Yoyo The figure below shows a crude model of a yoyo. A massless string 4. suspended vertically from a fixed point and the other end around a uniform cylinder of mass m and radius R. When the cylinder is released it. moves vertically down, rotating as the string unwinds. Write down the Lagrangian, using the distance x as your generalized coordinate. Find the Lagrange equation of motion and show that the cylinder accelerates downward with 2q/3. wrapped several times 1 Hints: You need to remember from your introductory physics course that the total kinetic energy of a body like the yoyo is T = mu Iw, where v is the velocity of the center of mass, I is the moment of inertia (for a uniform cylinder, I =mR2) and w is the angular velocity about the CM. You can express w in terms of å.] x R

Yoyo The figure below shows a crude model of a yoyo. A massless string 4. suspended vertically from a fixed point and the other end around a uniform cylinder of mass m and radius R. When the cylinder is released it. moves vertically down, rotating as the string unwinds. Write down the Lagrangian, using the distance x as your generalized coordinate. Find the Lagrange equation of motion and show that the cylinder accelerates downward with 2q/3. wrapped several times 1 Hints: You need to remember from your introductory physics course that the total kinetic energy of a body like the yoyo is T = mu Iw, where v is the velocity of the center of mass, I is the moment of inertia (for a uniform cylinder, I =mR2) and w is the angular velocity about the CM. You can express w in terms of å.] x R

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

See similar textbooks

Similar questions

Assume that you do not know about the kinetic energy or Newton’s Laws of motion. Suppose instead of deriving the Euler-Lagrange equations, we postulate them. We define the basic law of mechanics to be these equations and ask ourselves the question: What is the Lagrangian for a free particle? (This is a particle in empty space with no forces acting on it. Be sure to set up an inertial reference system.) The simplest choice might be to guess it must be proportional to v2,where v→ is the particle velocity in an inertial frame K. Take L = v2. A second inertial frame K′ moves at constant velocity −V0→ with respect to K, so that the transformation law of velocities isv'→ = v→ + V0→. Prove that L' = v'2 is a possible choice for the Lagrangian in the frame K′. Explain how this proves that all inertial frames are equivalent. With this approach we prove the equivalence of inertial frames from the form of the Lagrangian, instead of postulating this equivalence at the start, which is usual way…
Write generalized coordinates and Lagrange equation and equation of motion for this system. k C.G.
A force of F = 30 N is used to drag a crate d = 4 m across a floor. The force is directed at an angle upward from the crate so that the vertical component of the force is Fv = 18 N and the horizontal component is Fh = 24 N a) What is the work done by the horizontal component of the force? (b) What is the work done by the vertical component of the force? (c) What is the total work done by the 30-N force?
Using Lagrange's equation, obtain the equation of motion for a particle in plane polar coordinates.
When a light ray passes from a layer with index n1 = 1.5 to an adjacent layer with index n2, the ray deviates 10° from its initial direction as shown below. If the angle of incidence is e1 = 30°, then determine the value of the index n2. n2 n2 = 1.05 n2 = 1.26 n2 = 1.17 n2 = 1.48 n2 = 1.37
Suppose a mass M is attached to a rod of length L (also mass M) and free to pivot around a frictionless pin. The rod hangs vertically in gravity 9 as a (massless) horizontal spring with spring constant is connected to the midpoint of the rod, and connected to the wall at the opposite end. a) What is the Lagrangian L (0,0) of this system? Use the angular displacement to characterize the rod's orientation. b) Apply the Euler-Lagrange equation to get a 2nd-order differential equation in 8. c) Now assume is small. Show that the equation reduces to the harmonic oscillator equation + w²0=0 d) If you release the mass from rest after displacing it a small amount, at what frequency would the system oscillate?
Please prove the highlighted sections are true.
Derive using the Lagrangian and Lagrange's equation only. Please draw a diagram
How many generalised coordinates does an object in free fall have. Explain using the lagrangian theory
Problem 3 A ball of mass m, slides over a sliding inclined plane of mass M and angle a. Denote by X, the coordinate of O' with re- m spect to 0, and by (x.y) the Coordinate of m with respect M a O' to O'(see figure below) 1. Calculate the degree of freedom of the system 2. Find the velocity of m with respect to O. 3. Write the expression of the Lagrangian function 4. Derive the Euler Lagrange equations 5. Find z" and X" in ters of the masses (m,M), angle a and g
Lagrangian Dynamics Ep = 0 A pendulum of length / and mass m is mounted on a block of mass M. The block can move freely without friction on a horizontal surface as shown in the adjacent figure H. 1. Find the velocity of mass m, w.r.t the origin O 2. Write the Lagrangian of the system 3. Derive the Euler Lagrange equations
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?