STATIC AND DYNAMICS

584 INDEX angular momentum about a non-Newtonian frame, 284 linear damping coefficient, c, pointC, 341 product rule, 283 257 string skills, 28, 42 free body diagrams, 85 summary box, 9 logarithmic decrement, 258 Sturmey-Archer hub, 516 triple cross product, 433 measurement, 258 geometry of, 437 overdamping, 258 tensegrity structure, 133, 138 vector equations solutions, 258 three-force members, 111 reduction of into scalar equations, underdamping, 258 triple cross product, 433 unforced harmonic oscillator 117 summary, 259 geometry of, 437 Vector skills for mechanics , 7–76 two-force bodies, 109 vectors and scalars, 8 wires vectors, matrices, and linear algebraic free body diagrams, 85 undamped harmonic oscillator natural frequency of vibration, f , equations, 117–118 work of a force, 291 243 velocity oscillation frequency, λ, 242 Zero-force members, 132 period of vibration, 242 absolute solution, 242 general motion of a rigid body, 500 underdamping, 258 unforced damped harmonic oscillator, as a function of position, 225 circular motion at constant rate, 257–259 critical damping, 258 354 linear damping coefficient, c, 257 2D, 354 logarithmic decrement, 258 circular motion at constant rate, overdamping, 258 solutions, 258 derivation, 355 underdamping, 258 circular motion at variable rate, unforced harmonic oscillator summary, 259 357, 432 unit vectors general motion polar coordinates, 355 cartesian coordinates, 281 variable rate circular motion path coordinates, 551 2-D and 3-D, 378, 433 polar coordinates, 548, 549 acceleration, 432 general motion of a rigid body, 498 acceleration derivation, 357 moving frames energy, 472 absolute velocity, 563 examples, 365 one-dimensional motion, 219 extended bodies in 3-D, 470 relative geometry of, 471 general motion of a rigid body, linear momentum balance, 416 rotation of a rigid body about a 500 fixed axis, 472 relative motion of points on a rigid simple pendulum, 365 velocity, 432 body, 379, 435 Vibrations vector cross product, 34 forced oscillations dot product, 24 frequency response, 266 finding components using, 25 how to write, 9 forced oscillations and resonance, identities, 39 264–266 mixed triple product, 40 products of three vectors, 40 resonance rate of change of a, 281 loudspeaker, 266 frame dependency, 283 undamped harmonic oscillator energy, 245 oscillation frequency, 242 period of vibration, 242 solution, 242 unforced damped harmonic oscil- lator, 257–259 critical damping, 258

