Set 4 Circles and Newton

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 Dv a v2 v2 v1

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. s q Quick Review - Radians

7. Changing Velocity in Uniform Circular Motion • The change in the velocity vector is due to the change in direction • The vector diagram shows Dv = vf- vi

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 • q is tangent to the circle • The total acceleration is

16. r r v 12 2 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.

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