1. AP Physics ED Uniform Circular Motion glazert.co.uk

2. Centripetal Acceleration Falkirk Wheel, Falkirk, Scotland Designed to connect the Forth and Clyde Canal and Union Canal in Falkirk, Scotland Only rotating boatlift in the world. gizmodo.com

3. Centripetal Acceleration It takes just 22.5 kW to power the electric motors to turn the wheel. – Equal to the energy needed to boil 8 kettles of water!! techeblog.com

4. Centripetal Acceleration The wheel… – Overall diameter of 35 m – Two opposing arms each 15 m from the axis of rotation. – Shape of a Celtic- inspired double-headed axe. – Two diametrically opposite water-filled caissons holding 80,000 gallons of water each. undiscoveredscotland.co.uk

5. Centripetal Acceleration Caissons weigh the same whether or not they are carrying their maximum capacity of canal barges. According to Archimedes’ principal floating objects displace their combined weight in water. Despite its enormous mass the caissons remain perfectly balanced and rotate 180 o in less than 4 minutes. images.businessweek.com

6. Centripetal Acceleration Now THAT’S engineering! According to Newton’s 1 st Law… “an object in motion remains in motion at constant speed and direction unless acted upon by an outside force.” sterry.eu

7. Centripetal Acceleration The inertial tendency of the caisson is to move in a straight line but is prevented by a force acting towards the center of rotation to keep it moving in a circle. The wheel arms provide a centripetal (center seeking) force. sterry.eu FcFc

8. Centripetal Acceleration According to Newton’s 2 nd Law… “Acceleration is directly proportional to the net force and in the same direction as the net force…” The magnitude and direction of the resulting acceleration can be determined from a vector point of view. r r vivi vfvf

9. Centripetal Acceleration Consider motion at constant speed at a uniform distance r from the center of a circle. By definition the object is experiencing acceleration by virtue of the changing direction. r r vivi vfvf

10. Centripetal Acceleration r r vivi vfvf -v i vfvf ΔvΔv ΔvΔv

11. Centripetal Acceleration r r vivi vfvf -v i vfvf ΔvΔv ΔvΔv

12. Centripetal Acceleration The magnitude of the centripetal acceleration can be determined by considering similar triangles… r r vivi vfvf Δr -v i vfvf ΔvΔv

13. Centripetal Acceleration r r vivi vfvf -v i ΔvΔv Δr s

14. Centripetal Acceleration Considering a more instantaneous situation Δv becomes… Recall acceleration… Magnitude of centripetal acceleration… v

15. Centripetal Force According to Newton’s 2 nd Law… – “Acceleration is directly proportional to the net force and in the same direction as the net force…” – Meaning… if a centripetal acceleration is present it must result from a centripetal force!

16. Total Acceleration Both tangential (a t ) and centripetal acceleration (a c, or a r (radial)) contribute to the total acceleration experienced on a roller coaster. panda.unm.edu acac acac atat atat a total

17. Total Acceleration The magnitude and direction of the total acceleration can be determined by: panda.unm.edu acac acac atat atat a total

18. Lesson Summary Centripetal Acceleration – Center seeking Total Acceleration…

19. Example #1 Example 4.8 A ball tied to the end of a string 0.50 m in length swings in a vertical circle under the influence of gravity. When the string makes an angle θ = 20° with the vertical, the ball has a speed of 1.5 m/s. a.Find the magnitude of the radial component of acceleration at this instant. b.What is the magnitude of the tangential acceleration when θ = 20°? c.Find the magnitude and direction of the total acceleration at θ = 20°.

20. Example #2 Quick Quiz 4.3 a.Draw a motion diagram showing velocity and acceleration vectors for an object moving with constant speed counterclockwise around a circle. Draw similar diagrams for an object moving counterclockwise around a circle but b. slowing down at constant tangential acceleration and c.Speeding up at constant tangential acceleration.

21. Example #3 Quick Quiz 4.4 Draw a motion diagram showing the position, velocity, acceleration (total) vectors for a pendulum that, from an initial position of 45˚ to the right of vertical, swings in an arc to a final position of 45˚ to the left of vertical. (Use the center of the circle as the origin for position vectors.)