TABLE I Momenta and energy Linear Momentum Angular Momentum Kinetic Energy What system L (a) L˙ = d L (b) H C (c) H˙ C = d (d) EK (e) In General dt dt HC L x ıˆ + L yˆ + Lzkˆ L˙ x ıˆ + L˙ yˆ + L˙ zkˆ HCx ıˆ + HCyˆ + HCzkˆ H˙Cx ıˆ + H˙Cyˆ + H˙Czkˆ = mtot a cm = r cm/C × a cmmtot + H˙ cm = mtot vcm = r cm/C × vcmmtot + H cm = (no simple general expression) 1 vc2m + EK/cm (1) “F = ma” 2 mtot = d (m rcm) dt tot = d (no such thing) dt One Particle P mPvP mPaP r P/C × v P mP r P/C × a PmP 1 m PvP2 (2) 2 (3) System of mi vi mi ai r i/C × v i mi r i/C × ai mi (4) Particles 1 vi2mi (5) all particles i all particles i all particles all particles 2 Continuum all particles v dm a dm r /C × v dm r /C × a dm System of Systems 1 v2 dm (eg. rigid bodies) all mass all mass all mass all mass 2 all mass mi vi mi ai H Ci H˙ Ci EKi all sub-systems all sub-systems all sub-systems all sub-systems all sub-systems Rigid Bodies r cm/C × vcmmtot + [Icm ] · ω r cm/C × a cmmtot 1 vc2m H cm + [Icm ] · ω˙ + ω × H cm 2 mtot One rigid body (6) (2D and 3D) mtot vcm mtot a cm H˙ cm + 1 ω · [Icm] · ω (7) mtot a cm 2 mtot a cm mtot a cm E K /cm 2D rigid body r cm/C × vcmmtot + Izczm ωkˆ r cm/C × a cmmtot + Izczm ω˙ kˆ 1 vc2m H cm H˙ cm 2 mtot in x y plane mtot vcm with ω = ωkˆ 1 Izczm ω2 + 2 E K /cm One rigid body mtot vcm [IC] · ω = H C [IC] · ω˙ + ω × H C 1 ω · C · ω (8) if mtot vcm IzCz ωkˆ IzCzoω˙ kˆ 2 [I ] C is a fixed point “M = Iα” (2D and 3D) 1 IzCz ω2 (9) 2 2D rigid body if C is a fixed point with ω = ωkˆ The table has used the following terms: H cm = ri/cm × (mi v i ) angular momentum about the center of mass mtot =total mass of system, H˙ cm = ri/cm × (mi ai ) rate of change of angular mi = mass of body or subsystem i, momentum about the center of mass r cm/C = the position of the center of mass relative to point C, ω is the angular velocity of a rigid body, ω˙ = α is the angular acceleration of the rigid body, v i = velocity of the center of mass of sub-system or [Icm] is the moment of inertia matrix of the rigid body particle i, relative to the center of mass, and [Io] is the moment of inertia matrix of the rigid body ai = acceleration of the center of mass of sub-system relative to a fixed point (not moving point) on the body. i, H Ci = angular momentum of subsystem i relative to point C. H˙ Ci = rate of change of angular momentum of sub- system i relative to point C.

Table II Summary of methods of calculating velocity and acceleration Method Position Velocity Acceleration r or r P or r P/O In general, as measured v or v P or v P/F a or a P or a P/F relative to the fixed frame F . vx ıˆ + vyˆ + vzkˆ = r˙x ıˆ + r˙yˆ + r˙zkˆ rx ıˆ + ryˆ + rzkˆ ax ıˆ + ayˆ + azkˆ = v˙x ıˆ + v˙yˆ + z˙zkˆ vReˆ R + vθ eˆθ + vzkˆ = r¨x ıˆ + r¨yˆ + r¨zkˆ Cartesian Coordinates = R˙eˆR + Rθ˙eˆθ + z˙kˆ Polar Coordinates/ ReˆR + zkˆ veˆt aReˆ R + aθ eˆθ + azkˆ Cylindrical not used = (R¨ − Rθ˙2)eˆR + (Rθ¨ + 2R˙θ˙)eˆθ + z¨kˆ Coordinates at eˆt + aneˆn Path Coordinates = v˙eˆt + (v2/ρ)eˆn a P /F + a P/B + 2ωB × v P/B = Using data from a v P /F + v P/B = a P /F moving frame B with r¨ O /O + ωB × ωB × r P /O + ω˙ B × r P /O origin at O and r O /O + r P/O + B r¨ P/O +2ωB × v P/B angular velocity r˙ O /O + ωB × r P /O + B r˙ P/O a P/B relative to the fixed v P /F v P/B frame of ωB . The point P is glued to B and instantaneously coincides with P. ‘the 5-term acceleration formula’ Some facts about path coordinates The path of a particle is r (t). eˆt ≡ d r (s) , eˆt = d r (t) dt = v , κ ≡ d eˆt = d eˆt 1 , eˆn = κ | , eb ≡ eˆt ×eˆn, ρ = 1 | . ds dt ds v ds dt v |κ |κ Summary of the direct differentiation method P In the direct differentiation method, using moving frame B, we calculate v P by using a combination of the product rule of differentiation and the facts that ıˆ˙ = ωB × ıˆ , ˆ˙ = ωB × ˆ , and kˆ˙ = ωB × kˆ , as follows: vP = d y y' r P/O dt r P O r P x' d O' B = dt r O /O + r P/O = d (xıˆ + yˆ + zkˆ) + (x ıˆ + y ˆ + z kˆ ) r O /O dt F = (x˙ıˆ + y˙ˆ + z˙kˆ) + (x˙ ıˆ + y˙ ˆ + z˙ kˆ ) + x x (ωB × ıˆ ) + y (ωB × ˆ ) + z (ωB × kˆ ) but stop short of identifying these three groups of three terms as v P = v O /O + r˙ rel + ωB × r P/O . We could calculate a P similarly and would get a similar formula with 15 non-zero terms (3 for each term in the ‘five-term’ acceleration formula).

