Physics 1408-002 Principles of Physics Sung-Won Lee [email protected] Lecture 18 – Chapter 11 & 12 – March 24, 2009 Announcement I Lecture note is on the web Handout (6 slides/page) http://highenergy.phys.ttu.edu/~slee/1408/ *** Class attendance is strongly encouraged and will be taken randomly. Also it will be used for extra credits. HW Assignment #7 is placed on MateringPHYSICS , before spring break and is due by 11:59pm on Wendseday, 3/25 Exam 1I Average = 58 %!Chapter 11 Angular Momentum General Rotation •! Angular Quantities •! Angular Momentum—Objects Rotating About a Fixed Axis •! Vector Cross Product; Torque as a Vector •! Angular Momentum of a Particle •! Angular Momentum and Torque •! Conservation of Angular Momentum •! The Spinning Top and Gyroscope •! Rotating Frames of Reference; Inertial Forces; The Coriolis Effect 11-1 Angular Momentum—Objects Rotating About a Fixed Axis The rotational analog of linear momentum is angular momentum, L: Then the rotational analog of Newton’s second law is: In the absence of an external torque, angular momentum is conserved: 0 and constant.dLL Idt!= = =This means: Therefore, if an object’s moment of inertia changes, its angular speed changes as well. 11-1 Angular Momentum—Objects Rotating About a Fixed Axis A skater doing a spin on ice, illustrating conservation of angular momentum: (a) I is large and ! is small; (b) I is smaller so ! is larger. Angular momentum is conserved.11-2 Vector Cross Product; Torque as a Vector Torque can be defined as the vector product of the force and the vector from the point of action of the force to the axis of rotation: Here, is the position vector from the particle relative to O. r!•! The torque vector lies in a direction perpendicular to the plane formed by the position vector and the force vector The Vector Product & Torque Using Determinants •! The cross product can be expressed as!•! Expanding the determinants gives!Torque Vector Example •! Given the force!! = ?"m)ˆ00.5ˆ00.4(N)ˆ00.3ˆ00.2(jirjiF+=+=j!11.3 Angular Momentum •! Consider a particle of mass m located at the vector position r and moving with linear momentum p The angular momentum of a particle about a specified axis is given by: If we take the derivative of , we find: !LSince we have: Torque and Angular Momentum •! The torque is related to the angular momentum –! Similar to the way, force is related to linear momentum •! This is the rotational analog of Newton’s 2nd Law ! and L must be measured about the same origin!•! The SI units of angular momentum: [(kg.m2)/s] •! Both magnitude and direction of L depend on choice of origin •! The magnitude of L = mvr sin"#!!" is the angle between p and r •! The direction of L is perpendicular to the plane formed by r & p L = r x pAngular Momentum of a Particle •! The vector L = r x p is pointed out of the diagram •! The magnitude is L = mvr sin 90o = mvr sin 90o is used since v is perpendicular to r •! A particle in uniform circular motion has a constant angular momentum about an axis through the center of its path Angular Momentum of a System of Particles •! Total angular momentum of a system of particles is defined as the vector sum of the angular momenta of individual particles Ltot = L1 + L2 + …+ Ln = %Li •! Differentiating with respect to time!•! Any torques associated with the internal forces acting in a system of particles are zero. 11.3 L of a Rotating Rigid Object •! Each particle of the object rotates in the xy plane about the z-axis w/ angular speed of $ •! The angular momentum of an individual particle: •! L and $ are directed along the z-axis Li = miri2$#•! To find the angular momentum of the entire object, add the angular momenta of all the individual particles •! This also gives the rotational form of Newton’s 2nd-Law Angular Momentum of a Bowling Ball •! The momentum of inertia of the ball: I = 2/5MR 2 •! The angular momentum of the ball: Lz = I$ •! The direction of the angular momentum is in the positive z-direction 11.4 Conservation of Angular Momentum •! The total angular momentum of a system is constant in both magnitude and direction if the resultant external torque acting on the system is “0”. –! Net torque = 0 -> means that the system is isolated!! •! Ltot = constant or Li = Lf •! For a system of particles, Ltot = "Ln = constant •! If the mass of an isolated system undergoes redistribution, the moment of inertia (I) changes. –! The conservation of angular momentum requires a compensating change in the angular velocity. –! Ii $i = If $f •! This holds for rotation about a fixed axis and for rotation about an axis through the center of mass of a moving system •! The net torque must be zero in any case 11-5 Angular Momentum and Torque for a Rigid Object An Atwood machine consists of two masses, mA and mB, which are connected by an inelastic cord of negligible mass that passes over a pulley. If the pulley has radius R0 and moment of inertia I about its axle, determine the acceleration of the masses mA and mB, and compare to the situation where the moment of inertia of the pulley is ignored.11-6 Conservation of Angular Momentum If the net torque on a system is constant, The total angular momentum of a system remains constant if the net external torque acting on the system is zero. Conservation Law Summary •! For an isolated system - (1) Conservation of Energy: Ei = Ef (2) Conservation of Linear Momentum: pi = pf (3) Conservation of Angular Momentum: Li = Lf Summary of Chapter 11 •! Angular momentum of a rigid object: •! Newton’s second law: •!Angular momentum is conserved. •! Torque: Summary of Chapter 11 •! Angular momentum of a particle: •! Net torque: •! If the net torque is zero, the vector angular momentum is conserved. Chapter 12 Static Equilibrium; Elasticity and Fracture •! The Conditions for Equilibrium •! Solving Statics Problems •! Stability and Balance •! Elasticity; Stress and Strain •! Fracture •! Trusses and Bridges / Arches and DomesQuiz •! A ball and box have the same mass and are moving with the same velocity across a horizontal floor. The ball rolls without slipping and the box slides without friction. They encounter an upward slope in the floor. Which one makes it farther