Particle on a Helical Wire A smooth wire is bent into the shape of a helix5.with cylindrical polar coordinates p= R and : = dó, where R and A are constants andthe z axis is vertically up (and gravity vertically down). Using z as your generalizedcoordinate, write down the Lagrangian for a bead of mass m threaded on the wire. Findthe Lagrange equation and hence the bead's vertical acceleration . In the limit thatR 0, what is 2? Does this make sense?

Write an expression for Lagrangian in terms of cylindrical polar co-ordinates

Answered: Particle on a Helical Wire A smooth…

Similar questions

Please find the answer using [ijk] notation and matrices. At the very least find the inertia about C. Please use parallel axis theorem. 1. Calculate the moments of inertia of the system consisting of a slender, ho- mogeneous straight rod of length I and mass m with a thin disk of radius r and mass m about a) A and b) C, respectively. Use xyz - frame as the system - xed frame. Thin disk is rigidly attached to the rod perpendicularly and C is the center of the disk. Assume the constant density for the rod and disk system. In the gure, the rod is on the zy - plane and has an angle \theta relative to the y-axis Z Rod A 日 Thin disk +- y Figure 1: Rod with disk attached
Question 4 A particle of mass m moves in a central force field with potential V(r). The La- grangian in terms of spherical polar coordinates (r,0,¢) is 1 L = ;m (² +r²e² + ² sin² 0¢? ) – V(r). 1. Find the momenta (pr, pe, pp) conjugate to (r,0,4). 2. Find the Hamiltonian H(r,0,4,,pr, Po, Pp). 3. Write down the explicit Hamiltonian's equations of motion. ? Answer.
**Problem 3.23 Consider the following Hermitian matrix: 2 i 1 T = -i i 1 -i 2 (a) Calculate det(T) and Tr(T). (b) Find the eigenvalues of T. Check that their sum and product are consistent with (a), in the sense of Equation 3.82. Write down the diagonalized version of T. (c) Find the eigenvectors of T. Within the degenerate sector, construct two linearly independent eigenvectors (it is this step that is always possible for a Hermitian matrix, but not for an arbitrary matrix-contrast Problem 3.18). Orthogonalize them, and check that both are orthogonal to the third. Normalize all three eigenvectors. (d) Construct the unitary matrix S that diagonalizes T, and show explicitly that the similarity transformation using S reduces T to the appropriate diagonal form.
Express the Lagrangian for a free particle moving in a plane in a plane polar coordinates. From this proves that, in terms of radial and tangential components, the acceleration inpolar coordinates isa = (¨r − rθ˙2) er + (rθ¨ + 2 r˙ θ˙) eθ(where er and eθ are unit vectors in the positive radial and tangential directions).
For the mass distributions in Problems 5 to 7, find the inertia tensor about the origin, and find the principal moments of inertia and the principal axes. 5. Point masses 1 at (1, 1, 1) and at 1, 1, 1). - 6. Point masses 1 at (1,1, –2) and 2 at (1, 1, 1). 7. Mass of uniform density 1, bounded by the coordinate planes and the plane x + y + z = 1.
Write down the inertia tensor for a square plate of side ? and mass ? for a coordinate system with origin at the center of the plate, the z-axis being normal to the plate, and the x- and y- axes parallel to the edges.
For the frame of Prob. 7.17, determine the magnitude and location of the maximum bending moment in member BC.(Reference to Problem 7.17):A 5-in.-diameter pipe is supported every 9 ft by a small frame consisting of two members as shown. Knowing that the combined weight of the pipe and its contents is 10 lb/ft and neglecting the effect of friction, determine the magnitude and location of the maximum bending moment in member AC.
Using Lagrangian formalism, solve the following problem: A solid cylinder is released from rest to roll without slipping down a ramp of slope ϕ. a) What is the best choice of generalized coordinates to be used?
Answer number 7 Please use any neceecasary diagrams to further explain steps in soluton
Consider a 3-body system again. This time, let the particles be the Sun, the Earth, and the Moon. (a) Write the relative vector equations of motion for the Moon relative to the Earth. (b) At the very instance where the following right angle triangle geometry is achieved, compute the dominant and perturbing accelerations. Compare the magnitudes of both accelerations, is it reasonable to model the Moon's motion by considering a two-body problem with the Earth? Explain your answer.