The rigid rotor is formulated in terms of the angles φ and θ and a fixed...
Prob. 4 Assume that on the xy-plane, vectors P and Q make angles θ and φ with respect to the r-axis. Use the basic properties of the dot-product of vectors, show that cos(θ + φ)-cos θ cos φ-sin θ sin φ. Also, use the basic properties of the cross-product, show that sin (e+ φ)-cos θ sin o + cos θ sin o.
Determine the angles θ and φ so that the resultant force is directed along the positive x axis and has a magnitude -20 N. J0 N 30 N
2. A rigid rotor is constrained to move about a fixed axis, with respect to which its moment of inertia is I. The time-independent Schrödinger equation for this system is: 21 do where W is the energy, ф is the angular displacement of the rotor from a fixed direction (0 ps 2 ) , and 111(pf dф gives the probability of finding the rotor at an angular position between ф and ф+49 State the general solution of this equation and...
Coordinate system in rotation. Consider the Minkowski space in spherical coordinates (t, r, θ, φ) and perform a coordinate transformation to a rotational system given by t '= t, r' = r, θ '= θ, φ' = φ + ωt. (a) Find the metric in the new coordinates and all the Christoffel symbols. (b) Take θ' = θ = π/2. Write the equations of the geodesic and compare with the equations d²xi'/ dt'² = f^i, find the value of the...
All multiple choice questions pertain to the same question below. A rigid rotor is spinning in free space. At its maximum z-velocity, our rotor attains an me = 15. b) What is the total angular momentum for our rotor? O a. 15.498 b.240 R O C. 15 h O d. V2208
1.18. Points P and P' have spherical coordinates (r,0,y) and (r,θ,φ), cylindrical coordinates (p, p, z) and (p',p',z'), and Cartesian coordinates (x, y, z) and (x',y',z'), respectively. Write r - r in all three coordinate systems. Hint: Use Equation 1.2) with a r r and r and r' written in terms of appropriate unit vectors.
From Acheson: Elementary Fluid Dynamics
Equation 7.3 and 6.12 (Slow Flow Equations)
7.2. A rigid sphere of radius a is immersed in an infinite expanse of viscous fluid. The sphere rotates with constant angular velocity Ω. The Reynolds number R-Ωα2/v is small, so that the slow flow equations apply (see eqns (7.3) and (6.12)). Using spherical polar coordinates (r, θ, φ) with θ-0 as the rotation axis, show that a purely rotary flow u us(r, e, s possible provided that...
(a) Consider a particle which starts moving around from the origin in a 3-dimensional space. De- termine the velocity vector v(t) in terms of φ and θ if it is constantly moving at the speed 5m/s, along the direction (φ,0). Here, φ denotes the angle between the z-axis and the projection of the position vector r(t) on the xy-plane; meanwhile θ denotes the angle between the z-axis and r(t). You may assume that (φ, θ) are fixed over time at...
2. Consider a particle of mass M attached to a rigid massless rod of fixed length R whose other end is fixed at the origin. The rod is free to rotate about its fixed point. (a) Give an argument why the Hamiltonian for the system may be written as 21 21 with/-MR2 (b) If the particle carries charge q, and the rotor is placed in a constant magnetic field B, what is the modified Hamiltonian? (e) What is the energy...
1- 5. Two particles each of mass m are fixed at the end of a rigid rod of length 2a. This rod lies in the xy plane and is free to rotate in that plane about an axis passing through the midpoint of the rod and perpendicular to it (that is, parallel to the z-axis). Neglect the inertial properties of the rod in the rest of this question z-axis 1. Derive the classical expression for the kinetic energy of the...