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Rotational Equilibrium and Dynamics: Torque and Moment of Inertia

This chapter explores the concepts of torque, equilibrium, and rotational dynamics, including the calculation of torque, moments of inertia, and rotational kinetic energy. It also provides examples and demonstrations to illustrate these concepts.

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Rotational Equilibrium and Dynamics: Torque and Moment of Inertia

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  1. Chapter 8 Rotational Equilibrium and Rotational Dynamics

  2. Wrench Demo

  3. Torque • Torque, t , is the tendency of a force to rotate an object about some axis • F is the force • d is the lever arm (or moment arm) • Units are Newton-m

  4. Direction of Torque • Torque is a vector quantity • Direction determined by axis of twist • Perpendicular to both r and F • In 2-d, torque points into or out of paper • Clockwise torques point into paper. Defined as negative • Counter-clockwise torques point out of paper. Defined as positive

  5. Non-perpendicular forces • Only the y-component, Fsinf,produces a torque

  6. Non-perpendicular forces • F is the force • r is distance to point where F is applied • Φ is the angle between F and r

  7. Torque and Equilibrium • Forces sum to zero (no linear motion) • Torques sum to zero (no rotation)

  8. Meter Stick Demo

  9. Axis of Rotation • Torques require point of reference • Point of reference can be anywhere • Use same point for all torques • Pick the point to make problem least difficult

  10. Example Given M = 120 kg. Neglect the mass of the beam. a) Find the tensionin the cable b) What is the force between the beam andthe wall

  11. Solution a) Given: M = 120 kg, L = 10 m, x = 7 m Find: T Axis Solve for T = 824 N

  12. Solution b) Given: M = 120 kg, L = 10 m, x = 7 m, T = 824 N Find f f Solve for f = 353 N

  13. Alternative Solution b) Given: M = 120 kg, L = 10 m, x = 7 m Find f Axis f Solve for f = 353 N

  14. Another Example Given: W=50 N, L=0.35 m, x=0.03 m Find the tension in the muscle W x L F = 583 N

  15. Center of Gravity • Gravitational force acts on all points of an extended object • However, it can be considered as one net force acting on one point,the center-of-gravity, X. WeightedAverage

  16. Example Consider the 400-kgbeam shown below.Find TR

  17. Solution Find TR to solve for TR, choose axis here X = 2 m L = 7 m TR = 1 121 N

  18. One Last Example Given: x = 1.5 m, L = 5.0 m, wbeam = 300 N, wman = 600 N Find: T x L

  19. Solution Given: x , L , wbeam ,wman Find: T First, find Ty Ty = 330 N Now, find T x L T = 413 N

  20. Baton DemoMoment-of_Inertia Demo

  21. Torque and Angular Acceleration • When t  0, rigid body experiences angular acceleration • Relation between t and a is analagous to relation between F and a Moment of Inertia

  22. Moment of Inertia • This mass analog is called the moment of inertia, I, of the object • r is defined relative to rotation axis • SI units are kg m2

  23. More About Moment of Inertia • I depends on both the mass and its distribution. • If an object’s mass is distributed further from the axis of rotation, the moment of inertia will be larger.

  24. Demo: Moment of Inertia Olympics

  25. Moment of Inertia of a Uniform Ring • Divide ring into segments • The radius of each segment is R

  26. Example What is the moment of inertia of the following point masses arranged in a square? a) about the x-axis? b) about the y-axis? c) about the z-axis?

  27. Solution a) Find I about the x-axis? Given: M2=2, M3=3 kg, L=0.6 m r = 0.6·sin(45) First, find distance to 2-kg masses =0.72 kg·m2

  28. Solution b) Find I about the y-axis? r = 0.6·sin(45) Same as before, except you use the 3-kg masses =1.08 kg·m2

  29. Solution c) Find I about the z-axis? r = 0.6·sin(45) Use all the masses =1.8 kg·m2

  30. Other Moments of Inertia

  31. Example Treat the spindle as a solid cylinder. a) What is the moment of Inertia of the spindle? b) If the tension in the rope is 10 N, what is the angular acceleration of the wheel? c) What is the acceleration of the bucket? d) What is the mass of the bucket? M

  32. Solution a) What is the moment of Inertia of the spindle? Given: M = 5 kg, R = 0.6 m M = 0.9 kgm2

  33. Solution b) If the tension in the rope is 10 N, what is a? Given: I = 0.9 kg m2, T = 10 N, r = 0.6 m Solve for a a=6.67 rad/s2 c) What is the acceleration of the bucket? Given: r=0.6 m, a = 6.67 rad/s M a=4 m/s2

  34. Solution d) What is the mass of the bucket? Given: T = 10 N, a = 4 m/s2 M = 1.72 kg M

  35. Example A cylindrical space station of radius R = 12 m and mass M = 3400 kg is designed to rotate around the axis of the cylinder. The space station’s moment of inertia is 0.75 MR2. Retro-rockets are fired tangentially to the surface of space station and provide an impulse of 2.9x104 N·s. a) What is the angular velocity of the space station after the rockets have finished firing? b) What is the centripetal acceleration at the edge of the space station?

  36. Solution Given: M = 3400, R = 12, I = 0.75 MR2 = 3.67x105 F·t = 2.9x104. a) Find: w Solve for w. w= 0.948 rad/s b) Find centripetal acceleration = 10.8 m/s2

  37. Rotational Kinetic Energy Each point of a rigid body rotates with angular velocity w. Including the linear motion KE due to rotation KE of center-of-mass motion

  38. Example What is the kinetic energy of the Earth due to the daily rotation? Given: Mearth=5.98 x1024 kg, Rearth = 6.63 x106 m. First, find w = 7.27 x10-5 rad/s = 2.78 x10-29

  39. Example A solid sphere rolls down a hill of height 40 m. What is the velocity of the ball when it reaches the bottom? (Note: We don’t know r or m!) m cancels v = 23.7 m/s

  40. Suitcase Demo

  41. Angular Momentum Rigid body Point particle Analogy between L and p

  42. Rotating Chair Demo

  43. Angular Momentum and Kepler’s 2nd Law • For central forces, e.g. gravity, t = 0and L is conserved. • Change in area in Dt is:

  44. Example A 65-kg student sprints at 8.0 m/s and leaps onto a 110-kg merry-go-round of radius 1.6 m. Treating the merry-go-round as a uniform cylinder, find the resulting angular velocity. Assume the student lands on the merry-go-round while moving tangentially.

  45. Solution Known: M, R, m, v0 Find: wF First, find L0 Next, find Itot Now, given Itot and L0, find w = 2.71 rad/s

  46. Example Two twin ice skaters separated by 10 meters skate without friction in a circle by holding onto opposite ends of a rope. They move around a circle once every five seconds. By reeling in the rope, they approach each other until they are separated by 2 meters. a) What is the period of the new motion? b) If each skater had a mass of 75 kg, what is the work done by the skaters in pulling closer?

  47. Solution a) Find: Tf Given: R0=5 m, Rf=1 m, T0=5 s First, find expression for initial L (2 skaters) Next, apply conservation of L = T0/25 = 0.2 s

  48. Solution b) Find: W Given: m=75 kg, R0=5 m, Rf=1 m, T0=5 s, Tf=0.2 s First, find moments of inertia Next, find values for w Finally, find change in KE = 7.11x105 J

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