11/16/2009Physics 1001 Physics 100 Fall 2005 Lecture 11

11/16/2009Physics 1002 Radians and Other Angular Units Define radian  R R  = 1 radian (rad.) 360 o = 2  rad 180 o =  rad 90 o =  /2 rad etc. 1 rad  57.3 o

11/16/2009Physics 1003 Vector or cross products Note that this yields another vector (actually an axial vector) Contrast with the scalar product: yielding a scalar 

11/16/2009Physics 1004 ‛ x A x C  To the plane of AB Right Hand Rule Direction of advance of a right hand screw

11/16/2009Physics 1005 Note that the vector product is not commutative Again look at Right Hand Rule ‛ x A x C A × B B × A

11/16/2009Physics 1006 Relation between Angular and Linear Kinematics v r s  r  arc length v = r  a  r  Different points on a rigid body do not Have the same s, v, a but do have the same  Direction of  is given by the right hand rule x y z  r  Right hand Fingers are direction of  Thumb is direction of Note: product is not commutative s v

11/16/2009Physics 1007 P r  out v Apply Right Hand Rule: Rotate  into r, right thumb points in direction of v Point rotating a distance r from an axis

11/16/2009Physics 1008 Magnitude Direction Vector vs. Axial Vector

11/16/2009Physics 1009 Rotational Dynamics y x  axis F 1 F 2 F 3 F4F4 Forces pointing thru the axis (F 1  F 4 )  do not affect rotation Forces  x, with lever arm ≠ 0, ( F 2, F3), have maximal effect and the effect is proportional to the length of the lever arm.

11/16/2009Physics 10010

11/16/2009Physics Torque (m·N) N = r F sin  = r F when r  F Torque is the rotational analog of force. (m·N) N = r F sin  = r F when r  F

11/16/2009Physics Into paper

11/16/2009Physics Into the paper

11/16/2009Physics Example: Compound Wheel, R 1 =30 cm, R2=50 cm Net torque acts to accelerate rotation clockwise. Right Hand Rule

11/16/2009Physics Moment of Inertia a. k. a. rotational inertia Consider a collection of particles each having kinetic energy K i = ½ m i v i 2 = ½ m i r i 2  2 Then the total kinetic energy of the rigid body is K = ½ (  m i r i 2 )  2 Define I = (  m i r i 2 ) Moment of inertia (kg m 2 ) Rotational analog of mass Rigid body

11/16/2009Physics 10016

11/16/2009Physics 10017

11/16/2009Physics Newton’s First Law for Rotation massless rod An object at rest or in uniform rotational motion, in the absence of externally imposed torques, remains at rest or in uniform rotational motion. If you don’t do anything, nothing happens

11/16/2009Physics Statics and Equilibrium Statics: A special case of dynamics in which bodies remain at rest in static equilibrium. For a rigid body to be in equilibrium both linear and angular accelerations must be 0. Translational equilibrium: or the 3 scalar equations

11/16/2009Physics Rotational equilibrium: Center of Gravity and Center of Mass: For g uniform, i. e. small heights above Earth’ surface the C. M. and the C. G. are coincident. similarly for y and z again 3 scalar equations for the torques around 3  axes. A translational force like gravity acts like the total force is acting downward at the C. G./C. M.

11/16/2009Physics Experimental Determination of the C. G.: Suspend an object. Extending the line of suspension downward will pass thru the C. G. A a Hang from another point B b Dashed lines intersect at the C. G. C. G.

11/16/2009Physics Examples of Equilibrium: m F1F1 F2F2 F 1 = -F 2 translational F=0 F1F1 F2F2 r N 1 = -N 2 = F 1 r = -F 2 r N = 0 rotational x

11/16/2009Physics Stable, Unstable and Neutral Equilibrium Earth’s gravitational field g Potential energy as a function of position U unstable stable unstable neutral stable neutral

11/16/2009Physics For an extended body F or F  C. M. is raised stable C. M. falls unstable C. M. neither rises nor falls neutral unstable stable Pendulum with rigid cable

11/16/2009Physics Tip Over Angle: mgmg mgmg SlidesTips over  Arrows (  ) refer to direction of mg from the CM not the magnitude.

11/16/2009Physics Newton’s Second Law for Rotation massless rod (translational)

11/16/2009Physics Angular momentum N = I  Or N =  L /  t Where L = I  is called the angular momentum L is another conserved quantity e. g. ice skater, piano stool, bicycle stability, gyroscopes, stellar collapse, etc.

11/16/2009Physics Conservation of angular momentum No net torque, decrease I =>  must increase ad vice versa

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11/16/2009Physics Kepler’s Laws I. Planetary orbits are ellipses with the sun at one focus. II. Equal fractional areas of the ellipses are swept out in equal times. f f’

11/16/2009Physics Newton’s 3rd Law for rotation Action vs. reaction Twist something, something twists on you Action reaction pairs couples A pair of forces with equal magnitudes, opposite directions and different lines of action.

11/16/2009Physics F F A B C What is the torque produced about the 3 axes A, B and C? A couple produces a torque that does not depend on the location of the axis.

11/16/2009Physics Example: Stellar Collapse Inner core of a larger star collapses into a neutron star of very small radius r = r sun = 7 x 10 5 km, m = 2 m sun,, T = 10 d, r n-star = 10 km Assume no mass is lost in collapse. What is n-star’ rate of rotation?

11/16/2009Physics Rotational energy and work Work: Exert a torque through an angular displacement Potential energy is increased by the doing of work Kinetic energy of rotation This makes conservation of the total energy a little more complicated

11/16/2009Physics Wheel rolling down incline h v,  v  r m, I Wheel is at rest at top. Initial energy is all potential. U = mgh At bottom wheel is rolling and has lost all its potential energy w.r.t. ground. Total energy is all kinetic. Kinetic energy has two parts, translational and rotational Conservation of total energy here requires

11/16/2009Physics DisplacementRotation VelocityAngular velocity AccelerationAngular acceleration MassMoment of inertia ForceTorque MomentumAngular momentum EnergyRotational energy