Mathematics of Physics and Engineering
66 Systems of Point Masses rotation of the frame 0\ relative to this translated frame O'. We can now combine relation (2.1.44) f ...
Non-Rigid Systems of Points 67 with position vector 1 n We will see that some information about the motion of S can be obtained ...
68 Systems of Point Masses Typically, each Fj is a sum of an external force F\ ' from outside of S and an internal system force ...
Non-Rigid Systems of Points 69 From (2.2.7) it follows that AT n n 3 = 1 j=l 3 = 1 since rj x rj = 0. Now, we again assume that ...
70 Systems of Point Masses where TJE) = rj x Ff] is the external torque of FJ£) and T{ 0 E) is the total torque on the system S ...
Non-Rigid Systems of Points 71 where Vj(t) = rj(t) - rcM(t). Hence, (2.2.13) becomes L 0 = M rCM x rCM + LCM, (2.2.15) that is, ...
(^72) Systems of Point Masses Tj — rfe, we therefore find ^f = tJ x Ff> = £T&, = T&, (2.2.18) where TCM, is the exter ...
Rigid Systems of Points 73 (1738-1822), who also discovered the planet Uranus (1781) and infra-red radiation (around 1800). 2.2. ...
74 Systems of Point Masses system. Define the corresponding rotation vector u> according to (2.1.39) on page 63. Let us apply ...
Rigid Systems of Points 75 and it is therefore natural to introduce the quantity ICM = 2 mjll*j||^2! which is called the moment ...
76 Systems of Point Masses It is therefore natural to introduce the following notations: n n n j=i j=\ j=\ n n •*xy = -*yx = / j ...
Rigid Systems of Points 77 Remembering that our goal is an equation of the type (2.2.27), we have to continue the investigation ...
(^78) Systems of Point Masses EXERCISE 2.2.11. Consider four identical point masses m at the vertices of a square with side a. D ...
Rigid Bodies 79 the principal axes frame, we have DLcM{t) = LCMx{t) i 4- LCMy(t) j + LCMz(t)k. On the other hand, since the matr ...
(^80) Systems of Point Masses body occupies the region H in R^3 , then the mass M of the body is given by the triple (or volume) ...
Rigid Bodies 81 where v(t) = r(t) - rcM(t), and is (2.2.14) replaced by LCM(t) = llf(t) x «() P(r(t)) dV, (2.2.40) TC(t) which i ...
(^82) Systems of Point Masses tion p. The principal axes frame, with center at the center of mass and basis vectors i, j, k, is ...
Rigid Bodies 83 EXERCISE 2.2.16.C Verify that the vectors i, j, k define a principal axis frame. Hint: Iy = Iz = Iz = 0, as seen ...
84 Systems of Point Masses If a « 0 and 6 is so small that sin# « #, then equation (2.2.45) becomes {21/3)0+ g0 = 0. (2.2.46) Co ...
Lagrange's Equations 85 2.3.1 Lagrange's Equations We first illustrate Lagrange's method for a point mass moving in the plane. T ...
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