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Circular motion & relative velocity Announcements: • Prelectures from smartphysics are now being counted. • Tutorials tomorrow – pages 13-17 in red book. • CAPA due Friday at 10pm Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/ 1 Clicker question 1 Set frequency to BA A flare is dropped from an airplane flying at uniform velocity (constant speed in a straight line). Neglecting air resistance, the flare will A: quickly lag behind the plane B: remain vertically under the plane C: move ahead of the plane D: it depends how fast the plane is flying. The horizontal velocity of the flare will remain constant, and equal to what it began with, namely, the airplane's forward velocity. It will fall and remain under the plane all the time. 2 Centripetal acceleration A particle goes around a circle of radius R with constant speed so are perpendicular to their respective radials and have the same length as do the two radials, we identify two similar triangles so or and average acceleration is Instantaneous acceleration: 3 Centripetal acceleration In order to maintain constant speed, the acceleration vector for uniform circular motion must always be perpendicular to the velocity vector, i.e. pointing to the center of the circle. If you're in car going around a curve, you feel as if you're being thrown to the outside. Most people conclude since you feel this way, the acceleration must be outward. You're trying to go in a straight line, it's the car that's turning. Thus the accel. on you is not throwing you out. What you feel is the door or the seat belt on you, pulling you in. This acceleration prevents you from flying out of the car in the 4 straight line you'd like to go in. It's called centripetal acceleration. Clicker question 2 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? A X B E D the acceleration vector C points in the same direction as 5 Nonuniform circular motion What is the acceleration for circular motion with varying speed (nonuniform circular motion)? Can divide acceleration vector into two parts Tangential acceleration is related to change in speed and is parallel to the velocity vector with magnitude Radial acceleration is perpendicular to the velocity vector and points to the center of the circle with magnitude Since they are perpendicular: 6 Circular motion problem A child is on the outside of a merry-go-round with a diameter of 3 m. Starting from rest, another kid spins the merry-go-round up to 60 revolutions per minute in 3 seconds at a constant linear (tangential) acceleration. What is the acceleration 1 second into the spin up? The final frequency is The final speed is Tangential acceleration during spin up: Speed at 1 second: The radial acceleration at 1 second is 7 Clicker question 3 Set frequency to BA Q. A race car travels around the track shown at constant speed. Over which portion of the track is the magnitude of the acceleration the smallest? 2 A. B. C. D. 4 From 1 to 2 From 3 to 4 Different portion of track Impossible to tell 3 1 Car at constant speed so no linear (tangential) acceleration Centripetal acceleration is and v is constant so minimum acceleration occurs when R is largest. This is in the straight sections where R is infinite and so a=0. 8 Relative velocity Roger Clemens is on a moving walkway which moves at 2 m/s and throws his fastest pitch which he knows is 45 m/s. What speed is measured by an observer with a radar gun on the walkway? 45 m/s How about an observer with a radar gun off the end of the walkway? 47 m/s This is the principle behind relative velocity. It just comes down to vector addition 9 Relative velocity Reference frame defines a coordinate system and a velocity Define reference frame E as motionless relative to the Earth Define reference frame W as motionless relative to the walkway Velocity of Clemens’ pitch relative to the walkway is Velocity of walkway reference frame with respect to the Earth reference frame is Velocity of Clemens’ pitch with respect to the Earth reference frame is 10 Relative velocity In vector form we have A row boat in still water has a speed of vb It heads directly east across a river of width w which is flowing south at a speed of vr If vb=1.0 m/s, vr=0.5 m/s, and w=50 m, where and when will the boat land? Easy solution: so the trip takes 50 seconds and the boat lands 25 m downstream of the point directly across from the starting point 11 Relative velocity A row boat with speed of vb heads directly east across a river of width d flowing south at vr. For vb = 1.0 m/s, vr = 0.5 m/s, and w = 50 m, when and where will the boat land? Distance traveled is Trip takes 12 Clicker question 4 Set frequency to BA Q. To cross straight across a river (that is, end up at a place directly across from the starting point), will the presence of a current cause the trip to take a longer, shorter, or same amount of time? A. Trip will take a longer time if there is a current B. Trip will take a shorter time if there is a current C. Trip will take the same amount of time D. Not enough information to solve the problem 13 Relative velocity A row boat with speed of vb wants to reach a point directly across a river with a current of vr. For vb = 1.0 m/s, vr = 0.5 m/s, and river width of w = 50 m, when will the boat land? To go straight across we need a vertical velocity of zero so and so the crossing time is , longer than the no-current time of 50 s 14 New subject: Forces • Colloquially the idea of a force might be the thing that makes other things move – Although this is not entirely correct it summarizes the basic idea • Might also think of a force that surrounds and permeates us and binds the galaxies together – This also has some relevance 15 Types of forces Contact forces Examples are pushing a block, pulling a rope, hitting a ball as well as the force of friction Long range forces Examples are gravity and the force that causes a magnet to attract some metal objects 16