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Physics 114A - Mechanics Lecture 24 (Walker: Ch. 10.4-6) Rotational Inertia February 25, 2014

Physics 114A - Mechanics Lecture 24 (Walker: Ch. 10.4-6) Rotational Inertia February 25, 2014. John G. Cramer Professor Emeritus, Department of Physics B451 PAB jcramer@uw.edu. Announcements. HW#7 is due at 11:59 PM on Thursday, February 27. HW#8 is due at 11:59 PM on Thursday, March 6.

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Physics 114A - Mechanics Lecture 24 (Walker: Ch. 10.4-6) Rotational Inertia February 25, 2014

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  1. Physics 114A - MechanicsLecture 24 (Walker: Ch. 10.4-6)Rotational InertiaFebruary 25, 2014 John G. Cramer Professor Emeritus, Department of Physics B451 PAB jcramer@uw.edu

  2. Announcements • HW#7 is due at 11:59 PM on Thursday, February 27.HW#8 is due at 11:59 PM on Thursday, March 6. • My office hours are 12:30-1:20 PM on Tuesdays and 2:30-3:20 PM on Thursdays, both in the “114” area of the Physics Study Center on the Mezzanine floor of PAB A (this building). Physics 114A - Lecture 24

  3. Lecture Schedule (Part 3) We are here. Physics 114A - Lecture 24

  4. Example: The Microhematocrit In a microhematocrit centrifuge, small samples of blood are placed in heparinized capillary tubes (heparin is an anticoagulant). The tubes are rotated at 11,500 rpm, with the bottom of the tubes 9.07 cm from the axis of rotation. (a) Find the linear speed at the bottomof the tubes. (b) Find the centripetal acceleration atthe bottom of the tubes. Physics 114A - Lecture 24

  5. Example:Starting Up a Microhematocrit Suppose a microhematocrit centrifuge is starting up with a constant angular acceleration of a = 95.0 rad/s2. (a) What is the magnitude of the centripetal, tangential, and total acceleration of the bottom of a tube when the angular speed is w = 8.00 rad/s? (b) What angle f does the total acceleration vector make with the direction of motion? Physics 114A - Lecture 24

  6. Rolling Motion If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds: Physics 114A - Lecture 24

  7. Rolling Motion We may also consider rolling motion to be a combination of pure rotational and pure translational motion: Physics 114A - Lecture 24

  8. Rolling Motion We may also consider rolling motion at any given instant to be a pure rotation at rate w about the point of contact of the rolling object. Physics 114A - Lecture 24

  9. Example: A Rolling Tire A car with tires of radius 32 cm drives on a highway at a speed of 55 mph. (a) What is the angular speed w of the tires? (b) What is the linear speed vtop of the top to the tires? Physics 114A - Lecture 24

  10. Rotational Kinetic Energy& Moment of Inertia For this point mass m, the kinetic energy is: Define the moment of inertia of a point mass as: Then the kinetic energy is: Physics 114A - Lecture 24

  11. Rotational Kinetic Energy& Moment of Inertia We generalize this to any rotating object, which will have a kinetic energy: Here I, the moment of inertia, is given by Physics 114A - Lecture 24

  12. Example: A Dumbbell Use the definition of momentof inertia to calculate that of adumbbell-shaped object withtwo point masses m separatedby a distance of 2r and rotatingabout a perpendicular axis throughtheir center of symmetry. Physics 114A - Lecture 24

  13. Example:Nose to the Grindstone A grindstone of radius r = 0.610 m is being used to sharpen an axe. If the linear speed of the stone is 1.50 m/s and the stone’s kinetic energy is 13.0 J, what is its moment of inertia I ? Physics 114A - Lecture 24

  14. Moment of Inertia of a Hoop All of the mass of a hoop is at the same distance R from the center of rotation, so its moment of inertia is the same as that of a point mass rotated at the same distance. Physics 114A - Lecture 24

  15. Moments of Inertia Moments of inertia of various regular objects can be calculated: Physics 114A - Lecture 24

  16. More Moments of Inertia Physics 114A - Lecture 24

  17. Clicker Question 1 The T-shaped object is rotated about each axis shown. For which axis does it have the largest moment of inertia? Physics 114A - Lecture 24

  18. Example:Estimating the Moment of Inertia Estimate the moment of inertia of a thin rod of mass M and length L about an axis perpendicular to the rod and through one end. Perform this estimation by modeling the rod as three point masses, each representing one third of the rod. Physics 114A - Lecture 24

  19. The Parallel-Axis Theorem If you know the moment of inerita Icm of an object about its center of mass, you can calculate I about any other parallel axis by adding Md2, where M is the object’s mass and d is the separation of the axes. Physics 114A - Lecture 24

  20. Example: Applyingthe Parallel Axis Theorem A thin uniform rod of mass M and length L on the x axis has one end at the origin. Use the parallel axis theorem to find the moment of inertia about an axis of rotation parallel to the y axis and through the center of the rod. Physics 114A - Lecture 24

  21. Kinetic Energy of Rolling Trick: Instead of treatingthe rotation and translation separately, combine them by considering that instantaneously the system is rotating abut the point of contact. Physics 114A - Lecture 24

  22. Conservation of Energy The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies: The second equation makes it clear that the kinetic energy of a rolling object is a multiple of the kinetic energy of translation. Physics 114A - Lecture 24

  23. Example: Like a Rolling Disk A 1.20 kg disk with a radius 0f 10.0 cm rolls without slipping. The linear speed of the disk is v = 1.41 m/s. (a) Find the translational kinetic energy. (b) Find the rotational kinetic energy. (c) Find the total kinetic energy. Physics 114A - Lecture 24

  24. Rolling Down an Incline Physics 114A - Lecture 24

  25. Clicker Question 2 Which of these two objects, of the same mass and radius, if released simultaneously, will reach the bottom first? Or is it a tie? (a) Hoop; (b) Disk; (c) Tie; (d) Need to know mass and radius. Physics 114A - Lecture 24

  26. Compare Heights A ball is released from rest on a no-slip (high-friction) surface, as shown. After reaching the lowest point, it begins to rise again on a frictionless surface. When the ball reaches its maximum height on the frictionless surface, it is higher, lower, or the same height as its release point? The ball is not spinning when released, but will be spinning when it reaches maximum height on the other side, so less of its energy will be in the form of gravitational potential energy. Therefore, it will reach a lower height. Physics 114A - Lecture 24

  27. Example: Spinning Wheel A block of mass m is attached to a string that is wrapped around the circumference of a wheel of radius R and moment of inertia I, initially rotating with angular velocity w that causes the block to rise with speed v . The wheel rotates freely about its axis and the string does not slip. To what height h does the block rise? Physics 114A - Lecture 24

  28. End of Lecture 24 • Before Thursday, read Walker, Chapter 10.6. • Homework Assignment 7 is due at 11:59 PM on Thursday, February 27. Physics 114A - Lecture 24

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