Second Law of Rotation

1. Second Law of Rotation • Newton’s First Law of Rotation • Newton’s Second Law of Rotation • Rotational Inertia / Moment of Inertia • Rotational Inertia of extended objects • Examples

2. Newton’s First Law of Rotation • Translational Equilibrium • Rotational Equilibrium • Both required for equilibrium (Ch 9) • Seesaw example

3. Newton’s Second Law of Rotation • Derive for one point: • Newton’s Second Law for translation • Newton’s Second Law for Rotation r F

4. Rotational 2nd Law for many points • For collection of objects (or extended object) • Internal torques redistribute to individual particles r1 m1 r2 m2

5. Rotational Inertia • “Moment of Inertia” or “Rotational Inertia” • Keeps cropping up in all equations! • Like “Rotational Mass” • Product of mass and how its distributed around axis • Units Kg m2 • Simple for point object, requires integral calculus for extended objects.

6. Rotational Inertia for Disk • For point or hollow ring • For disk or cylinder • Integrate from 0 to R • Put in density

7. Rotational Inertia for various objects • Don’t memorize – If you need these we’ll provide them

8. Summary - Rotational Inertia for objects • For a point object or a ring • all mass concentrated at one radius • For any other object • mass distributed at different radii • Don’t worry, we’ll give you the fraction

9. Example – Potter’s wheel • Torque • Angular acceleration • Time

10. Example – pulley and rope • Torque from tension • Total Torque • Angular acceleration If wheel was uniform disk • Moment of inertia

11. Example – pulley and bucket • Rotation 2nd Law for pulley • Translation 2nd Law for bucket (note sign change to sync direction) • Connection between translational and angular acceleration

12. pulley and bucket (cont) • Rotation and Translation 2nd Law • Eliminating FT • Rearranging • Solving for α

13. pulley and bucket (cont) • Putting in numbers • Acceleration of bucket

14. Review - Atwood’s Machine • Example 4-13 • Free body diagram for each block • Tensions same by 3rd law • Accelerations locked together (- sign) • Equation of motion for each • Solve for acceleration • Solve for tension • But what about Pulley?

15. Atwood’s Machine with Pulley • Tensions can now be different • 2nd Law for m1 • 2nd Law for m2 (reverse direction) • 2nd Law for pulley • Linear acceleration of blocks and angular acceleration of pulley synchronized!

16. Atwood’s Machine with Pulley (2) • Substitute T2 and T1 in pulley equation • Write a in terms of α • Rearrange

17. Atwood’s Machine with Pulley (3) • Angular acceleration of pulley • Linear acceleration of blocks • Note if I = 0 (same as before!)

18. Atwood’s Machine with Pulley (4) • Velocity after blocks have moved distance h (we’ll need this when we do it by energy)

19. Rotational energy contribution • Energy • Just gets added in with all other energy! • E = ½ m1 v12 + ½ m1 v12 • E = ½ m1 r12 ω2 + ½ m2 r22 ω2 • E = ½ I ω2 • Etot = ½ mv2 + ½ I ω2 r1 m1 r2 m2

20. Rotation – Energy - Examples • Sphere down incline (8-13) • mgh = ½ mv2 + ½ Iω2, ω = v/R, I = 2/5 MR2 • v = sqrt(10/7gh) • Demonstrate ramp • Atwood’s machine • Ei = m2gh • Ef = m1gh + ½ m1v2 + ½ m2v2 + ½ Iω2 • v2 = 2gh(m2 – m1)g / (I/r2 + m1 + m2) • Compare with 2nd law derivation