Calculate the energy, corrected to first order, of a harmonic oscillator with potential 1 V(x) =kx + ox + ox°
Q: When the pendulum is released from rest, what is the maximum speed, v in m/s, the mass reaches?
A: Here we have a very simple problem, which can be done using conservation of energy.
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- A spring of length l = 20 cm and spring constant k = 5000 N/m has one end attached to a wall and the other end attached to a blockwall and the other end attached to a block as shown in Fig. 1(b). The end with the block is displaced from its equilibrium position(relaxed spring), compressing it by 5.0 cm, and the system is released from rest. Assuming thatthere is no dissipation during the process the motion is described from Hooke's law. Constructthe graph of the force as a function of the position of the block considering its coordinate system located (a) on the wallthe wall; (b) in the equilibrium position. (c) Discuss the difference between Hooke's law in the form F = -kx and F = -k∆sand which of the above situations is described by which form. Is this the force of the block on the spring or of the spring onthe block? (d) Explain the motion of the block and the sign of the equation. This same spring is placed vertically on theground and a 10.2 kg block is held 15 cm above the spring,…Answer both Q1: An object undergoes simple harmonic motion. As itmomentarily passes through the equilibriumposition, which statement is true about its potentialenergy U and kinetic energy K?minimum U, minimum Kmaximum U, maximum Kminimum U, maximum Kmaximum U, minimum KNone of the above 2. An object undergoes simple harmonic motion. When itstops at its turnaround points, which of thefollowing statements is true about its potential energy U andkinetic energy K?minimum U, maximum Kminimum U, minimum Kmaximum U, minimum Kmaximum U, maximum KNone of the aboveShow 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
- A laminar boundary layer profile may be assumed to be approximately of the form u/U₁ = f(n) = f(y/6) i) Use an integral analysis with the following two-segment velocity profile, f(n)=(n/6)(10-3n-13), for 0≤17 ≤0.293 and ƒ (7) = sin (лn/2) for 0.293≤ n ≤1, to find expressions for the displacement thickness &, the momentum thickness e, the shape factor H, the skin-friction coefficient a, and the drag coefficient CD.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 TA certain oscillator satisfies the equation of motion: ä + 4x = 0. Initially the particle is at the point x = V3 when it is projected towards the origin with speed 2. 2.1. Show that the position, x, of the particle at any given time, t, is given by: x = V3 cos 2t – sin 2t. (Note: the general solution of the equation of motion is given by: x = A Cos 2t + B Sin 2t, where A and B are arbitrary constants)
- An m-mass pendulum is connected to a spring with the spring constant k through the massless rope along I as in the picture. x, i wiwww. k point y =0 m The spring is then given a deviation of x, so it moves at a speed of x. Assume point y = 0 is parallel to the spring, so that the potential energy of the pendulum negative value. If the pendulum has a deviation of 0, specify: Lagrangian equation of the systemConsider 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?A simple harmonic oscillator is at equilibrium when the mass is at position x =0. The mass ispulled to x = +12 cm and released from rest.Rank the speed of the mass when it is at the following positions from least to greatest.
- 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]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.Suppose that you have a potential V (x) x2 + 6x – 8. Using a Taylor Series around Xo = 3, approximate the potential as a harmonic oscillator. O + (= – 3)? 7-2 (포-3)2 | (x – 3)? ||