1. Set 4 Circles and Newton February 3, 2006

2. Where Are We Today –Quick review of the examination – we finish one topic from the last chapter – circular motion We then move on to Newton’s Laws New WebAssign on board on today’s lecture material –Assignment – Read the circular motion stuff and begin reading Newton’s Laws of Motion Next week –Continue Newton –Quiz on Friday Remember our deal!

3. Remember from the past … Velocity is a vector with magnitude and direction. We can change the velocity in three ways –increase the magnitude –change the direction –or both If any of the components of v change then there is an acceleration.

4. Changing Velocity v1v1 v2v2 v2v2 vv a

5. Uniform Circular Motion Uniform circular motion occurs when an object moves in a circular path with a constant speed An acceleration exists since the direction of the motion is changing –This change in velocity is related to an acceleration The velocity vector is always tangent to the path of the object

6. Quick Review - Radians  s

7. Changing Velocity in Uniform Circular Motion The change in the velocity vector is due to the change in direction The vector diagram shows v = v f - v i

8. The acceleration Centripetal Acceleration

9. Centripetal Acceleration The acceleration is always perpendicular to the path of the motion The acceleration always points toward the center of the circle of motion This acceleration is called the centripetal acceleration

10. Centripetal Acceleration, cont The magnitude of the centripetal acceleration vector was shown to be The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion

11. Period The period, T, is the time required for one complete revolution The speed of the particle would be the circumference of the circle of motion divided by the period Therefore, the period is

12. Tangential Acceleration The magnitude of the velocity could also be changing In this case, there would be a tangential acceleration

13. Total Acceleration The tangential acceleration causes the change in the speed of the particle The radial acceleration comes from a change in the direction of the velocity vector

14. Total Acceleration, equations The tangential acceleration: The radial acceleration: The total acceleration: –Magnitude

15. Total Acceleration, In Terms of Unit Vectors Define the following unit vectors –r lies along the radius vector –  is tangent to the circle The total acceleration is

16. A ball on the end of a string is whirled around in a horizontal circle of radius 0.300 m. The plane of the circle is 1.20 m above the ground. The string breaks and the ball lands 2.00 m (horizontally) away from the point on the ground directly beneath the ball's location when the string breaks. Find the radial acceleration of the ball during its circular motion. 12 2 r r v

17. A pendulum with a cord of length r = 1.00 m swings in a vertical plane (Fig. P4.53). When the pendulum is in the two horizontal positions = 90.0° and = 270°, its speed is 5.00 m/s. (a) Find the magnitude of the radial acceleration and tangential