2. object moves in a circular path about an external point (“revolves”)

3. According to Newton’s First Law of Motion, objects move in a straight line at a constant speed unless a net force makes them change speed and/or direction. An external force is therefore necessary to make an object follow a circular path. This force is called a CENTRIPETAL (“center seeking”) FORCE.

4. Since every unbalanced force causes an object to accelerate in the direction of that force (Newton’s Second Law), a centripetal force causes a CENTRIPETAL ACCELERATION. This acceleration results from a change in direction, and does not imply a change in speed, although speed may also change.

5. Centripetal force and acceleration may be caused by: gravity – planets orbiting the sungravity – planets orbiting the sun friction – car rounding a curvefriction – car rounding a curve a rope or cord – swinging a mass on a stringa rope or cord – swinging a mass on a string r m In all cases, a mass m moves in a circular path of radius r with a linear speed v. The time to make one complete revolution is known as the period, T. v The linear speed v is the circumference divided by the period. v = 2  r/T

6. The formula for centripetal acceleration is: ac = v2/rac = v2/rac = v2/rac = v2/r and centripetal force is: Fc = mac = v2/rFc = mac = mv2/rFc = mac = v2/rFc = mac = mv2/r m = mass in kg v = linear velocity in m/s F c = centripetal force in N r = radius of curvature in m a c = centripetal acceleration in m/s 2

7. Go to the computer simulation found at http://www.mhhe.com/physsci/physical/giambatt ista/banked_curve/banked_curve.html http://www.mhhe.com/physsci/physical/giambatt ista/banked_curve/banked_curve.html http://www.mhhe.com/physsci/physical/giambatt ista/banked_curve/banked_curve.html to investigate the motion of a car rounding a banked curve.

8. It is common to have students perform lab activities in which they investigate centripetal force as a function of radius, mass, and linear velocity. My experience has been that timing and measurement errors, along with friction in the experimental apparatus, lead to poor results for most student groups. See the following links for descriptions of these types of lab activities: http://www.batesville.k12.in.us/physics/phynet/mechanics/Circ ular%20Motion/labs/cf_and_speed.htm http://www.batesville.k12.in.us/physics/phynet/mechanics/Circ ular%20Motion/labs/cf_and_speed.htm http://www.sethi.org/classes/elabs/lab_04.htmlhttp://www.sethi.org/classes/elabs/lab_04.html (simulated lab activity) http://prettygoodphysics.wikispaces.com/file/view/Centripetal+ Force+Lab.pdf

9. object moves in circular path about an internal point or axis (“rotates” or “spins”)

10. The amount that an object rotates is its angular displacement. angular displacement, , is given in degrees, radians, or rotations. 1 rotation = 360 deg = 2  radians The time rate change of an object’s angular displacement is its angular velocity. angular velocity, , is given in deg/s, rad/s, rpm, etc...

11. The time rate change of an object’s angular velocity is its angular acceleration. angular acceleration, , is given in deg/s 2, rad/s 2, rpm/s, etc... Formulas for rotational motion follow an exact parallel with linear motion formulas. The only difference is a change in variables and a slight change in their meanings.

12. Constant LINEAR v f = v i + at d = v av t v av = (v f + v i )/2 d = v i t + 0.5at 2 v f 2 = v i 2 + 2ad ROTATIONAL  f  =  i +  t  =  av t ω av =  (  f  +  i )/2  =  i t  +  0.5  t 2  f 2  =  i 2  +  2 

13. Just as a net force causes an object to undergo linear acceleration, a net torque causes an object to undergo a rotational (angular) acceleration. Torque is defined to be the product of the force and the perpendicular distance from the force to the rotational axis or point (lever arm or moment arm). More precisely, torque is a vector that is the cross (vector) product of the moment arm (r) and the force (F) vectors.

14. The magnitude of the torque is found by multiplying the force, the perpendicular distance of the force from the pivot point or axis, and the sine of the angle between these two. Torque = rFsin  The official direction of the torque is given by a “right hand rule,” but we can sometimes specify the direction as clockwise or counterclockwise. Learn more about torque at these and other web sites: http://hyperphysics.phy-astr.gsu.edu/hbase/torq2.html http://en.wikipedia.org/wiki/Torque http://www.comfsm.fm/~dleeling/physics/torque.html http://www.physics.uoguelph.ca/tutorials/torque/index.html

15. Rotational Inertia : the tendency of an object to resist any changes in its rotational state Rotational Inertia depends on the mass of the object and the distribution of the mass (shape of object) Newton’s 1 st Law of Motion: Restated for Rotational Motion The rotational velocity (angular speed and angular direction) of an object remains constant unless acted on by an unbalanced torque.

16. Rotational Inertia Values (“Moments of Inertia”) figure copied from http://hyperphysics.phy-astr.gsu.edu/hbase/mi.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/mi.html

17. Newton’s 2 nd Law of Motion: Restated for Rotational Motion A net torque causes an object to have an angular acceleration in the same direction as the net torque. The angular acceleration is directly proportional to the net torque and inversely proportional to the object’s rotational inertia.

18. Angular (Rotational) Momentum: the product of rotational inertia and angular velocity The conservation of angular momentum leads to some incredibly interesting phenomena, as changes to the shape of the rotating object leads to changes in its rotational inertia, which will result in changes to its rotational velocity so that angular momentum is conserved.

19. Angular (Rotational) Kinetic Energy: the product of ½ the rotational inertia and angular velocity squared An object that rolls down an incline has gravitational potential energy (mgh) converted into both linear kinetic energy (0.5mv 2 ) and rotational kinetic energy (0.5 I  2 ).

20. Investigate circular and rotational motion using the PhET simulation found at http://phet.colorado.edu/en/simulation/rotation. http://phet.colorado.edu/en/simulation/rotation