Rotational Motion. A. Angular Kinematics. 1. Angles. The Ecliptic Circle. Rotational Motion. Physics: Mechanics. A. Angular Kinematics

Transcription

1 Physics: Mechanics Rotational Motion Dr. Bill Pezzaglia A. Angular Kinematics Rotational Motion (Angular Kinematics & Dynamics) B. Angular Dynamics C. Conservation Laws Updated: 0July08 A. Angular Kinematics 3. Angles 4. Angles. Angular Velocity 3. Angular Acceleration Perhaps 000 BC people changed from a nomadic culture to agrarian, settling in Sumer. Sumerians needed a calendar to tell them when to plant food. Surviving Babylonian Cuneiform Clay ablets of astronomical positions of sun & planets he Ecliptic Circle he Babylonians determined the exact path of the sun through the zodiac constellations 6 x

2 Hammurabi [80-70 BC] 7 (a) Angles in sexagesimal 8 school of scribes defines sexagesimal numbers (base 60). Degrees: 360=circle Arcminutes (minutes of arc): 60 = Arcseconds (seconds of arc): 60 = (b) Angle in Radians 9 (c) Angular Displacement 0 Circle: 360= radians Or radian =7.3 Analogous to linear displacement Arc length formula is easy in radians: s=r Its messier in degrees: s R 7.3 Definition: how far its rotated: = f - I here is a wrap around ambiguity, e.g. if you look at the clock and it says 3, and later it says, is it hours later or 4 hours later?. Angular Velocity (b) Units of Angular Velocity (a) Definition: (analogous to average velocity) Expressed in terms of period for one complete revolution: For constant angular velocity we have simple equation t t SI units: radians per second Other systems used: Degrees/second Rpm: revolutions per minute (example, convert rpm to rads/sec): min rads 60sec rev t rev rads min 0. 0 sec

3 (c) angential Velocity 3 3. Angular Acceleration 4 he (tangential) speed at a spot distance r from axis can be calculated as the total distance traveled (circumference of circle) in one period: (a) Definition t r vt v r t SI Units: rads/sec Kinematic equations for rotation under constant angular acceleration are completely analogous to those for linear acceleration: t t B. Angular Dynamics. orque. Moment of Inertia 3. Newton s Law for rotation overview 76 Leonhard Euler derives rotational analogy to Newton s nd law 6. orque 7 (b) Define orque 8 (a) Law of Lever : Archimedes of Syracuse 87- BC Define torque =(lever arm)(force) (effort arm)(effort Force)=(load arm)(loadforce) SI Units: NewtonMeter=Nm Note: lever arm also called moment arm. It is the perpendicular distance from line of force to the axis SIGN CONVENION: CW: Clockwise is negative torque CCW: Counterclockwise is positive Give me a place to stand on, and I will move the Earth. 3

4 (c) Center of Gravity (Center of Mass) 9 (d) Find Center of Gravity (Mass) 0 he total force on an object can be considered to act upon the center of mass of a body. Pappus of Alexandria showed how to find centroid of a triangle. If the force is gravity, we call it the center of gravity A tilted block will fall over due to torque from center of gravity. If you hang an object from several points and draw vertical lines, they will intersect at the center of gravity. (why?) (e) Equilibrium. Rotational Inertia A system will be in equilibrium if: Resistance to torque, i.e. resistance to being rotated. Sum of forces (in any direction) is zero Sum of torques (about any axis) is zero Hence the lever law simply states that the clockwise torque of the load (about fulcrum) is cancelled by counter-clockwise torque from the effort. (effort arm)(effort Force)=(load arm)(loadforce) Generically, depends upon mass and square of size Requires calculus, so we just give you a table of results: 3. Rotational Dynamic Law 3 C. Conservation Laws 4 76 Leonhard Euler derives rotational analogy to Newton s nd law Definition of orque Definition of angular acceleration I r F t. Angular Momentum. Conservation of 3. Rotational Energy 4

5 . Angular Momentum Definition orque causes a change in angular momentum Angular momentum changes if angular acceleration OR change in moment of inertia (e.g. star shrinks in size) L I L r p rmv L t I I t. Conservation otal angular momentum of a system is conserved if there is no net external torque. Example: he sun rotates in 7 days. When sun expands to red giant (0x bigger in size), the rotation rate will decrease by a factor of 00, so the period of rotation will increase by 00 to 700 days. I I MR MR R R 6 3. Rotational Kinetic Energy Definition For rolling motion, So, an object with bigger moment of inertia (hollow cylinder) will roll slower than a ball down an inclined plane due to more energy going into rotation. K rot v cm r mgh I 7 mv I References, Notes Hewitt also covers centripetal & centrifugal force in chapter 8. we move it later. 8 Demonstrations 9 rainwheels (cups) Centroid demo Moment of inertia samples Inclined plane (rolling objects) Rotating platform (consv L) Gyroscope/precession