**Quartic Oscillations**Consider a point particle of mass \( m \) (e.g., marble whose radius is insignificant compared to any other length in the system) located at the equilibrium points of a curve whose shape is described by the quartic function:\[y(x) = A \left( \frac{x^4}{4} - Bx^2 + B^2 \right), \quad A, B > 0. \tag{1}\]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 \( \vec{g} = -g\hat{y} \), with \( \hat{y} \) 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)** 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 quantity \(\delta\). Which stationary locations will show oscillatory motion with a small amplitude? Provide an explanation with words based on the figure in (b). No need to use equations ... yet. Neglect friction.

Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Answered: Consider a point particle of mass m…

Similar questions

Let G(u, v) = (3u + v, u - 2v). Use the Jacobian to determine the area of G(R) for: (a) R = [0, 3] x [0, 5] (b) R = [2, 5] x [1, 7]
Find a vector equation for the line through the point (-4, 8, 2) perpendicular to the plane –7x – 7y + 7z = 16. with -o < t<∞ 7(1) =
A block of mass m = 240 kg rests against a spring with a spring constant of k = 550 N/m on an inclined plane which makes an angle of θ degrees with the horizontal. Assume the spring has been compressed a distance d from its neutral position. Refer to the figure. (a) Set your coordinates to have the x-axis along the surface of the plane, with up the plane as positive, and the y-axis normal to the plane, with out of the plane as positive. Enter an expression for the normal force, FN, that the plane exerts on the block (in the y-direction) in terms of defined quantities and g. (b) Denoting the coefficient of static friction by μs, write an expression for the sum of the forces in the x-direction just before the block begins to slide up the inclined plane. Use defined quantities and g in your expression. (c) Assuming the plane is frictionless, what will the angle of the plane be, in degrees, if the spring is compressed by gravity a distance 0.1 m? (d) Assuming θ = 45 degrees and the…
Consider the schematic of the single pendulum. M The kinetic energy T and potential energy V may be written as: T = ²m²²8² V = -gml cos (0) аас dt 80 The Lagrangian L is given by L=T-V, and the Euler-Lagrange equations for the motion of the pendulum are given by the following second order differential equation in : ас 80 = 11 = 0 Write down the second order ODE using the specific T and V defined above. Please write this ODE in the form = f(0,0). Notice that this ODE is not linear! Now you may assume that l = m = g = 1 for the remainder of the problem. You may still suspend variables to get a system of two first order (nonlinear) ODEs by writing the ODE as: w = f(0,w) What are the fixed points of this system where all derivatives are zero? Write down the linearized equations in a neighborhood of each fixed point and determine the linear stability. You may formally linearize the nonlinear ODE or you may use a small angle approximation for sin(0); the two approaches are equivalent.
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?
Note: Because the argument of the trigonometric functions in this problem will be unitless, your calculator must be in radian mode if you use it to evaluate any trigonometric functions. You will likely need to switch your calculator back into degree mode after this problem.A massless spring is hanging vertically. With no load on the spring, it has a length of 0.24 m. When a mass of 0.59 kg is hung on it, the equilibrium length is 0.98 m. At t=0, the mass (which is at the equilibrium point) is given a velocity of 4.84 m/s downward. At t=0.32s, what is the acceleration of the mass? (Positive for upward acceleration, negative for downward)
Calculate the energy, corrected to first order, of a harmonic oscillator with potential:
(a) What does the quadrupole formula (P) = = = (Qij Q³ ³) compute? Reason the answer. (b) A point mass m undergoes a harmonic motion along the z-axis with frequency w and amplitude L, x(t) = y(t) = 0, z(t) = L cos(wt). Show that the only non-vanishing component of the quadrupole moment tensor is = Im L² cos² (wt). (c) Use the quadrupole formula to compute the power radiated by the emission of gravitational waves. (Hint: recall that (cos(t)) = (sin(t)) = 0 and (cos² (t)) = (sin² (t)) = ½½ for a given frequency 2.)