down at the same initial speed. Neglecting air resistance, the ball to hit the ground below the cliff with the greater speed is the one initially thrown A. Upward B. Downward C. Neither, they both hit with the same speed For the one thrown upward, what is its speed at it passes the person on the way down? From v y v y0 g( y y ) 0 it is clear v y v y0 for y y 0 Also, v y does not depend on the sign of v y0. 5

6 Clicker question Set frequency to BA Q. A bullet is fired horizontally from a rifle on the Moon (no air) with initial speed v o. Assuming a level and endless ground, which statement of A-C, if any, is false? A. During the flight, the minimum speed of the bullet is v 0. B. During its entire flight the acceleration is constant. C. The flight time increases as v 0 increases. OR D. All of the above statements are true. A. v x v x0 v 0 and v v x + v y so v v 0 B. Acceleration is due to gravity and is constant downward C. Let y 0 be the gun height so the bullet lands at y0. Since v y0 0, y 1 g t so t y /, independent of v moon 0 g moon 6

7 Clicker question 3 Set frequency to BA Q. True or False: If the speed of an object is constant, the acceleration must be zero. A. True B. False C. Didn t read the question and randomly chose an answer. Remember that with constant speed there can still be an acceleration that is changing the direction of the velocity. 7

8 Centripetal acceleration We learned that when the speed is constant, there can still be acceleration if the velocity direction is changing (like going around a curve). This is called centripetal acceleration or radial acceleration. Radial acceleration is perpendicular to the velocity vector and has a magnitude of v! a! rad a rad v r To have constant speed, the acceleration vector for uniform circular motion must be perpendicular to the velocity vector (otherwise the speed will change). X 8

9 Uniform circular motion problem Some high performance sport cars can turn without skidding with an acceleration of 1.1g. At 100 mph, how sharp a corner can these cars take (without skidding). The radial acceleration is a rad 1.1g m/s 10.8 m/s Need to convert 100 mph to SI units: 100 miles hr 1 hr 3600 s 580 ft 1 mile 1 in 1 ft.54 cm 1 in 1 m 100 cm 44.7 m/s The radial acceleration is related to speed by a rad v Solving for radius gives r v a rad 44.7 m/s ( ) 10.8 m/s 185 m r 9

10 Clicker question 4 Set frequency to BA Q. An object is moving along a circular path and is slowing down, as shown. Which arrow best represents the object s acceleration vector at point X? E D A C B!! a Δv Δt same direction as Δv! Since the acceleration is in the X Δv! To go around a circle requires perpendicular acceleration and to slow down requires acceleration opposite the velocity. v! v! 1 10

11 Nonuniform circular motion What is the acceleration for circular motion with varying speed (nonuniform circular motion)? Can divide acceleration vector into two parts a tan a rad Tangential acceleration is related to v! 1 change in speed and is parallel to the! d v velocity vector with magnitude atan dt Radial acceleration is perpendicular to the velocity vector and points to the center of the circle with magnitude a v rad r! Since they are perpendicular: a a tan + a rad a! v! 11