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Rotational Kinematics

Rotational Kinematics. In rotational motion, position is represented by an angle, such as q , and a radius, r. x = 3. p /2. r. q. p. 0. angular. 3p /2. Position. In translational motion, position is represented by a point, such as x. x. 0. 5. linear.

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Rotational Kinematics

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  1. Rotational Kinematics

  2. In rotational motion, position is represented by an angle, such as q, and a radius, r. x = 3 p/2 r q p 0 angular 3p/2 Position • In translational motion, position is represented by a point, such as x. x 0 5 linear

  3. Angular displacement is represented by Dq, which is not a vector, but behaves like one for small values. Counterclockwise is considered to be positive. Dx = - 4 p/2 Dq p 0 angular 3p/2 Displacement • Linear displacement is represented by the vector Dx. x 0 5 linear

  4. This distance s is related to the angular displacement Dq by the equation s = rDq. s Dq Tangential and angular displacement • A particle that rotates through an angle Dq also translates through a distance s, which is the length of the arc defining its path. r

  5. The instantaneous velocity has magnitude vT = ds/dt and is tangent to the circle. The same particle rotates with an angular velocity w = dq/dt. The direction of the angular velocity is given by the right hand rule. Tangential and angular speeds are related by the equation v = r w. vT s Dq vT Speed and velocity r w is outward according to RHR

  6. Tangential acceleration is given by aT = dvT/dt. This acceleration is parallel or anti-parallel to the velocity. Angular acceleration of this particle is given by a = dw/dt. Angular acceleration is parallel or anti-parallel to the angular velocity. Tangential and angular accelerations are related by the equation a = r a. s Dq Acceleration vT r Dq vT w is outward according to RHR Don’t forget centripetal acceleration. a = at2+ac2

  7. What is the direction of the instantaneous velocity, v? What is the direction of the angular velocity, w? What is the direction of the tangential acceleration, aT? What is the direction of the angular acceleration a? What is the direction of the centripetal acceleration, ac? What is the direction of the overall acceleration, a, of the particle? Problem: Assume the particle is speeding up. What changes if the particle is slowing down?

  8. First Kinematic Equation • v = vo + at (linear form) • Substitute angular velocity for velocity. • Substitute angular acceleration for acceleration. •  = o + t (angular form)

  9. Second Kinematic Equation • x = xo + vot + ½ at2(linear form) • Substitute angle for position. • Substitute angular velocity for velocity. • Substitute angular acceleration for acceleration. • q = qo + ot + ½ t2(angular form)

  10. Third Kinematic Equation • v2 = vo2 + 2a(x - xo) • Substitute angle for position. • Substitute angular velocity for velocity. • Substitute angular acceleration for acceleration. • 2 = o2 + 2(q - qo)

  11. Practice problem The Beatle’s White Album is spinning at 33 1/3 rpm when the power is turned off. If it takes 1/2 minute for the album’s rotation to stop, what is the angular acceleration of the phonograph album? (-0.12 rad/s2) Hwk: Chpt 10 # 1-4, 9,10,13,16,17

  12. Rotational Energetics

  13. Inertia and Rotational Inertia • In linear motion, inertia is equivalent to mass. • Rotating systems have “rotational inertia”. • I = mr2 (for a system of particles) • I: rotational inertia (kg m2) • m: mass (kg) • r: radius of rotation (m) • Solid objects are more complicated; we’ll get to those later. See page 278 for a “cheat sheet”.

  14. Sample Problem • A 2.0-kg mass and a 3.0-kg mass are mounted on opposite ends a 2.0-m long rod of negligible mass. What is the rotational inertia about the center of the rod and about each mass, assuming the axes of rotation are perpendicular to the rod?

  15. Kinetic Energy • Bodies moving in a straight line have translational kinetic energy • Ktrans = ½ m v2. • Bodies that are rotating have rotational kinetic energy • Krot = ½ I w2 • It is possible to have both forms at once. • Ktot = ½ m v2 + ½ I 2

  16. Practice problem A 3.0 m long lightweight rod has a 1.0 kg mass attached to one end, and a 1.5 kg mass attached to the other. If the rod is spinning at 20 rpm about its midpoint around an axis that is perpendicular to the rod, what is the resulting rotational kinetic energy? Ignore the mass of the rod.

  17. Rotational Inertia

  18. Rotational Inertia Calculations • I = mr2 (for a system of particles) • I =  dm r2 (for a solid object) • I = Icm + m h2 (parallel axis theorem) • I: rotational inertia about center of mass • m: mass of body • h: distance between axis in question and axis through center of mass

  19. Practice problem A solid ball of mass 300 grams and diameter 80 cm is thrown at 28 m/s. As it travels through the air, it spins with an angular speed of 110 rad/second. What is its • translational kinetic energy? • rotational kinetic energy? • total kinetic energy?

  20. Practice Problem Derive the rotational inertia of a long thin rod of length L and mass M about a point 1/3 from one end • using integration of I =  r2 dm • using the parallel axis theorem and the rotational inertia of a rod about the center.

  21. Practice Problem Derive the rotational inertia of a ring of mass M and radius R about the center using the formula I =  r2 dm.

