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- One-dimensional crystal assembled from three identical atoms connected to one another and to the walls by identical springs (as shown in the fig. 1). Find the normal modes of the given system. Plot the relative displacements versus time for all three normal modes. k m m k m O w Ow Fig. 1Find the amplitude (if one exists), period, and phase shift of each function. Graph each function. Be sure to label key points. Show at least two periods.can you pls help me solve this.. our topic is Oscillations. 1. A simple pendulum of length 3m is set into simple harmonic motion at an angle 20 degrees. Determine: a) Amplitude of the wave (A = Ltheta) b) period (T = 2pi radical length / g) c) Angular frequency (w = radical k/m = radical g/l) d) Maximum velocity and acceleration of the motion (vmax = w radical A^2 - x^2 ; amax = -w^2x) •Pls answer with complete solution (3 dec places) •Sketch an illustration/diagram with labels (this is a must; for a better understanding) Thank you I will give your answer a LIKE for helping me...
- Two pendulums have equal length L, but different masses mị and m2. The pen- dulums are coupled by a spring with spring constant K. The pendulums can only move in the plane of the figure. Find the frequencies of small oscillations around the equilibrium point. Use arrows on a picture like the one below to show the ap- proximate displacements corresponding to these modes. You do not need to find algebraic expressions for the displacements. Escape clause: if this problem is a little too hard, you will get partial credit for solving the special case mı = m2. L |L ml m2 kfunction of r. Consider a small displacement r= Ro+r' and use the binomial theorem: (1+x)" = 1+ nx + 2! п(п — 1) n(n — 1)(n — 2) з + -x' + 3! to show that the force does approximate a Hooke's law force. 71. Consider the der Waals potential van 12 U(r) = U. used model the to potential energy function of two molecules, where the minimum potential is at r = Ro. Find the force as aIt is known that a general solution for the position of a harmonic oscillator is x(t) = C cos (wt) + S sin (wt), where C, S, and ware constants. (Figure 2) Your task, therefore, is to determine the values of C, S, and w in terms of k, m,and init and then use the connection between x (t) and a (t) to find the acceleration. Figure L min x=0 Xinit 1 of 2 > Part A Combine Newton's 2nd law and Hooke's law for a spring to find the acceleration of the block a (t) as a function of time. Express your answer in terms of k, m, and the coordinate of the block x (t). ► View Available Hint(s) a (t) = Submit 17 ΑΣΦ Part B Complete previous part(s) Review | Constants ?
- I need some help with this. For the Harmonic Motion of a mass+spring given in the picture, answer the following question: Velocity is the change in displacement with respect to time, i.e. v = dx/dt. Write an expression for the velocity of the mass as a function of time.A mass of 10 kilograms is attached to the end of a spring. A force of 4 Newtons stretches the spring 10 cm. The mass is pulled 15 cm above the equilibrium point and then released with an initial downward velocity of 20 cm/sec. Determine the displacement of the mass as a function of time. Spring constant: Differential Equation and initial conditions: whose consta 50 cm/sec. Determine Displacement of the mass as a function of time: Niem. The system iThe motion of a mass and spring is described by the following equation: 11y"+4y+6y= 35 cos(yt) Identify the value of y that would produce resonance in the system. Give an exact value, but you don't need to simplify radicals. Y What is the amplitude of the steady-state solution when the system is at resonance? Round your answer to two decimal places. The amplitude is If there was no external force, the oscillation of the spring could be described in the form A sin(ßt + p). What is the value of B? Give an exact value, but you don't need to simplify radicals. В
- Read the problems carefully, provide the needed values and graphs of the functions, then answer the questions that follow. Modeling: Simple Harmonic Motion A minute hand sweeps around the face of a clock once every 60 minutes. This clock has a radius of 10 cm. Write an equation for the position of the horizontal diameter of the clock (or the value of the y-coordinate of the point at the end of the minute-hand) as a function of the time in minutes. Assume that when t = 0, the minute hand is at 12. First, use the your knowledge of the 30" – 60° – 90" relationship to fill out the table for the first half of the minute hand's rotation. Then plot those values on the graph and use what you've found to finish the graph for a full rotation of the minute hand. t, in minutes 0 5 10 15 20 25 30 y, in cm What type of function will best model this situation? Why? What is the amplitude? How did you determine this? What is the period? How did you determine this? Use these answers to write an equation…Problem 3: Cylinder on sliding plate A plate of mass Mp is attached to a spring (constant k). It can slide, frictionlessly, only in x. On the plate is a cylinder of radius R and mass MH which rolls on the plate without slipping. Using the Lagrangian formalism: A. Find the EOM(s) for the system B. Find the resonant frequency go MH R MpQ3: The time dependance of the position 2(t) of an underdamped oscillator (of drag co- efficient c and spring constant k) can be described by a sinusoidal curve with exponentially decaying amplitude. So, one can conventionally plot exponential-decay functions f(t) and -f(t) that "sandwich" x(t) and mimic its exponential decay. Both f(t) and its mirror image -f(t) touch r(t) at some time instances t„, and t,, respectively, where n = 1,2, 3, -... (See Fig. 3.4.3 in your assigned textbook). Following the same convention of the constants y and wa as we did in the class, suppose that #(t) is given by x(t) = Ae cos(wat), and, hence, f(t) is given by f(t) = Ae-. then solve the following (a) Find the first two instants of time to and t, at which f(t) and - f(t) touch x(t). (b) Since r(t) is oscillatory, it has multiple local maxima. Suppose that {Tn} are the time instances at which x(t) has a local maximum, where n = 1, 2, 3, -.., then find T1 and T2. (c) Discuss the results you obtained from…