1. 1 Circular Motion Suppose you drive a go cart in a perfect circle at a constant speed. Even though your speed isn’t changing, you are accelerating. This is because acceleration is the rate of change of velocity (not speed), and your velocity is changing because your direction is changing! Remember, a velocity vector is always tangent to the path of motion. v v v

2. 2 Tangential vs. Centripetal Acceleration 10 m/s 15 m/s So how do we calculate the centripetal acceleration ? ? ? Stay tuned! 18 m/s start finish Suppose now you drive your go cart faster and faster in a circle. Now your velocity vector changes in both magnitude and direction. If you go from start to finish in 4 s, your average tangential acceleration is: a t = (18 m/s - 10 m/s) / 4 s = 2 m/s 2 So you’re speeding up at a rate of 2 m/s per second. This is the rate at which your velocity changes tangentially. But what about the rate at which your velocity changes radially, due to its changing direction? This is your centripetal (or radial) acceleration.

3. 3 Centripetal Acceleration v 0 = v r r  v f = v Let’s find a formula for centripetal acceleration by considering uniform circular motion. By the definition of acceleration, a = (v f - v 0 ) / t. We are subtracting vectors here, not speeds, otherwise a would be zero. ( v 0 and v f have the same magnitudes.) The smaller t is, the smaller  will be, and the more the blue sector will approximate a triangle. The blue “triangle” has sides r, r, and v t (from d = v t ). The vector triangle has sides v, v, and | v f - v 0 |. The two triangles are similar (side-angle-side similarity). vfvf v0v0  v f - v 0  r r v tv t continued on next slide

4. 4 Centripetal Acceleration (cont.) v 0 = v r r  v f = v By similar triangles, v v  | v f - v 0 |  r r v tv t v r = v t v t So, multiplying both sides above by v, we have | v f - v 0 | t = r v 2v 2 ac =ac = Unit check: (m/s) 2 m = m2 / s2m2 / s2 m m s 2 =

5. 5 Centripetal acceleration vector always points toward center of circle. v acac atat v acac atat moving counterclockwise; speeding up moving counterclockwise; slowing down “Centripetal” means “center-seeking.” The magnitude of a c depends on both v and r. However, regardless of speed or tangential acceleration, a c always points toward the center. That is, a c is always radial (along the radius).

6. 6 Resultant Acceleration a c atat a The overall acceleration is the vector sum of the centripetal acceleration and the tangential acceleration. That is, a = a c + a t This is true regardless of the direction of motion. It holds true even when an object is not moving in a perfect circle. Note: The equation above does not include v. Vectors of different quantities cannot be added! moving counterclockwise while speeding up or moving clockwise while slowing down

7. 7 v acac acac Non-circular paths acac R2R2 R1R1 P 1 v P 2 Here we have an object moving along the brown path at a constant speed (a t = 0). a c changes, though, since the radius of curvature changes. At P 1 the path is approximated by the large green circle, at P 2 by the smaller orange one. The smaller r is, the bigger a c is. r v 2v 2 ac =ac =

8. 8 Angular Speed,  Linear speed is how fast you move, measured as distance per unit time. Angular speed is how fast you turn, measured as an angle per unit time. The symbol for angular speed is the small Greek letter omega, , which looks like a curvy “w”. Units for angular speed include: degrees per second; radians per second; and rpm (revolutions per minute). A 3 m 110  B Suppose an object moves steadily from A to B along the circle in 5 s. Then  = 110  / 5 s = 22  / s. The distance it covers is (110  / 360  )(2  )(3) = 5.7596 m. So its linear speed is v = (5.7596 m) / (5 s) = 1.1519 m / s.

9. 9 Arc length: s = r  r  s If  is in radians, then the arc length, s, is  times r. This follows directly from the definition of a radian. One radian is the angle made when the radius of a circle is wrapped along the circle. 1 radian r r When the arc length is as long as the radius, the angle subtended is one radian. (A radian is really dimensionless, since it’s found by dividing a length by a length.)

10. 10 v = r  M  C C´C´ The Three Stooges go to the park. Moe and Larry are on a merry-go- round ride of radius r that Larry is pushing counterclockwise, running at a speed v. In a time t, Moe goes from M to M´ and Curly goes from C to C´. Moe is twice as far from the center as Curly. The distance Moe travels is r , where  is in radians. Curly’s distance is ½ r . Both stooges sweep out the same angle in the same time, so each has the same angular speed. However, since Moe travels twice as far, his linear speed in twice as great. M´M´ v v / 2v / 2 s = r  stst r  t = =  t r v = r 

11. 11 Ferris Wheel Problem Schmedrick is working as a miniature ferris wheel operator (radius 2.1 m). He gets a little overzealous and cranks it up to 75 rpm. His little brother Poindexter flies out at point P, when he is 35  from the low point. At the low point the wheel is 1 m off the ground. A 1.5 m high wall is 27 m from the low point of the wheel. Does Poindexter clear the wall? P Strategy outlined on next slide

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