1. © 2007 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials. Figures Chapter 7 College Physics, 6 th Edition Wilson / Buffa / Lou

3. TRANSLATION (or linear) – motion along a path; measureable quantities include both position and velocity

4. ROTATIONAL – spinning at a constant rate about a vertical axis

5. CIRCULAR – set of motion along an orbit

6. Figure 7-1 Polar coordinates A point may be described by polar coordinates instead of Cartesian coordinates—that is, by (r,θ) instead of (x,y). For a circle, θ is the angular distance and r is the radial distance. The two types of coordinates are related by the transformation equations x = r cos θ and y = r sin θ.

7. Figure 7-2 Radian measure Angular displacement may be measured either in degrees or in radians (rad). An angle θ is subtended by an arc length s. When s = r, the angle subtending s is defined to be 1 rad. More generally, θ = s/r, where θ is in radians. One radian is equal to 57.3°.

8. Figure 7-3 Arc length—found by means of radians Given : Ө = 90 o = π/2 rad Find s : arc length s = rӨ = (256 m) ( π /2) = 402 m

9. Figure 7-4 Angular distance For small angles, the arc length is approximately a straight line, or the chord length. Knowing the length of the tanker, we can find how far away it is by measuring its angular size.

10. Learn by Drawing 7-1 The Small-Angle Approximation

11. object moves in circular path about an external point (“revolves”)

12. Figure 7-5 Angular velocity The direction of the angular velocity vector for an object in rotational motion is given by the right-hand rule: When the fingers of the right hand are curled in the direction of the rotation, the extended thumb points in the direction of the angular-velocity vector. Circular senses or directions are commonly indicated by (a) plus and (b) minus signs.

13. Centripetal force and acceleration may be caused by: gravity - planets orbiting the sun friction - car rounding a curve a rope or cord - swinging a mass on a string r m In all cases, a mass m moves in a circular path of radius r with a linear speed v. The time to make one complete revolution is known as the period, T. v The speed v is the circumference divided by the period. v = 2r/T

14. Figure 7-6 Tangential and angular speeds Tangential and angular speeds are related by v = rω, where ω is in radians per second. Note that all of the particles of an object rotating about a fixed axis travel in circles. All the particles have the same angular speed ω, but particles at different distances from the axis of rotation have different tangential speeds.

15. Figure 7-7 Uniform circular motion The speed of an object in uniform circular motion is constant, but the object’s velocity changes in the direction of motion. Thus, there is an acceleration.

16. Figure 7-8 Analysis of centripetal acceleration (a) The velocity vector of an object in uniform circular motion is constantly changing direction. (b) As Δt, the time interval for Δθ, is taken to be smaller and smaller and approaches zero, Δv (the change in the velocity, and therefore an acceleration) is directed toward the center of the circle. The result is a centripetal, or center-seeking, acceleration which has a magnitude of ac = v 2 /r.

17. Figure 7-9 Centripetal acceleration For an object in uniform circular motion, the centripetal acceleration is directed radially inward. There is no acceleration component in the tangential direction; if there were, the magnitude of the velocity (tangential speed) would change. Note: Centripetal acceleration depends on tangential speed (v) and radius (r).

18. According to Newton’s First Law of Motion, objects move in a straight line unless a force makes them turn. An external force is necessary to make an object follow a circular path. This force is called a CENTRIPETAL (“center seeking”) FORCE. Since every unbalanced force causes an object to accelerate in the direction of that force (Newton’s Second Law), a centripetal force causes a CENTRIPETAL ACCELERATION. This acceleration results from a change in direction, and does not imply a change in speed, although speed may also change.

19. The formula for centripetal acceleration (ac) is: ac = v2/r and centripetal force (Fc) is: Fc = mac = mv2/r m = mass in kg v = linear velocity in m/s Fc = centripetal force in N r = radius of curvature in m ac = centripetal acceleration in m/s2

20. Figure 7-11 Centripetal force (a) A ball is swung in a horizontal circle. (b) If the string breaks and the centripetal force goes to zero, what happens to the ball?

21. Figure 7-13 Ball on a string Integrated Ex. 7.9. A 1.0 m cord is used to suspend a 0.50 kg tetherball from the top of the pole. After several hits, the ball goes around the pole with a tangential speed of 1.1 m/s at angle of 20 o relative to the pole. What is the magnitude of the centripetal force? Given: L =1.0 m v t = 1.1 m/s m=0.50 kg Ө = 20 o

22. object moves in circular path about an internal point or axis (“rotates” or “spins”)

23. The amount that an object rotates is its angular displacement. angular displacement, , is given in degrees, radians, or rotations. 1 rotation = 360 deg = 2 radians The change in an object’s angular displacement over time i ii is its angular velocity. angular velocity, w, is given in deg/s, rad/s, rpm, etc...

24. The change in an object’s angular velocity over time is its angular acceleration. Angular acceleration, a, is given in deg/s2, rad/s2, rpm/s, etc... Formulas for rotational motion follow an exact parallel with linear motion formulas. The only difference is a change in variables and a slight change in their meanings.

25. TRANSLATIONAL v f = v i + at d = v av t v av = (v f + v i )/2 d = v i t + ½ at 2 v f 2 = v i 2 + 2ad ROTATIONAL  f  =  i +  t  =  av t ω av =  (  f  +  i )/2  =  i t  +  ½  t 2  f 2  =  i 2  +  2 