Introduction to and Andy Ruina and Rudra Pratap

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Chapter 0. Detailed Contents Detailed Contents 5 6.1 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 Box 6.1 ‘Zero-length’ springs . . . . . . . . . . . . . . . . 311 Box 6.2 A puzzle with two springs and three ropes. . . . . . 318 Box 6.3 How stiff a spring is a solid rod . . . . . . . . . . 319 Box 6.4 Stiffer but weaker . . . . . . . . . . . . . . . . . . 319 Box 6.5 2D geometry of spring stretch . . . . . . . . . . . 321 6.2 Force amplification . . . . . . . . . . . . . . . . . . . . . . 330 6.3 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 340 Box 6.6 Shears with gears . . . . . . . . . . . . . . . . . . 344 Problems for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . 351 7 Hydrostatics 360 Hydrostatics concerns the equivalent force and moment due to distributed pressure on a surface from a still fluid. Pressure increases with depth. With constant pressure the equivalent force has magnitude = pressure times area, acting at the centroid. For linearly-varying pressure on a rectangular plate the equivalent force is the average pressure times the area acting 2/3 of the way down. The net force acting on a totally submerged object in a constant density fluid is the displace weight acting at the centroid. 7.1 Fluid pressure . . . . . . . . . . . . . . . . . . . . . . . . . 361 Box 7.1 Adding forces to derive Archimedes’ principle . . . 364 Box 7.2 Pressure depends on position but not on orientation 365 Problems for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . 373 8 Tension, shear and bending moment 376 The ‘internal forces’ tension, shear and bending moment can vary from point to point in long narrow objects. Here we introduce the notion of graphing this variation and noting the features of these graphs. 8.1 Arbitrary cuts . . . . . . . . . . . . . . . . . . . . . . . . . 377 Problems for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . 393 Part III: Dynamics 396 9 Dynamics in 1D 396 The scalar equation F D ma introduces the concepts of motion and time derivatives to mechanics. In particular the equations of dynamics are seen to reduce to ordinary differential equations, the simplest of which have memorable analytic solutions. The harder differential equations need be solved on a computer. We explore various concepts and applications involving momentum, power, work, kinetic and potential energies, oscillations, collisions and multi-particle systems. 9.1 Force and motion in 1D . . . . . . . . . . . . . . . . . . . . 398 Box 9.1 What do the terms in F D ma mean . . . . . . . 403 Box 9.2 Solutions of the simplest ODEs . . . . . . . . . . . 408 Box 9.3 D’Alembert’s mechanics: beginners beware . . . . 411 9.2 Energy methods in 1D . . . . . . . . . . . . . . . . . . . . . 418
6 Chapter 0. Detailed Contents Detailed Contents Box 9.4 Particle models for the energetics of locomotion . . 429 9.3 Vibrations: mass, spring and dashpot . . . . . . . . . . . . . 436 Box 9.5 A cos.t/ C B sin.t/ D R cos.t / . . . . . 441 Box 9.6 Solution of the damped-oscillator equations . . . . 447 9.4 Coupled motions in 1D . . . . . . . . . . . . . . . . . . . . 461 Box 9.7 Normal modes: the math and the recipe . . . . . . 468 9.5 Collisions in 1D . . . . . . . . . . . . . . . . . . . . . . . . 476 Box 9.8 When equal rods collide the vibrations disappear . 481 9.6 Advanced: forcing & resonance . . . . . . . . . . . . . . . . 485 Box 9.9 A Loudspeaker cone is a forced oscillator. . . . . . 490 Box 9.10 Solution of the forced oscillator equation . . . . . 492 Box 9.11 The vocabulary of forced oscillations . . . . . . . 493 Problems for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . 501 10 Particles in space 514 This chapter is about the vector equation F * D m * a for one particle. Concepts and applications include ballistics and planetary motion. The differential equations of motion are set-up in cartesian coordinates and integrated either numerically, or for special simple cases, by hand. Constraints, forces from ropes, rods, chains floors, rails and guides that can only be found once one knows the acceleration, are not considered. Box 10.1 Newton’s laws in Newtonian reference frames . . 516 10.1 Dynamics of a particle in space . . . . . . . . . . . . . . . . 517 Box 10.2 The derivative of a vector depends on frame . . . 524 10.2 Momentum and energy . . . . . . . . . . . . . . . . . . . . 533 Box 10.3 Conservative forces and non-conservative forces 539 Box 10.4 Particle theorems for momenta and energy . . . . 541 10.3 Central-force motion and celestial mechanics . . . . . . . . . 545 Problems for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . 555 11 Many particles in space 562 This more advanced chapter concerns the motion of two or more particles in space. We will use * F D m * a for each particle. We will use Cartesian coordinates only. The start is the set up of “two-body” type problems which are easily generalized to 3 or more particles. The first section concerns smooth motions due to forces from gravity, springs, smoothly applied forces and friction. The second section concerns the sudden change in velocities when impulsive forces are applied. 11.1 Coupled particle motion . . . . . . . . . . . . . . . . . . . . 564 11.2 particle collisions . . . . . . . . . . . . . . . . . . . . . . . 572 Box 11.1 Effective mass . . . . . . . . . . . . . . . . . . . 574 Box 11.2 Energetics of collisions . . . . . . . . . . . . . . 575 Box 11.3 Coefficient of generation . . . . . . . . . . . . . 578 Box 11.4 A particle collision model of running . . . . . . . 579 Problems for Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . 584 12 Straight line motion 588
Page 1 and 2: Introduction to STATICS and DYNAMIC
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