2. NAZARIN B. NORDIN nazarin@icam.edu.my

3. What you will learn: Centripetal force: acceleration, centrifugal force/ acceleration, mass-radius polygons Centrifugal force applied to wheel balancing/ clutches, governors Curved tracks: vehicles overturning/ sliding on level track, vehicles on banked track

4. Angular displacement (  ) is usually expressed in radians, in degrees, or in revolutions. ANGULAR DISPLACEMENT

5. 57.3 0 One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. 1 rev = 360 0 = 2  radians (rad) 1 2 3 4 5 6 6 segments gets to here. 2  segments gets completely around.

6. Thus the angle  in radians is given in terms of the arc length l it subtends on a circle of radius r by The radian measure of an angle is a dimensionless number.

7. THE ANGULAR SPEED The angular speed (  ) of an object whose axis of rotation is fixed is the rate at which its angular coordinate, the angular displacement , changes with time. If  changes from  i to  f in a time t, then the average angular speed is

8.  = 2  f. f is the frequency in revolutions per second, rotations per second, or cycles per second. Accordingly,  is called the angular frequency. We can associate a direction with  and thereby create a vector quantity. The units of  are exclusively rad/s. Since each complete turn or cycle of a revolving system carries it through 2  rad

9. THE ANGULAR ACCELERATION The angular acceleration (  ) of an object whose axis of rotation is fixed is the rate at which its angular speed changes with time. If the angular speed changes uniformly from  i to  f in the time t, then the angular acceleration is constant and

10. The units of  are typically rad/s 2, rev/min 2, and such. It is possible to associate a direction with , and therefore with , thereby specifying the angular acceleration vector , but we will have no need to do so here.

11. Equations for uniformly accelerated angular motion are exactly analogous to those for uniformly accelerated linear motion. In the usual notation we have:

12. RELATIONS BETWEEN ANGULAR AND TANGENTIAL QUANTITIES: When a wheel of radius r rotates about an axis whose direction is fixed, a point on the rim of the wheel is described in terms of the circumferential distance l it has moved, its tangential speed v, and its tangential acceleration a T. These quantities are related to the angular quantities , , and , which describe the rotation of the wheel, through the relations:

13. provided radian measure is used for , , and . By simple reasoning, l can be shown to be the length of belt wound on the wheel or the distance the wheel would roll (without slipping) if free to do so. In such cases, v and a T refer to the tangential speed and acceleration of a point on the belt or of the center of the wheel.

14. An object moving in a circle with constant speed, v, experiences a centripetal acceleration with: *a magnitude that is constant in time and is equal to *a direction that changes continuously in time and always points toward the center of the circular path Uniform Circular Motion For uniform circular motion, the velocity is tangential to the circle and perpendicular to the acceleration

15. A circular motion is described in terms of the period T, which is the time for an object to complete one revolution. Period and Frequency The distance traveled in one revolution is The frequency, f, counts the number of revolutions per unit time. r

16. The moon’s nearly circular orbit about the earth has a radius of about 384,000 km and a period T of 27.3 days. Determine the acceleration of the Moon towards the Earth. Example of Uniform Circular Motion

17. Uniform Circular Motion : Newton’s 2nd Law: The net force on a body is equal to the product of the mass of the body and the acceleration of the body. * The centripetal acceleration is caused by a centripetal force that is directed towards the center of the circle.

18. ROTATIONAL INERTIA Law of inertia for rotating systems An object rotating about an axis tends to remain rotating at the same rate about the same axis unless interfered with by some external influence. Examples: bullet, arrow, and earth Demo – Football and spinning basketballDemo – Football and spinning basketball Demo - Whirly Tube (Zinger)Demo - Whirly Tube (Zinger) Demo – Whirly ShooterDemo – Whirly Shooter Demo - Disc GunDemo - Disc Gun Demo - Rubber BandsDemo - Rubber Bands

19. Demo - Inertia Bars Moment of inertia (rotational inertia) The sluggishness of an object to changes in its state of rotational motion Distribution of mass is the key. Example: Tightrope walker

20. CENTRIPETAL ACCELERATION Centripetal acceleration (a c ): A point mass m moving with constant speed v around a circle of radius r is undergoing acceleration. The direction of the velocity is continually changing. This gives rise to an acceleration a c of the mass, directed toward the center of the circle. We call this acceleration the centripetal acceleration; its magnitude is given by

21. Because v = r , we also have where  must be in rad/s.

22. THE CENTRIPETAL FORCE The centripetal force (F c ) is the force that must act on a mass m moving in a circular path of radius r to give it the centripetal acceleration v 2 /r. From F = ma, we have Where F c is directed toward the center of the circular path.

23. Centripetal force - center seeking force Examples: tin can and string, sling, moon and earth, car on circular path CENTRIPETAL FORCE Demo - Coin on clothes hangerDemo - Coin on clothes hanger Demo - String, ball, and tubeDemo - String, ball, and tube Demo - Loop the loopDemo - Loop the loop

24. CENTRIFUGAL FORCE Centrifugal force - center fleeing force Often confused with centripetal Examples: sling and bug in can Demo - Walk the Line Centrifugal force is attributed to inertia.

25. CENTRIFUGAL FORCE IN A ROTATING REFERENCE FRAME A frame of reference can influence our view of nature. For example: we observe a centrifugal force in a rotating frame of reference, yet it is a fictitious (pseudo) force. Centrifugal force stands alone (there is no action-reaction pair) - it is a fictitious force.

26. Another pseudo force - CoriolisCoriolis