Welcome back to Physics 211

1. Welcome back to Physics 211 Today’s agenda Recap 2D motion Motion in a circle Tangential and radial components of acceleration

2. Reminder • Exam next Thursday in class (22nd Sep.) • Practice exams, hw solns, formula summary online • MPHW2 available noon this Friday • Note: MPHW1 deadline extended to this Friday also

3. curved paths – constant speed • Velocity vector lies along tangent to path Dv vI vF Change in velocity vector – non-zero acceleration

4. A biker is riding at constant speed clockwise on the oval track shown below. Which vector correctly describes the acceleration at the point indicated?

5. Biker moving around oval at constant speed As point D is moved closer to C, angle approaches 90°.

6. A car is moving at constant speed clockwise around a peanut-shaped racetrack. Which vector correctly describes the acceleration at the point indicated?

7. Zoom vD vc Dv As D->C angle between vC and Dv -> 90 Note: a at 90 degrees to v and points ‘inside curve’

8. Acceleration vectors for car moving on peanut-shaped track at constant speed

9. Summary • For motion at constant speed instantaneous acceleration vector is at 90 degrees to velocity vector • Points ``inward’’ • Magnitude ?

10. Acceleration vectors for ball swung in a horizontal circle at constant speed v R v1 q q v2 What is the magnitude of the acceleration?

11. Magnitude … |Dv|=vsinq=vq if q small (radians) time to do revolution: 2pR/v number of revolutions/sec = v/2pR q=Dt (v/ 2pR) 2p |a|=|Dv/Dt|=vq/(qR/v) =v2/R

12. Radian measure • Alternative way of measuring angle • 2p radians = 3600 • To convert q (radians) to q (degrees) multiply by 180/p • Radian measure default in Physics small angle approx only good if use radians: sin x=tan x=x x small

13. Acceleration of object moving at constant speed on a circular path: Acceleration depends on radius of circle.

14. A ball is rolling counter-clockwise at constant speed on a circular track. One quarter of the track is removed. What path will the ball follow after reaching the end of the track?

15. demo • Circular track and ball – remove section

16. Two cars are moving at different constant speeds on a curved road. One after the other, they are passing the same point on the road: Car A at 18 mph; car B at 36 mph. If car A’s acceleration is 2 m/s2, car B’s acceleration is: 1. 1 m/s2 2. 2 m/s2 3. 4 m/s2 4. 8 m/s2

17. What is the magnitude of the acceleration of an object moving at constant speed if the path is curved but not a circle? “r” is theradius of curvatureof the path at a given point

18. Radius of curvature • The radius of a circle which just touches the curved path at that point. r Radius of curvature r

19. Acceleration vectors for object moving around oval at constant speed

20. speed changing ? • Consider acceleration for object on curved path starting from rest • v2/r=0, so no radial acceleration • But a is not zero ! It must be parallel to velocity • a=D|v|/Dt = rate of change of speed

21. Speed changing - 2 Alternatively consider path with very large radius of curvature  zero radial accel. Motion looks 1 dimensional. Therefore remaining acceleration must be along direction of velocity and measure rate of change of speed.

22. Acceleration vectors for object speeding up

23. Acceleration vectors for object speeding up:Tangential and radial components

24. Summary Components of acceleration vector: Along direction of velocity: (Tangential acceleration) “How much does speed of the object increase?” Perpendicular to direction of velocity: (Radial acceleration) “How quickly does the object turn?”

25. Ball going through loop-the-loop

26. Pendulum

27. Acceleration vector for object speeding up from rest at point A

28. 2D Components • To study 2D motion can resolve all vectors into components. 2 ways: • Cartesian (projectile motion) • Radial/tangential – circular motion, general motion on curved paths • Either can be used. Choose easiest