1 8 Rotational Motion describing rotation & “2 nd Law” for rotation conserved rotations center of mass & stability Homework: RQ: 1,2, 6, 7, 13, 14, 16, 18, 20, 23, 32. Ex: 22, 40, 47. Problem: 3.

2 Describing Rotation speed on rotating object ~ distance from center (v ~ r) however, all points on object have same #rotations/second (v/r is same) rotational speed = v/r “direction”: clockwise or counter-clockwise

3 torque torque = force x lever-arm lever-arm determined by force-line as shown below

4 torque examples large torque: lever-arms = r zero torque: lever-arms = 0

5 rotational inertia translational inertia depends only on the mass of the object rotational inertia depends on mass, shape, and size. bigger objects tend to have higher rotational inertias e.g. tightwalker uses long pole for balance.

6 high rotational inertia improves stability of rotating objects Examples: turntable platter, bicycle wheel, frisbee

7 Example Torque Meter stick balanced from 50cm Hang 200grams from 30cm Lever-arm = 20cm ?Torque? Force x lever-arm = weight of 200 grams x 20cm ~ 200grams x 20cm = 4000gram-cm Where can we put 100grams so that it balances the meter stick?

8 Pract. Phys. p.42 #2. 2a) twice the lever-arm, W = 250N CCW-torque = 250Nx4m = 1000mN CW-torque = 500Nx2m = 1000mN Rotational Equilibrium is state when CCW and CW torques are equal. We also say that the “sum of the torques” is zero for Rot. Eq.

9 Cont. 2b) 4/3 weight ratio, what is lever-arm? CCW-torque = 300Nx4m=1200Nm CW-torque = 400Nx??m = 1200Nm => 3m balances the see-saw.

10 Cont. 2c) weight of board balances child. CCW-torque = 600Nx1m = 600Nm CW-torque = ??Nx3m = 600Nm ??N = 600Nm/3m = 200N

11 angular momentum linear momentum: (linear inertia) x (linear velocity) = mv angular momentum: (rotational inertia) x (rotational velocity) conserved when net external torque is zero

12 isolated rotation (rotat. inertia) x (rotat. velocity) = constant skater, diver, gymnast “tuck” to spin faster, “open” to spin slower

13 center of mass average location of object’s mass at center of symmetrical objects (hoop, ball, rod) affects stability (rollover risk ~ height of center of mass)

14 center of mass closer to more massive object equal to “center of gravity” (for small system)

15 stability tendency to maintain position center of mass must be above base

16 wine rack where is the center of mass?

17 equilibrium condition of F net = 0 & net torque = 0 stable objects are in equilibrium not all equilibriums are stable

18 centripetal force force required to maintain curved motion points toward center of curvature

19 centrifugal force (p44 #1, #3) not a force on object opposite of centripetal apparent force due to inertia e.g. ball (or water) in a spinning bucket “feels” a force holding it to the bottom.

20 Practicing Physics P46 circular motion P48 rotational dynamics

21 Summary angular velocity = v/r torque = force x lever-arm rotational inertia = ‘rotational mass’ angular momentum of isolated systems is conserved definitions: center of mass, stability, equilibrium centripetal & centrifugal forces