11.26** A bead of mass m is threaded on a frictionless circular wire hoop of radius R and mass m(same mass). The hoop is suspended at the point A and is free to swing in its own vertical plane asshown in Figure 11.20. Using the angles 1 and 2 as generalized coordinates, solve for the normalfrequencies of small oscillations, and find and describe the motion in the corresponding normal modes.[Hint: The KE of the hoop is 167, where I is its moment of inertia about A and can be found usingthe parallel axis theorem.]02mFigure 11.20 Problem 11.26

Answered: Image /qna-images/answer/2f13532c-1bd9-49ed-85c2-4f8fb45d4545.jpg

Answered: 11.26** A bead of mass m is threaded on…

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter11: Dynamics Of Rigid Bodies

Section: Chapter Questions

Problem 11.9P

See similar textbooks

Similar questions

The fixing element B receives a horizontal motion given by xB = bcos(wt). Derive the equation of motion for the mass m and define the critical angular frequency we at which the oscillations of the mass become excessively large. B xB= b cos cit
A particle of mass m is suspended from a support by a light string of length which passes through a small hole below the support (see diagram below). The particle moves in a vertical plane with the string taut. The support moves vertically and its upward displacement (measured from the ring) is given by a function z = h(t). The effect of this motion is that the string-particle system behaves like a simple pendulum whose length varies in time. I [Expect to use a few lines to answer these questions.] a) Write down the Lagrangian of the system. b) Derive the Euler-Lagrange equations. z=h(t) c) Compute the Hamiltonian. Is it conserved?
Prob.1 (i) State the required conditions of simple harmonic motion (SHH). (ii) Consider the torsional pendulum with a moment of inertia l and torsion constant K. If the pendulum starts its oscillation with an initial angle Oi and angular de velocity wj at t = 0. Obtain the equation of motion and angular frequency of dt oscillation w. for this pendulum and discuss that it can be classified as SHH. (iii) Show that the torsional angle and angular velocity o of the pendulum for all time can be expressed as O(t) = 0; cos wt + sin wt w(t) = -w0; sin wt +w¡ cos wt .
Prob.1 (i) State the required conditions of simple harmonic motion (SHH). (ii) Consider the torsional pendulum with a moment of inertia I and torsion constant K. If the pendulum starts its oscillation with an initial angle O, and angular at t = 0. Obtain the equation of motion and angular frequency of dt de velocity j %3D oscillation w. for this pendulum and discuss that it can be classified as SHH. Show that the torsional angle and angular velocity a of the pendulum for all time can be expressed as (iii) sin wt 0 (t) = 0; coS wt + w(t) = -w0; sin wt + w cos wt .
i need the answer quickly
A simple pendalum consists of a point mass m suspended by a (weight less cord or rod of length 1, as shown, and swinging in a vertical plane under the action of gravity. Show that for small oscillations (small 0), both 0 and r are sinusoidal functions of time, that is, the motion is simple harmonic. Hint: Write the differential Hquation F/= ma for the particle m. sin 0 0 for small 6, and show that = A sin wt is a solution of your equation. What are A and w? 13. Use the approximation
A particle of mass m is suspended from a support by a light string of length which passes through a small hole below the support (see diagram below). The particle moves in a vertical plane with the string taut. The support moves vertically and its upward displacement (measured from the ring) is given by a function z = h(t). The effect of this motion is that the string-particle system behaves like a simple pendulum whose length varies in time. I b) [Expect to a few lines to wer these questions.] a) Write down the Lagrangian of the system. Derive the Euler-Lagrange equations. z=h(t) Compute the Hamiltonian. Is it conserved?
For a simple pendulum of length l, the angle theta between rest position and deflected position is the only generalized coordinate. Kinetic energy of the bob of mass m is T = 1/2 ml2 theta prime2, and its potential energy, V = mgl (1- Cos theta). Display the Lagrangian, L of this system, and hence determine the equation of motion for the simple harmonic motion for the smal angle theta.
Quartic oscillations Consider a point particle of mass m (e.g., marble whose radius is insignificant com- pared to any other length in the system) located at the equilibrium points of a curve whose shape is described by the quartic function: x4 y(x) = A ¹ Bx² + B² B²), (1) Where x represents the distance along the horizontal axis and y the height in the vertical direction. The direction of Earth's constant gravitational field in this system of coordinates is g = −gŷ, with ŷ a unit vector along the y direction. This is just a precise way to say with math that gravity points downwards and greater values of y point upwards. A, B > 0. (a) Find the local extrema of y(x). Which ones are minima and which ones are maxima? (b) Sketch the function y(x). (c) What are the units of A and B? Provide the answer either in terms of L(ength) or in SI units. (d) If we put the point particle at any of the stationary points found in (a) and we displace it by a small quantity³. Which stationary locations…
The following problem involves an equation of the form = f(y). dy dt Sketch the graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one as asymptotically stable or unstable. Draw the phase line, and sketch several graphs of solutions in the ty-plane. dy = y(y − 1)(y — 2), yo ≥0 dt The function y(t) = 0 is no equilibrium solution at all. ▼ The function y(t) = 1 is Choose one The function y(t) = 2 is Choose one
Send answers
An object attached to a spring vibrates with simple harmonic motion as described by the figure below. A coordinate plane is shown with t (s) on the horizontal axis and x (cm) on the vertical axis. A curve is shown to make one and a half complete oscillations along t. The curve begins at the origin moving with a steep slope. The curve is moving with increasing x and decreasing slope until it is horizontal and at its maximum at (1, 2). From (1, 2) the slope of the curve becomes negative and steadily decreases until it crosses the t-axis at (2, 0) with a steep negative slope. From (2, 0) the curve continues below the t-axis with increasing slope until it is horizontal and at its minimum at (3, −2). From (3, −2) the slope of the curve steadily increases until the curve crosses the t-axisat (4, 0) with a steep slope. From (4, 0) one oscillation is complete and the curve repeats the same pattern, decreasing slope until the maximum at (5, 2) and continuing decreasing slope until crossing the…

SEE MORE QUESTIONS

Recommended textbooks for you

Classical Dynamics of Particles and Systems

Physics

ISBN:

9780534408961

Author:

Stephen T. Thornton, Jerry B. Marion

Publisher:

Cengage Learning

Classical Dynamics of Particles and Systems

Physics

ISBN:

9780534408961

Author:

Stephen T. Thornton, Jerry B. Marion

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS