Chapter Review PDF

Chapter Review 81 CHAPTER REVIEW A scalar is a positive or negative number; e.g., mass and temperature. A 2 A vector has a magni...

Chapter Review 81 CHAPTER REVIEW A scalar is a positive or negative number; e.g., mass and temperature. A 2 A vector has a magnitude and direction, where the arrowhead represents the sense of the vector. Multiplication or division of a vector by a scalar will change only the magnitude of 2A the vector. If the scalar is negative, the 1.5 A sense of the vector will change so that it A acts in the opposite sense. 0.5 A If vectors are collinear, the resultant is R R = A + B simply the algebraic or scalar addition. A B Parallelogram Law Two forces add according to the parallelogram law. The components form the sides of the parallelogram and the resultant is the diagonal. a Resultant FR F1 To find the components of a force along b F2 any two axes, extend lines from the head Components of the force, parallel to the axes, to form the components. To obtain the components of the FR = 2F 21 + F22 - 2 F1F2 cos uR FR F1 resultant, show how the forces add by u2 tip-to-tail using the triangle rule, and F1 F2 FR u1 uR then use the law of cosines and the law of = = sin u1 sin u2 sin uR F2 sines to calculate their values.  82 C h a p t e r 2    F o r c e V e c t o r s Rectangular Components: Two Dimensions y Vectors Fx and Fy are rectangular components of F. 22 F Fy x Fx The resultant force is determined from the algebraic sum of its components. y y F2y (FR)x = Fx F1y FR (FR)y (FR)y = Fy F2x F1x u x  x FR = 2(FR)2x + (FR)2y F3x (FR)x (FR)y F3y u = tan-1 2 2 (FR)x Cartesian Vectors F F F The unit vector u has a length of 1, no units, and u= u it points in the direction of the vector F. F 1 A force can be resolved into its Cartesian components along the x, y, z axes so that z F = Fx i + Fy j + Fz k. Fz k F The magnitude of F is determined from the F= 2Fx2 + Fy2 + Fz2 positive square root of the sum of the squares of u its components. g b a Fy j y The coordinate direction angles a, b, g are F Fx Fy Fz u= = i + j+ k determined by formulating a unit vector in the F F F F Fx i direction of F. The x, y, z components of u = cos a i + cos b j + cos g k u represent cos a, cos b, cos g. x  Chapter Review 83 The coordinate direction angles are related so that only two of the three cos2 a + cos2 b + cos2 g = 1 angles are independent of one another. To find the resultant of a concurrent force 2 FR = F = Fx i + Fy j + Fz k system, express each force as a Cartesian vector and add the i, j, k components of all the forces in the system. z Position and Force Vectors (zB  zA)k A position vector locates one point in space r = (x B - x A )i relative to another. The easiest way to B + (y B - y A )j formulate the components of a position r vector is to determine the distance and + (z B - z A )k A direction that one must travel along the y x, y, and z directions—going from the tail to (xB  xA)i (yB  yA)j the head of the vector. x z If the line of action of a force passes r F F = Fu = F a b through points A and B, then the force r r B acts in the same direction as the position vector r, which is defined by the unit u vector u. The force can then be expressed A as a Cartesian vector. y x Dot Product A The dot product between two vectors A and B yields a scalar. If A and B are A # B = AB cos u expressed in Cartesian vector form, then u the dot product is the sum of the products = A xB x + A yB y + A zB z B of their x, y, and z components. A#B A The dot product can be used to determine A u = cos-1 a b the angle between A and B. AB u ua a a Aa  A cos u ua The dot product is also used to determine the projected component of a Aa = A cos u ua = (A # u a)u a vector A onto an axis aa defined by its unit vector ua.