1. Lecture 15: Rotational Motion

2. Questions of Yesterday 1) A piece of clay traveling north with speed v collides perfectly inelastically with an identical piece of clay traveling east with speed v. What direction does the resultant piece of clay travel? a) north b) east c) 45 o N of E d) 45 o S of W 2) Ball 1 of mass m, traveling with speed v, collides with Ball 2 of mass 2m and comes to rest, what is the speed of Ball 2 after the collision? a) 2v b) v c) v/2 d) v/(√2)

3. Linear Motion  x = x f - x i Displacement: xtxt v = Velocity:Acceleration: vtvt a = Constant a Equations: v = v 0 + at  x = v 0 t + 1/2at 2 v 2 = v 0 2 + 2a  x  F = ma Force (2nd Law): p = m*v Momentum:  p = F  t Impulse: Equations/Concepts valid for straight line motion between points in space (x-y plane)

4. Circular Motion Circumference = 2  r How do you define “position” and “displacement” when motion is circular? r  s s = r*  Arc length: Angle Unit = Radian 2  Radians = 360 o  = 0  =  / 2  = 2   =   = 3  / 2 srsr  = Angular Position:

5. Circular Motion r ii titi srsr  = Angular Position: ff tftf  =  f -  i Angular Displacement:  av  =  f -  i t f - t i   t = Average Angular Velocity: SI Units: Radians (rad) SI Units: Radians per second (rad/s) lim  t -> 0  =   t Instantaneous Angular Velocity

6. Circular Motion r ii titi ff tftf  av  =  f -  i t f - t i   t = Average Angular Acceleration: SI Units: Radians per second squared (rad/s 2 ) lim  t -> 0  =   t Instantaneous Angular Acceleration

7. Constant Angular Acceleration Linear Motion with Constant a: v = v 0 + at  x = v 0 t + 1/2at 2 v 2 = v 0 2 + 2a  x  =  0 +  t Rotational Motion with Constant  :  =  0 t + 1/2  t 2  2 =  0 2 + 2 

8. Rotational Motion Which position has a greater angular displacement in a given time interval? What about angular speed? Angular acceleration? 

9. Rotational Motion Which position has a greater angular displacement in a given time interval? What about angular speed? Angular acceleration? 

10. Angular and Linear Quantities r titi  tftf srsr = Displacement: Direction of linear velocity v of an object moving in a circular path is always TANGENT to the path ss vT=rvT=r Tangential Speed: aT=raT=r Tangential Acceleration:

11. Centripetal Acceleration r  If you’re jogging on a circular track with constant tangential speed is your acceleration ZERO? Why or Why not? vfvf v f - v i t f - t i a av = vivi During circular motion at constant speed your direction is constantly changing so you still have an acceleration CENTRIPETAL ACCELERATION Acceleration associated with constant speed circular motion

12. Centripetal Acceleration r  vfvf Centripetal Acceleration always points towards the CENTER of the circle v f - v i t f - t i a av = vivi -vi-vi vfvf vv 

13. Centripetal Acceleration Centripetal Acceleration always points towards the CENTER of the circle r  vfvf vivi -vi-vi vfvf vv r srsr vvvv = vtvt a av = v2rv2r ac=ac= =r2=r2 Similar Triangles ss

14. Centripetal Acceleration What if your tangential speed is NOT constant? r  vfvf vivi -vi-vi vfvf vv r Acceleration has both tangential and centripetal components! a = (a c 2 + a T 2 ) 1/2 vv vcvc vTvT v2rv2r ac=ac= aT=raT=r

15. Rotational Motion: Practice Problem A race car starts from rest on a circular track of radius 400 m. The car’s speed increases at the constant rate of 0.500 m/s 2. At the point where the magnitudes of the centripetal and tangential accelerations are equal, what is… the tangential speed of the car? the angular speed of the car? the distance traveled? the number of revolutions made? the elapsed time?

16. Centripetal Force  F = ma If an object is accelerating what do know about it (think Newton’s 2nd law)? Can an object be moving in a circular path if no forces are acting on? If an object is undergoing constant speed circular motion what direction is the net force acting on the object? mv 2 r  F c = ma c =

17. Centripetal Force  F T = ma T What if an object undergoing circular motion and changing its tangential speed? mv 2 r  F c = ma c = -vi-vi vfvf vv  a acac aTaT FF FTFT FCFC  F = ma Just like linear motion (∑F x = ma x, ∑F y = ma y )… must split vector equation into perpendicular components!!

18. Centripetal Force As you round the bend at constant speed in what direction.. is your net acceleration? Why? Is your net force? Why? do you feel yourself being pulled? Why? Remember Newton’s 1st law?? What force is acting on you and your car to let you round the bend?

19. Centripetal Force Remember Newton’s 1st law?? What force is acting on you and your car to let you round the bend? N FgFg f As you round the bend at constant speed in what direction.. is your net acceleration? Why? Is your net force? Why? do you feel yourself being pulled? Why?

20. Practice Problem Suppose that a 1800-kg car passes over a bump in a roadway that follows the arc of circle of radius 20.0 m. What force does the road exert on the car as the car passes the highest point of the bump if the car travels at 9.00 m/s? What is the maximum speed the car can have without losing contact with the road as it passes this highest point?

21. Questions of the Day 1) You are going through a vertical loop on roller coaster at a constant speed. At what point is the force exerted by the tracks on you (and the cart you are in) the greatest? a) at the highest point b) at the lowest point c) halfway between the highest and lowest point d) the force is equal over the whole loop 2) You are on a merry-go-round moving at constant speed. If you move to the outer edge of the merry-go-round, what happens to the net centripetal force keeping you on the merry-go-round? a) it increases b) it decreases c) it stays the same d) there is no net centripetal force acting on you