  22. Torque and Angular Acceleration I

  23. Equilibrium • Equilibrium occurs when there is no net force and no net torque on a system. • Static equilibrium occurs when nothing in the system is moving or rotating in your reference frame. • Dynamic equilibrium occurs when the system is translating at constant velocity and/or rotating at constant rotational velocity. • Conditions for equilibrium: • St = 0 • SF = 0

  24. Hinge (rotates) r Direction of rotation F Torque Torque is the rotational analog of force that causes rotation to begin. Consider a force F on the beam that is applied a distance r from the hinge on a beam. (Define r as a vector having its tail on the hinge and its head at the point of application of the force.) Arotation occurs due to the combination of r and F. In this case, the direction is clockwise. What do you think is the direction of the torque? Direction of torque is INTO THE SCREEN.

  25. Calculating Torque • The magnitude of the torque is proportional to that of the force and moment arm, and torque is at right angles to plane established by the force and moment arm vectors. What does that sound like? •  = r  F •  : torque • r: moment arm (from point of rotation to point of application of force) • F: force

  26. Practice Problem F What must F be to achieve equilibrium? Assume there is no friction on the pulley axle. 3 cm 2 cm 2 kg 10 kg

  27. Torque and Newton’s 2nd Law • Rewrite SF = ma for rotating systems • Substitute torque for force. • Substitute rotational inertia for mass. • Substitute angular acceleration for acceleration. • S = I  • : torque • I: rotational inertia • : angular acceleration

  28. Practice Problem A 1.0-kg wheel of 25-cm radius is at rest on a fixed axis. A force of 0.45 N is applied tangent to the rim of the wheel for 5 seconds. • After this time, what is the angular velocity of the wheel? • Through what angle does the wheel rotate during this 5 second period?

  29. Rotational Dynamics Workshop

  30. Sample problem • Derive an expression for the acceleration of a flat disk of mass M and radius R that rolls without slipping down a ramp of angle q.

  31. Practice problem Calculate initial angular acceleration of rod of mass M and length L. Calculate initial acceleration of end of rod.

  32. Sample problem Calculate acceleration. Assume pulley has mass M, radius R, and is a uniform disk. m2 m1

  33. Rotational Dynamics Lab Tuesday, December 2, 2008 Wednesday, December 3, 2008

  34. Announcements • Pass forward these assignments: • Exam repair must be done this week by Thursday. You must do repairs in the morning (7:00 AM) or lunch. One shot only, so plan on staying the whole hour.

  35. Demonstration • A small pulley and a larger attached disk spin together as a hanging weight falls. DataStudio will collect angular displacement and velocity information for the system as the weight falls. The relevant data is: • Diameter of small pulley: 3.0 cm • Mass of small pulley: negligible • Diameter of disk: 9.5 cm • Mass of disk: 120 g • Hanging mass: 10 g • See if we can illustrate Newton’s 2nd Law in rotational form.

  36. Demonstration calculations

  37. Rotational Dynamics Lab

  38. Work and Power in Rotating Systems

  39. Practice Problem What is the acceleration of this system, and the magnitude of tensions T1 and T2? Assume the surface is frictionless, and pulley has the rotational inertia of a uniform disk. T1 mpulley = 0.45 kg rpulley = 0.25 m T2 m1 = 2.0 kg m2 = 1.5 kg 30o

  40. Work in rotating systems • W = F • Dr (translational systems) • Substitute torque for force • Substitute angular displacement for displacement • Wrot = t • Dq • Wrot : work done in rotation •  : torque • Dq: angular displacement • Remember that different kinds of work change different kinds of energy. • Wnet = DK Wc = -DU Wnc = DE

  41. Power in rotating systems • P = dW/dt (in translating or rotating systems) • P = F • v (translating systems) • Substitute torque for force. • Substitute angular velocity for velocity. • Prot = t • w (rotating systems) • Prot : power expended •  : torque • w: angular velocity

  42. Conservation of Energy • Etot = U + K = Constant • (rotating or linear system) • For gravitational systems, use the center of mass of the object for calculating U • Use rotational and/or translational kinetic energy where necessary.

  43. Practice Problem A rotating flywheel provides power to a machine. The flywheel is originally rotating at of 2,500 rpm. The flywheel is a solid cylinder of mass 1,250 kg and diameter of 0.75 m. If the machine requires an average power of 12 kW, for how long can the flywheel provide power?

  44. Practice Problem A uniform rod of mass M and length L rotates around a pin through one end. It is released from rest at the horizontal position. What is the angular speed when it reaches the lowest point? What is the linear speed of the lowest point of the rod at this position?

  45. Rolling without Slipping Rolling without slipping review Conservation of Energy review Introduction to angular momentum of a particle Thursday, December 5, 2008

  46. Rolling without slipping • Total kinetic energy of a body is the sum of the translational and rotational kinetic energies. • K = ½ Mvcm2 + ½ I 2 • When a body is rolling without slipping, another equation holds true: • vcm =  r • Therefore, this equation can be combined with the first one to create the two following equations: • K = ½ M vcm2 + ½ Icm v2/R2 • K = ½ m 2R2 + ½ Icm2

  47. Sample Problem A solid sphere of mass M and radius R rolls from rest down a ramp of length L and angle q. Use Conservation of Energy to find the linear acceleration and the speed at the bottom of the ramp.

  48. Sample Problem • A solid sphere of mass M and radius R rolls from rest down a ramp of length L and angle q. Use Rotational Dynamics to find the linear acceleration and the speed at the bottom of the ramp.

  49. Angular Momentum of Particles

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