Category: Classical Mechanics

Thornton & Marion, Classical Dynamics, Fifth Edition Chapter 1. Matrices, Vectors, and Vector Calculus Problem 11. Triple scalar product The problem asks you to show that 1) triple scalar product can be written as 2) the product is unaffected by an interchange of operators or by a change in the order. 3) a geometrical interpretation,…

Thornton & Marion, Classical Dynamics, Fifth Edition Chapter 1. Matrices, Vectors, and Vector Calculus Problem 10. Differentiation of a vector The problem asks you to calculate vector derivatives to get velocity and acceleration of a particle. The problem gives position vector of a particle in a plain elliptical orbit. We should know about calculation of…

Thornton & Marion, Classical Dynamics, Fifth Edition Chapter 1. Matrices, Vectors, and Vector Calculus Problem 09. Vector calculation The problem asks you to evaluate some vector calculation. The problem gives Two vectors 1) 2) We should know about some vector calculus, such as 1) addition and subtraction 2) dot product and cross product Solution  

Thornton & Marion, Classical Dynamics, Fifth Edition Chapter 1. Matrices, Vectors, and Vector Calculus Problem 08. An equation of a plane in vector form The problem asks you to show that the given equation is the equation of a plane. This problem assumes 1) be a vector from the origin to a fixed point 2)…

Chapter 1. Matrices, Vectors, and Vector Calculus Problem 07. Scalar Product of two (unit) vectors The problem asks you to 1) Find the vectors describing the diagonals 2) The angle between diagonal vectors This problem assumes A unit cube in the Cartesian (Rectangular) coordinate system We should know about 1) Position vectors 2) Scalar Product…

Thornton & Marion, Classical Dynamics, Fifth Edition Chapter 1. Matrices, Vectors, and Vector Calculus Problem 06. Another proof of the orthogonality condition The problem asks you to prove that Equation (1.15) can be obtained by using the fact that the transformation matrix preserves the length of the line segment. This problem assumes that the coordinate…

Thornton & Marion, Classical Dynamics, Fifth Edition Chapter 1. Matrices, Vectors, and Vector Calculus Problem 05. The determinant of the transformation matrix The problem asks you to show that This problem assumes the transformation matrix to be a two-dimensional orthogonal matrix. You should know about 1) Calculation of a determinant 2) Properties of the orthogonal…

Chapter 01. Matrices, Vectors, and Vector Calculus Problem 04. Transposed and Inverse Matrix The problem asks you to prove the properties of transposed and inverse matrix. You should know about 1) Transposed matrix 2) Inverse matrix Solution (a) Let , then By definition of transpose, Therefore, (b) i) For orthogonal matrices, , so we can…

Chapter 01. Matrices, Vectors, and Vector Calculus Problem 03. Transformation Matrix The problem asks you to find a transformation matrix that satisfying some condition. This problem gives (or assumes) This matrix rotates a rectangular coordinate through an angle of 120 degrees about an axis making equal angles with the original one. You should know about…

Chapter 01. Matrices, Vectors, and Vector Calculus Problem 02. Trigonometric properties of direction cosines Solution For Equation (1.10): Proof) Using Figure 1-4 (a) in the book, let the length of a point P on the line from the origin be , then Therefore, it becomes For Equation (1.11): Proof) In Figure 1-4 (b) in the…