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Physics 1501: Lecture 19 Today ’ s Agenda

Physics 1501: Lecture 19 Today ’ s Agenda. Announcements HW#7: due Oct. 21 Midterm 1: average = 45 % … Topics Rotational Kinematics Rotational Energy Moments of Inertia . Summary (with comparison to 1-D kinematics). Angular Linear.

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Physics 1501: Lecture 19 Today ’ s Agenda

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  1. Physics 1501: Lecture 19Today’s Agenda • Announcements • HW#7: due Oct. 21 • Midterm 1: average = 45 % … • Topics • Rotational Kinematics • Rotational Energy • Moments of Inertia

  2. Summary (with comparison to 1-D kinematics) Angular Linear And for a point at a distance R from the rotation axis: • x = Rv = R a = R

  3. Example: Wheel And Rope • A wheel with radius R = 0.4m rotates freely about a fixed axle. There is a rope wound around the wheel. Starting from rest at t = 0, the rope is pulled such that it has a constant acceleration a = 4m/s2. How many revolutions has the wheel made after 10 seconds? (One revolution = 2 radians) a R

  4. = 0 + 0(10) + (10)(10)2 = 500 rad Wheel And Rope... • Use a = Rto find : = a / R = 4m/s2 / 0.4m = 10 rad/s2 • Now use the equations we derived above just as you would use the kinematic equations from the beginning of the semester. a  R

  5. m4 m1 r1  r4 m3 r2 r3 m2 Rotation & Kinetic Energy • Consider the simple rotating system shown below. (Assume the masses are attached to the rotation axis by massless rigid rods). • The kinetic energy of this system will be the sum of the kinetic energy of each piece:

  6. which we write as: Rotation & Kinetic Energy... • So: but vi = ri v1 m4 v4 m1 r1  r4 v2 m3 r2 r3 m2 v3 Define the moment of inertia about the rotation axis I has units of kg m2.

  7. Lecture 19, Act 1Rotational Kinetic Energy • I have two basketballs. BB#1 is attached to a 0.1m long rope. I spin around with it at a rate of 2 revolutions per second. BB#2 is on a 0.2m long rope. I then spin around with it at a rate of 2 revolutions per second. What is the ratio of the kinetic energy of BB#2 to that of BB#1? A) 1/4 B) 1/2 C) 1 D) 2 E) 4 BB#1 BB#2

  8. Rotation & Kinetic Energy... • The kinetic energy of a rotating system looks similar to that of a point particle:Point Particle Rotating System v is “linear” velocity m is the mass.  is angular velocity I is the moment of inertia about the rotation axis.

  9. Moment of Inertia • So where • Notice that the moment of inertia I depends on the distribution of mass in the system. • The further the mass is from the rotation axis, the bigger the moment of inertia. • For a given object, the moment of inertia will depend on where we choose the rotation axis (unlike the center of mass). • We will see that in rotational dynamics, the moment of inertia I appears in the same way that mass m does when we study linear dynamics !

  10. Calculating Moment of Inertia • We have shown that for N discrete point masses distributed about a fixed axis, the moment of inertia is: where r is the distance from the mass to the axis of rotation. Example: Calculate the moment of inertia of four point masses (m) on the corners of a square whose sides have length L, about a perpendicular axis through the center of the square: m m L m m

  11. I = 2mL2 Calculating Moment of Inertia... • The squared distance from each point mass to the axis is: Using the Pythagorean Theorem so L/2 m m r L m m

  12. m m I = mL2 m m Calculating Moment of Inertia... • Now calculate I for the same object about an axis through the center, parallel to the plane (as shown): r L

  13. I = 2mL2 Calculating Moment of Inertia... • Finally, calculate I for the same object about an axis along one side (as shown): r m m L m m

  14. Calculating Moment of Inertia... • For a single object, I clearly depends on the rotation axis !! I = 2mL2 I = mL2 I = 2mL2 m m L m m

  15. Lecture 19, Act 2Moment of Inertia • A triangular shape is made from identical balls and identical rigid, massless rods as shown. The moment of inertia about the a, b, and c axes is Ia, Ib, and Ic respectively. • Which of the following is correct: a (a)Ia > Ib > Ic (b)Ia > Ic > Ib (c)Ib > Ia > Ic b c

  16. dm r Calculating Moment of Inertia... • For a discrete collection of point masses we found: • For a continuous solid object we have to add up the mr2 contribution for every infinitesimal mass element dm. • We have to do anintegral to find I :

  17. dr r R L Moments of Inertia • Some examples of I for solid objects: Solid disk or cylinder of mass M and radius R, about a perpendicular axis through its center.

  18. Moments of Inertia... • Some examples of I for solid objects: Solid sphere of mass M and radius R, about an axis through its center. R Thin spherical shell of mass M and radius R, about an axis through its center. R

  19. R Thin hoop of mass M and radius R, about an axis through a diameter. Moments of Inertia • Some examples of I for solid objects: Thin hoop (or cylinder) of mass M and radius R, about an axis through its center, perpendicular to the plane of the hoop. R

  20. Parallel Axis Theorem • Suppose the moment of inertia of a solid object of mass M about an axis through the center of mass is known, = ICM • The moment of inertia about an axis parallel to this axis but a distance R away is given by: IPARALLEL = ICM + MR2 • So if we know ICM , it is easy to calculate the moment of inertia about a parallel axis.

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