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Parcels of air (small volumes of air) in a stable atmosphere (where the temperature increases with height) can oscillate up and down, due to the restoring force provided by the buoyancy of the air parcel. The frequency of the oscillations are a measure of the stability of the atmosphere. Assuming that the acceleration of an air parcel can be modeled as
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- A body of mass m is suspended by a rod of length L that pivots without friction (as shown). The mass is slowly lifted along a circular arc to a height h. a. Assuming the only force acting on the mass is the gravitational force, show that the component of this force acting along the arc of motion is F = mg sin u. b. Noting that an element of length along the path of the pendulum is ds = L du, evaluate an integral in u to show that the work done in lifting the mass to a height h is mgh.arrow_forwardIf a mass m is placed at the end of a spring, and if the mass is pulled downward and released, the mass-spring system will begin to oscillate. The displacement y of the mass from its resting position is given by a function of the form y = c,cos wt + c2 sin wt (1) where w is a constant that depends on spring and mass. Show that set of all functions in (1) is a vector space.arrow_forwardConsider a particle with total energy E is oscillating in a potential U(x) = A|x|" with A>0 and n>0 in one dimension. Which one of the following gives the relation between the time-period of oscillation T and the total energy E: 1/n-1/2 (a) T∞ E¹ 0 (b) Tx Eº (c) TxE" (d) To Enarrow_forward
- 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 ω1arrow_forwardConsider a non-linear oscillator governed by the differential equation x''(t) - ϵx[x'(t)] + x = 0, where the initial conditions are given by x(0)=xo and v(0)=0. Employ the Lindstedt method to expand up to the second order. Specifically, determine the three sets until the order of ϵ^2. Next, calculate the values of ω1 and ω2, and use them for x(t) up to the 2nd order.arrow_forwardAn object of mass 3 grams is attached to a vertical spring with spring constant 27 grams/secʻ. Neglect any friction with the air. (a) Find the differential equation y" = f(y, y') satisfied by the function y, the displacement of the object from its equilibrium position, positive downwards. Write y for y(t) and yp for y' (t). y" : -9y Σ (b) Find r1, r2, roots of the characteristic polynomial of the equation above. r1,r2 = Зі, - 3і Σ (b) Find a set of real-valued fundamental solutions to the differential equation above. Y1(t) = cos(3t) Σ Y2(t) = sin(3t) Σ (c) At t = 0 the object is pulled down 1 cm and the released with an initial velocity downwards of 3/3 cm/sec. Find the amplitude A > 0 and the phase shift o E (-1, 7| of the subsequent movement. A = Σφ Σarrow_forward
- The gravitational force on a particle of mass m inside the earth at a distance r from the center (r < the radius of the earth R) is F = −mgr/R (Chapter 6, Section 8, Problem 21). Show that a particle placed in an evacuated tube through the center of the earth would execute simple harmonic motion. Find the period of this motion.arrow_forwardThe graph shows the displacement from equilibrium of a mass-spring system as a function of time after the vertically hanging system was set in motion at time t = 0. Assume that the units of time are seconds, and the units of displacement are centimeters. The first t-intercept is (0.75, 0) and the first minimum has coordinates (1.75,-4). (a) What is the period T of the periodic motion? T = seconds (b) What is the frequency f in Hertz? What is the angular frequency w in radians / second? f = Hertz W = radians / second (d) Determine the amplitude A and the phase angle y (in radians), and express the displacement in the form y(t) = A cos(wt - y), with y in meters. y(t) = meters (e) with what initial displacement y(0) and initial velocity y'(0) was the system set into motion? y(0) = meters y'(0) = meters / second ***arrow_forwardAnisotropic Oscillator Consider a two-dimensional anisotropic oscillator is rational (that is, ws/wy = p/q 4. Prove that if the ratio of the frequencies w and wy where p and q are integers with gcd(p, q) = 1, where gcd stands for greatest common divisor) then the motion is periodic. Determine the period. Parrow_forward
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- Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning