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Reminder

Reminder. Please take advantage of my office hours MWF 11-noon By appointment: gblazey@nicadd.niu.edu Help room 2 nd floor Faraday The University has Retention Programs that can help, let me know if you are interested. Unit 4: Circular Motion and Gravity.

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  1. Reminder • Please take advantage of my office hours • MWF 11-noon • By appointment: gblazey@nicadd.niu.edu • Help room 2nd floor Faraday • The University has Retention Programs that can help, let me know if you are interested. Physics 253

  2. Unit 4: Circular Motion and Gravity • We’ve fully investigated linear motion and forces. But this is somewhat limited. • Now it’s time to take a look at circular motion and apply Newton’s Laws • To do so we’ll collect the various sections of the book on circular motion. • Beginning with uniform circular motion, angular variables, and the equations of motion for angular motion (3-9, 10-1, 10-2) • Then we’ll move onto: • Applications of Newton’s Laws (5-2, 5-3, 5-4) • Harmonic Motion (14-1, 14-2) • The Universal Law of Gravitation, Satellites, and Kepler’s Law’s (Chapter 6) Physics 253

  3. Uniform Circular Motion (Section 3-9) • Definition: Motion of an object moving in a circle at constant speed. • Examples: • Ball on a string • Satellite in orbit around the earth. • Stars around a galaxy’s center (a black hole?) • Characteristics: • Magnitude of velocity constant • Direction under continuous change • Consequently there is an acceleration Physics 253

  4. The Direction of the Acceleration of Circular Motion • Consider the figure at right • During a small time interval Dt, a particle moves from A to B • It covers a small arc labeled Dl and subtends a small angle Dq. • The change in the velocity vector is just given by v2-v1=Dv Physics 253

  5. Now consider taking Dt to the limit. • As it becomes very small then Dl and Dq become very small. • Also the two vectors v2 and v1 become almost parallel. Accordingly Dv will be perpendicular to both. • This can only occur if Dv points toward the center of the circle. • Recall that • Which means a points toward the center as well • Accordingly its called centripetal or radial acceleration aR Physics 253

  6. What about the magnitude? • The triangle ABC is congruent or geometrically similar to the triangle abc. • Thus: • As shown w hen Dt approaches zero, the arc equals the chord. • In that limit a c b Physics 253

  7. But since then gives, The basic result is then: an entirely geometric result. In words: An object moving in a circle of radius r with constant speed v has an acceleration toward the center of the circle with magnitude aR=v2/r. aR=v2/r Physics 253

  8. Other Characteristics of Circular Motion • Acceleration and velocity are perpendicular • Frequency, f, is the number of revolutions per second • Period, T, time required for one revolution. • T = 1/f • v=2pr/T Physics 253

  9. The moon orbits the earth at a radius of 384,000km (3.84x108m) and with a period of 27.3 days(2.36x106s). What is the centripetal acceleration? The circumference of the circle describing the orbit is 2pr. The velocity would be that circumference divided by the period T or 2pr/T. So aR=v2/r = (2pr/T)2/r = 4p2r/T2 We can just substitute our values for r and T: aR=4p2r/T2= 4(3.14)2(3.84x108m) = (2.36x106s)2 2.72x10-3 m/s2 Presumably this is due to gravity? Compare to 9.80m/s2 The Moon’s Radial Acceleration. Physics 253

  10. Angular Quantities (Section 10-1) • As we will see, there is a close parallelism between the variables of linear motion and those of angular motion. • The motion of a rigid body can be described with both translation motion and rotational motion. • Consider the disk at the right undergoing purely rotational motion, that is all points move in a circle about the axis of rotation, which is projecting from the screen. Physics 253

  11. We use R rather than r to indicate distance from the axis of rotation. Physics 253

  12. Indicating Angular Position of a Point • Given by an angle with respect to an axis. • A point P “moves through” angle q as it travels along arc l. • Angles can be given in degrees or more conveniently in radians. • One radian is the angle subtended by an arc equal to the radius. • Note that one radian is the same angle for any sized circle. 1 rad Physics 253

  13. More on Radians • By definition then q = l/R where R is the radius of a circle, and l is the arc length subtended by q. • Note radians are dimensionless! • Radians are easily related to degrees since the 360o arc length of a complete circle is 2pR: 360o =l/R =2pR/R = 2p rads  1 rad = 57.3o Physics 253

  14. Angular Variables of Motion: w and a • Consider the wheel, it’s angular displacement after a bit of rotation is given by: • In complete analogy with average velocity the average angular velocity, w, is defined as: • And the instantaneous angular velocity is Physics 253

  15. We can also define average and instantaneous angular acceleration in analogy to linear acceleration: • The units of w are rad/s and for a they are rad/s2. Physics 253

  16. The Relationship between Angular and Linear Velocity • Each point on a rotating rigid body has nonzero w and v. • The figure helps to under-stand the relationship between the two for P. • The magnitude of the linear velocity is given by Physics 253

  17. v=Rw Note that different radii have equal angular velocity but very different linear velocity Physics 253

  18. The Relationship between Angular and Linear Acceleration • If an object’s angular velocity changes there will also be angular acceleration. • Every point on the object will then undergo tangential acceleration. • But also recall there is a radial acceleration: The greater R the greater the acceleration – think “Crack the Whip” Physics 253

  19. Collecting Results • We can also write the angular velocity, w, in terms of the frequency, f. • Since • A frequency of 1 rev/sec = an angular velocity of 2p rads/sec, we can say: • f = w/2p or w=2pf • The unit of frequency rev/s is given the name hertz(Hz) and since revolutions are not a true unit (just a place keeper) 1Hz=1s-1. Physics 253

  20. Heinrich Rudolf Hertz (1857 - 1894) a German physicist and mechanician for whom the hertz, an SI unit, is named. In 1888, he was the first to satisfactorily demonstrate the existence of electromagnetic radiation by building an apparatus to produce UHF radio waves (300 MHz and 3 GHz) . Physics 253

  21. What is angular velocity? Speed @3.0cm from axis? Linear acc. at 3.0 cm.? How many 5.0mm bits can be written per second at 3.0cm? If the disk takes 3.6 s to reach speed what is the average acceleration? An Example: Parameters of a Hard Drive Rotating at 5400 rpm. Physics 253

  22. Equations of Motion for Rotational Motion • The definitions of average and instantaneous angular velocity and angular acceleration are identical to linear velocity and acceleration except for a variable change: • Recall the definitions of average and instantaneous velocity and acceleration led to the four equations of linear motion for constant acceleration. • An identical analysis for angular motion at constant angular acceleration would lead to the same four equations with the replacement: Physics 253

  23. Note since the equations are identical there is no need for a re-derivation, this is a pretty common technique! Physics 253

  24. An example: Back to the Hard-drive • How many revolutions did the hard-drive execute when accelerating from 0 to 5400 rpm in 3.6s? • Well we know w0=0, w=570 rad/s, a=160rad/s2 • We’re really after the total angle turned during this interval. Use the 2nd equation: Physics 253

  25. Solving Rotational Motion Problems • Draw the situation, showing direction of rotation. • Decide on positive and negative directions of motion. • Write down list of rotational kinematic variables, q, a, w, wo, and t. • Verify that 3 of 5 variables are known, then select appropriate equation. • If not enough information see if segments share information or look for constraints • If two solutions, exist choose the physical one. Physics 253

  26. A blender on "puree" has blades spinning at angular velocity of +375 rad/s. The blades are accelerated with the "blend" selection. They reach their final angular velocity after the blades have rotated through +44.0 rad (seven revolutions). The angular acceleration has a constant value of +1740 rad/s2. Find the final angular velocity of the blades. We’ll the picture here is pretty simple. Looking down on the blender we see the blades rotating counterclockwise at an initial angular speed of +375 rad/s. A counterclockwise acceleration kicks in at +1740 rad/s2. After a counterclockwise displacement of +44.0 rad the final angular velocity is reached. Here I’ve painted a picture for you and retained counterclockwise as the positive direction (Steps 1 and 2). Now let’s write the list (Step 3): Second Example: A blender Physics 253

  27. We have three variables and see that the final angular velocity is given by the last equation (Step 4): w2 = wo2 + 2aq Now it’s just plug-and-chug: w2 = (375 rad/s)2 + 2(1740 rad/s2)(44.0rad) = 2.97 x 105 rad2/s2 Taking the root, w = +/-542 rad/s. Since all the motion is in the positive direction the answer must also be positive (Step 6): w = +542 rad/s. Physics 253

  28. Where we are and where we are going… • So now we’ve got equations of motion for rotational as well as linear kinematics! • That means, just as we were with linear kinematics, we are now in a position to concentrate on the dynamics of uniform circular motion. • That is, Friday we’ll apply Newton’s 2nd laws to circular motion. Physics 253

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