Table III Object [I ] Point mass [I cm ] = m 000 000 y 000 O x  −x y  z y2 + z2 x2 + z2 −x z −yz  [I O ] = m  −x y −yz x2 + y2 −x z [I cm ] =  y/2cm + z/2cm −x/cm y/cm   −x/cm y/cm −x/cm z/cm x/2cm + z/2cm −y/cm z/cm  d m −x/cm z/cm −y/cm z/cm x/2cm + y/2cm General 3D body If the axes are principal axes of the body. y [I cm ] = A0 0 With A, B, C ≥ 0 and A + 0B0 B ≥ C, B + C ≥ A, and 0 0C A + C ≥ B. O [I o] =  y/2o + z/2o −x/o y/o  z  −x/o y/o −x/o z /o x x/2o + z/2o −y/oz/o  d m −x/o z /o − y/o z /o x/2o + y/2o  yc2m/o + zc2m/o −xcm/o ycm/o  [I o] = [I cm ] + m  −xcm/o ycm/o −xcm /o z cm /o xc2m/o + zc2m/o −ycm/ozcm/o  −xcm /o z cm /o − ycm /o z cm /o xc2m/o + yc2m/o The 3D Parallel Axis Theorem  y/2cm −x/cm y/cm 0  −x/cm y/cm x/2cm [I cm ] =  0  d m 0 0 x/2cm + y/2cm Izczm General 2D Body If the axes are principal axes of the body. y [I cm ] = A0 0 With A+ B = C (The perpen- 0B0 dicular axis theorem). Also, 0 0C A ≥ 0, B ≥ 0. zO x  y/2o −x/o y/o   −x/o y/o x/2o 0 [I o] = 0 0 d m 0 x/2o + y/2o d  yc2m/o −xcm/o ycm/o 0 xc2m/o [I o] = [I cm ] + m  −xcm/o ycm/o 0  0 0 xc2m/o + yc2m/o d2 The 3D Parallel Axis Theorem. The 2D thm concerns the lower right terms of these 3 matrices. General moments of inertia. The tableshows a point mass, a general 3-D body, and a general 2-D body. The most general cases of the perpeendicual axis theorem and the parallel axis theorem are also shown..

Object Table IV Examples of Moment of Inertia [I ] Uniform rod Izczm = 1 2, [I cm ] = 1 2 000 m m 010 y 001 12 12 O x 1 1 000 z m m 010 IzOz = 2, [I O ] = 2 001 3 3 Uniform hoop  1 0 0 yR 0 Izczm = m R2, [I cm ] = m R2  2 1 x 2 z 0 001 Uniform disk  1 0 0 y 0 R Izczm = 1 m R2, [I cm ] = m R2  4 1 2 4 x 0 z 0 0 1 2 Rectangular plate  b2 0 0 y 0 a2 0 Izczm = 1 m(a2 + b2), [I cm ] = 1 m b 12 12 0 0 a2 + b2 z x a Solid Box  b2 + c2 0  0 a2 + c2 0 y [I cm ] = 1 m 0 12 0 0 a2 + b2 c b x z a Uniform sphere y R [I cm ] = 2 m R2 100 z 5 010 001 x Moments of inertia of some simple objects. For the rod both the [I cm] and [I O ] (for the end point at O) are shown. In the other cases only [I cm] is shown. To calculate [I O ] relative to other points one has to use the parallel axis theorem. In all the cases shown the coordinate axes are principal axes of the objects.