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Concept of a Dyad and Dyadic: Consider Two Vectors a and B Dyad: It Consists of a Pair of Vectors a B for Two Vectors a a N D B

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1/11/2010 CHAPTER 1 Introductory Concepts • Elements of Vector Analysis • Newton’s Laws • Units • The basis of Newtonian Mechanics • D’Alembert’s Principle 1 Science of Mechanics: It is concerned with the motion of material bodies. • Bodies have different scales: Microscropic, macroscopic and astronomic scales. In mechanics - mostly macroscopic bodies are considered. • Speed of motion - serves as another important variable - small and high (approaching speed of light). 2 1 1/11/2010 • In Newtonian mechanics - study motion of bodies much bigger than particles at atomic scale, and moving at relative motions (speeds) much smaller than the speed of light. • Two general approaches: – Vectorial dynamics: uses Newton’s laws to write the equations of motion of a system, motion is described in physical coordinates and their derivatives; – Analytical dynamics: uses energy like quantities to define the equations of motion, uses the generalized coordinates to describe motion. 3 1.1 Vector Analysis: • Scalars, vectors, tensors: – Scalar: It is a quantity expressible by a single real number. Examples include: mass, time, temperature, energy, etc. – Vector: It is a quantity which needs both direction and magnitude for complete specification. – Actually (mathematically), it must also have certain transformation properties. 4 2 1/11/2010 These properties are: vector magnitude remains unchanged under rotation of axes. ex: force, moment of a force, velocity, acceleration, etc. – geometrically, vectors are shown or depicted as directed line segments of proper magnitude and direction. 5 e (unit vector) A A = A e – if we use a coordinate system, we define a basis set (iˆ , ˆj , k ˆ ): we can write A = Axi + Ay j + Azk Z or, we can also use the A three components and Y define X T {A}={Ax,Ay,Az} 6 3 1/11/2010 – The three components Ax , Ay , Az can be used as 3-dimensional vector elements to specify the vector. – Then, laws of vector-matrix algebra apply. – Tensors: scalar - an array of zero dimension vector - an array of one dimension 7 – quantities which need arrays of two or higher dimension to specify them completely - called tensors of appropriate rank. Again - to be a tensor, the object must also satisfy certain transformation properties of rotation and translation. Exs: Second-order tensors: stress at a point in deformable body - stress tensor has nine components (a 3x3 matrix in a representation when the basis is defined), inertia tensor (again, a 3x3 matrix in usual notation) expressing mass distribution in a rigid body.8 4 1/11/2010 • TYPES OF VECTORS: Consider a force F acting on a body at point B P. The force has a line F of action AB. This force P can lead to translation of the rigid body, rotation A of the rigid body about some point, as well as deformation of the body. 9 F F or P2 P1 The same force F is now acting at two different points P1 , P2 of the body, i.e., the lines of action are distinct. – same translational effect – the translational effect depends only on magnitude and direction of the force, not on its point of application or the line of action- free vectors 10 5 1/11/2010 B B F P1 F or P A A 2 The force F has the same line of action AB in the two cases. The points of application (P1 and P2) are different but moment about every point is the same → same rotational effect (as well as translational effect): effect of vector F depends on magnitude, direction as well as line of action - sliding vectors 11 If the body is deformable, the effect of force is different depending its point of application; whether the force acts at point P1 or P2. Thus, in such a case, the point of application is also crucial - bound vectors. B B F P or 1 F P A A 2 deformation 12 6 1/11/2010 • Equality of vectors: For free vectors A a n d B , A = B if and only if A a n d B have the same magnitude and the same direction. • Unit vectors: If A is a vector with magnitude A, A / A is a vector along A with unit length → e A = A /A or A =AeA. 13 e unit vector in A |1| eA the direction A = A e of A. • Addition of vectors: Consider two vectors A and B. Their addition is a vector C given by C = A + B . Also C = B + A (addition is commutative). The result is also a vector. 14 7 1/11/2010 C = A+B Graphically, one can use the parallelogram rule of vector addition. B C B For more than two vectors, one can add A sequentially - polygon of vectors. Consider The addition of vectors A, B, and D . C = A + B , E = (A + B) + D = C + D D or B E E = (A+ D)+ B C B = A+(B+ D). O A 15 • COMPONENTS OF A VECTOR: Consider the vector addition for A and B: graphically: C = A + B C B O A We can interpret A and B to be components of the vector C . Clearly, the components of C are non-unique. As an F B example, E and F are E C also components. O A We can make it more systematic. 16 8 1/11/2010 • Let e 1 , e 2 , e 3 - three linearly independent unit vectors (not necessarily orthogonal) and let A be a vector. We can write A = A 1 + A 2 + A 3 where Ai , i = 1, 2, 3 are components of A along the directions specified by unit vectors e i ,i = 1, 2, 3. e3 A Then : e 2 A3 A2 A= A1e1 +A2e2 +A3e3 A1 e1 A , i = 1, 2, 3 are unique scalar components i 17 Let B = B 1 e 1 + B 2 e 2 + B 3 e 3 be another vector, with components expressed in same unit vectors. We can then write the sum as C= A+B=(A1 +B1)e1 +(A2 +B2)e2 + (A3 + B3)e3 The components of the vector C are then → C1 = A1 + B1 , C2 = A2 + B2 and C3 = A3 + B3. 18 9 1/11/2010 • The more familiar case of unit vectors is the Cartesian coordinate system - (x, y, z) Let i , j , k - unit vectors along x, y and z directions. Then A=Ax i+Ay j+A2 k where Ax , Ay , Az are components of A Z A k along axes. z A Ay j Y Ax i X 19 •SCALAR PRODUCT: Definition: (DOT) For two vectors A and B, the dot product is defines as A ⋅ B = A B c o s θ Dot product is Commutative, i.e., A ⋅B = B⋅ A If 3 3 B A = Ai ei , B = Bi ei , i=1 i=1 θ A then A⋅B = A1B1 + A2B2 + A3B3 |B|cosθ provided the unit vectors are an orthogonal set, i.e., e1 ⋅ e 2 = e 2 ⋅ e3 = e3 ⋅ e1 = 0. 20 10 1/11/2010 •VECTOR PRODUCT: Let A , B be two vectors that make an angle θ with respect to each other. Then, the vector or cross product is defined as a vector C with magnitude C = A× B = A B sinθ . B Let k be the unit vector normal to the k θ plane formed by vectors A and B. It is A fixed by the right hand screw rule. Then A× B = A B sinθk 21 Some properties of cross product are : A×B=-B×A Consider unit vectors for the Cartesian coordinate system (x,y,z), (i, j, k): Then i i × j = j × k = k × i = 1 i × i = j × j = k × k = 0 k j i × j = k ; j× k = i right-hand rule k × i = j 22 11 1/11/2010 Now, consider cross-product again. When vectors A and B are expressed in component form: A=Ax i +Ay j+Az k B= Bxi+By j+Bzk, The cross product is evaluated by the operation i j k A × B = A x A y A z B x B y B z ≡ (Ay Bz − Az By )i + (Az Bx − Ax Bz ) j + (Ax By − Ay Bx )k 23 • SCALAR TRIPLE PRODUCT: Consider three vectors A, B, and C. The scalar triple product is given by R=A •(B×C) = (A×B)• C Note that the result is the same scalar quantity. It can be interpreted as the A volume of the C parallelepiped having the vectors A , B a n d C as B the edges. The sign can be +ve or -ve. 24 12 1/11/2010 • VECTOR TRIPLE PRODUCT: Consider vectors A, B, and C. Then, vector triple product is defined as a vector D, given by D = A × ( B × C ) Note that A×(B×C) ≠ (A×B)×C One can show that D=A×(B×C)=(A•C)B-(A•B)C • DERIVATIVE OF A VECTOR: Suppose that a vector A is a function of a scalar u, i.e., A = A ( u ) . We can then consider change in vector A associated with change in the scalar u. 25 Let A = A ( u ) and A (u + ∆u) ≡ A(u) + ∆A dA A(u) + ∆A -A(u) Then = lim du ∆u→0 ∆u dA ∆A or = lim du ∆u→0 ∆u This is the derivative of A with respect to u. Ex: The position vector r(t) for a particle moving depends on time. We define the dr ∆r velocity to be v(t) ≡ = lim dt ∆t→0 ∆t 26 13 1/11/2010 s ex: O’ z ∆r O P P’ r (s) x y OP ∆s Consider a particle P moving along a curved path.
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