Answered: Problem 5: Consider a 1D simple… | bartleby

Related questions

Question

Problem 5: Consider a 1D simple harmonic oscillator (without damping). (a) Compute
the time averages of the kinetic and potential energies over one cycle, and show that they
are equal. Why does this make sense? (b) Show that the space averages of the kinetic and
potential energies are
(T)₂ = k1²
KA² and (U),= KA².
Why is this a reasonable result?

Expert Solution

Step by step

Solved in 6 steps with 5 images

SEE SOLUTION Check out a sample Q&A here

Blurred answer

Similar questions

Show that the function x(t) = A cos ω1t oscillates with a frequency ν = ω1/2π. What is the frequency of oscillation of the square of this function, y(t) = [A cos ω1t]2? Show that y(t) can also be written as y(t) = B cos ω2t + C and find the constants B, C, and ω2 in terms of A and ω1
Two blocks of masses m1=1.0 kg and m2=3 kg are connected by an ideal spring of force constant k=4 N/m and relaxed length L. If we make them oscillate horizontally on a frictionless surface, releasing them from rest after stretching the spring, what will be the angular frequency ω of the oscillation? Choose the closest option. Hint: Find the differential equation for spring deformation.
Coupled Harmonic Oscillators X2 :0 eeeee Leeee NM ) Write down the 2nd law for each of the masses. Use coordinates x, aud x, for m and M, and i,, *2 and a, a2. Hirnt: the force from spring I on monly depends on x, , but the ferce from spring 2 on m depand s on (x2-x,) two differantial 2) To simplify, let k, -k,k and m-M. Rewrite egg equatious from i) right above each other. your 3) Detine two new variables, X (Greek"chi") - x,+X2 and Ax = X,-x 2 two di ferential equations to produce Then add and subtract your very simple (harmanic) ones. 4) Write down the solutiou to bothe equations using Wask+2k2 ond evefficients A, B, C, and D two new , but w. : un 5) Now assune k-10k2 (this means that the two masses are "weakly coupled"). Also assume X,(0) = -10cm, x,(0)=0, xz (0)= o, x,(0)=0. Solve for A,B, C, D, and solve for x (t).
Consider a thin hoop of mass (1.420 ± 0.001) kg and radius (0.250 ± 0.002) m. The moment of inertia for a thin hoop rotating about an axis going through its center is MR2 . Calculate the moment of inertia of this hoop and its uncertainty using error propagation rules.
Sketch f(y) versus y. Show that y is increasing as a function of t for y < 1 and also for y > 1. The phase line has upward-pointing arrows both below and above y = 1. Thus solutions below the equilibrium solution approach it, and those above it grow farther away. Therefore, ϕ(t) = 1 is semistable.
Consider the simple pendulum: a ball hanging at the end of a string. Derive the expression for the period of this physical pendulum, taking into account the finite size ball (i.e. the ball is not a point mass). Assume that the string is massless. Start with the expression for the period T'of a physical pendulum with small amplitude oscillati T = 2π The moment of inertia of the ball about an axis through the center of the ball is Here, I, is the moment of inertia about an axis through the pivot (fixed point at the top of the string, m is the mass of the ball, g is the Earth's gravitational constant of acceleration, and h is the distance from the pivot at the top of the string to the center of mass of the ball. Note, this pre-lab asks you to do some algebra, and may be a bit tricky. I mgh Iball = / mr² T
By using hamiltonian equations. Find the solution of harmonic oscillator in : A-2 Dimensions B-3 dimensions
Consider a simple harmonic oscillator with natural frequency, w = 1 and the displacement from equilibrium of the mass is denoted by x. At t=0, the mass is released from x=1m with speed 2m/s. What is the magnitude of x at time, t= s?
Problem 3: The motion of critically damped and overdamped oscillator systems is hardly "oscillatory". (a) To illustrate this, prove that a critically damped oscillator passes through the originx = 0 at most once, and determine the relationship between the initial conditions To and vo that is required for the oscillator to pass through the origin. (b) Do the same thing for the overdamped oscillator.
The Newton–Raphson method for finding the stationary point(s) Rsp of a potential energy surface V is based on a Taylor-series expansion around a guess, R0. Similarly, the velocity Verlet algorithm for integrating molecular dynamics trajectories [xt, vt] is based on a Taylor-series expansion around the initial conditions, [x0, v0]. Both of these methods rely strictly on local information about the system. How many derivatives do we need to compute in order to apply them?
Please, I want to solve the question correctly, clearly and concisely
Consider the solution tothe harmonic oscillator given above by x(t)=Ccos(wt−v) Prove tha tx(t0)=x(t0+2piw) In other words, the solution has the same value at time:t0 and at time:t0+2piw regardless of what value we have for ?0. The value 2piw is then the period T of the harmonic oscillator.

SEE MORE QUESTIONS