Download A Short Introduction to Theoretical Mechanics by A. Mous PDF

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By A. Mous

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Don’t forget the particular solution and the homogeneous solution. Convince the reader that your answer is correct. 8. Find the Fourier series expansion of the function, F0 ωt − F0 π F (t) = Ans: 0 ≤ ωT < 2π. ∞ F (t) = − 9. 1 2F0 sin nωt. πn • Write down a particular solution, with any constants expressed explicitly in terms of ω, k,m,b, and F0 , to the equation ∞ m¨ x + bx˙ + kx = − 1 2F0 sin nωt. πn • What is the solution to the homogeneous equation in terms of two arbitrary constants? ) 10. Find the one-dimensional motion of a particle if its potential energy is given by V (x) = −k0 x x<0 1 V (x) = k1 x2 2 x>0 Let the initial displacement be x2 > 0 and the initial velocity be zero.

E. ) • Let a particle move on a circle of constant radius b. What is the radial component of acceleration (physical)? What is the transverse component? • A bug crawls outward with constant speed v0 along the spoke of a wheel which is rotating with constant angular speed ω. Find the radial and transverse components of the physical acceleration as functions of time. Assume r = 0 at t = 0. (Ans: ar = −v0 tω 2 , aθ = 2v0 ω) 2. Cylindrical coordinates are defined by x = r cos θ, y = r sin θ, z = z. Calculate the metric tensor, physical velocity and physical acceleration in cylindrical coordinates.

Computer Project 4 (See Appendix) 19. Computer Project 5 (See Appendix) 20. Computer Project 6 (See Appendix) 21. 1 Definitions and Theorems Consider a particle which moves in a two or three-dimensional region of space under the influence of a force F. Consider two points, a and b in the region of space. 1 Definition The net force is the vector sum of all forces acting on the particle. 2 b a F · ds. Theorem Let a and b refer to any two points on the trajectory of a particle. The work done between these two points by the net force F(x, v, t) is equal to the increase in the kinetic energy of the particle.

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