9. Solve the coupled oscillator system such that m₁ = m² = 1, k₁ = k3 = 9,k₂ = 8. Assume the first mass is pushed 2 units to the right of equilibrium and the 2nd 2 units to the left of equilibrium, with zero initial velocities. Describe the behavior of the motions of the masses at each of the nodes.
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- A particle of mass m moving in one dimension has potential energy U(x) = U0[2(x/a)2 (x/a)4], where U0 and a are positive constants. (a) Find the force F(x), which acts on the particle. (b) Sketch U(x). Find the positions of stable and unstable equilibrium. (c) What is the angular frequency of oscillations about the point of stable equilibrium? (d) What is the minimum speed the particle must have at the origin to escape to infinity? (e) At t = 0 the particle is at the origin and its velocity is positive and equal in magnitude to the escape speed of part (d). Find x(t) and sketch the result.9. Solve the coupled oscillator system such that m₁ = m² = 1, k₁ = k3 = 9,k₂ = 8. Assume the first mass is pushed 2 units to the right of equilibrium and the 2nd 2 units to the left of equilibrium, with zero initial velocities. Describe the behavior of the motions of the masses at each of the nodes.Here are the equations of three oscillators. Which of these has the greatest maximum velocity? a. x(t)=3sin(4t+pie/2) b. x(t)=2sin(3t+pie/2) c. x(t)=10sin(t+pie/2) d. x(t)=sin(10t+pie/2)
- 3. Consider two coupled oscillators whose equations of motion are given by -3z1 +z, dt? d²r2 = I - 3r2, (a) Calculate the characteristic determinant for this system. Note: The characteristic determinant is det M-1, where M is the matrix for which x+Mx = 0 and 2 is the normal mode frequency. %3D (b) Compute the frequencies of the normal modes of oscillation. (c) Solve the cquations of motion subject to the initial conditions r (0) = 0.02 m, r>(0) (0) - -0.1 ms and (0) (d) Show that the normal coordinates for the system are given by yh = + 2 and y2 = -2. Use the initial conditions from (c) to find the solution in terms of the normal coordinates. = 0.02 m. 0.1 ms5). The motion of a particle is defined by the follouing equations: ex = x-2y and dt dy =5x-y with x(0)=2 and yco) =-1 dt a. Find x(t) and yit) b. If xlt) and yct) are perniodic, find their amplitudes and peiods. Co which apaph represents the moton best? 3. う人 フイ NIETHER A 2.vezsion1. Question 1: A mass m attached to a horizontal spring of spring constant k is oscillating on a frictionless surface. If this mass-spring system makes 180 oscillations during a time interval of 6 minutes, then the frequency of oscillations is:
- 1. Calculate the period of a pendulum. Write down the kinetic energy and potential energy for a pendulum within the small angle approximation ( cosø=1- Let the initial angle be ø, and let go (initial velocity=0). Use Eq. (4.57) with appropriate limits of integration to find the period of the pendulum. Compare it with known result. pxr dx If - l = Eq 4.57A tank contains 120 gallons of water and 45 oz of salt. Water containing a salt (1+=sin t) oz/gal flows into the tank at a rate of 8 10 concentration of gal/min, and the mixture in the tank flows out at the same rate. The long-time behavior of the solution is an oscillation about a certain constant level. What is this level? What is the amplitude of the oscillation? Round the values to two decimal places. 55 Oscillation about a level = oz, 0.125 Amplitude of the oscillation = 0z.4:31 .ull 4G O A docs.google.com 3 m/s 2.1 m/s 1.8 m/s In an oscillatory motion of a simple pendulum, the ratio of the maximum angular acceleration, e"max, to the maximum angular velocity, O'max, is 21t s^(-1). What is the time needed for the pendulum to complete two oscillations? 4 sec 0.25 sec 1 sec 2 sec 0.5 sec A traveling wave on a taut string with a tension force T, is given by the wave function: y(x,t) = 0.05sin(Ttx-10Tt), where x and y are in meters and t is in %3D ! seconds. If the linear mass density of the string is given by u= 0 01 ka/m
- A conical pendulum has length / and the angle made by the string with vertical is 0 = 38° . The mass of the object is m=140 g. If the period of the circula motion of the object is T=1.74s find the length of the string. Take g=10m/s. Round your answer to two decimal places. m6. Describe the motion of a particle of mass m, constrained to move on the surface of a cylinder of radius a (see figure below), directed towards the origin by a force which is proportional to the the distance of the particle from the origin. Show that particle is doing simple harmonic motion in the z- direction. Ignore the effect of gravitational force.5A = Material point of mass m moves under the influence of force F-kr = –krî With in other words, the mass m is at the tip of an isotropic harmonic oscillator with equilibrium position at the origin of the axes. a) Calculate the potential energy V(r) of m. b) To design qualitatively 1) the potential energy V(r) of the mass m, 2) its "centrifugal" dynamic energy (r) = 1² /2mr² where L is the measure of angular momentum of the mass m and r its distance from the origin of the axes, and 3) the active potential energy of U(r) = V (r)+ Vä(r). "