Write the equations that describe the simple harmonic motion of a particle moving uniformly around a circle of radius8units, with linear speed 3 units per second.
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- Our unforced spring mass model is mx00 + βx0 + kx = 0 with m, β, k >0. We know physically that our spring will eventually come to rest nomatter the initial conditions or the values of m, β, or k. If our modelis a good model, all solutions x(t) should approach 0 as t → ∞. Foreach of the three cases below, explain how we know that both rootsr1,2 =−β ± Sqrt(β^2 − 4km)/2mwill lead to solutions that exhibit exponentialdecay.(a) β^2 − 4km > 0. (b) β^2 − 4km =0. (c) β^2 − 4km >= 0.You are a coach for the Physics Olympics team participating in a competition overseas. You are given the following sample problem to present to your team of students, which you need to solve very quickly: A person is standing on the midline of a soccer field. At one end of the field, as shown, is a letter D, consisting of a semicircular metallic ring of radius R and a long straight metal rod of length 2R, the diameter of the ring. The plane of the ring is perpendicular to the ground and perpendicular to the midline of the field shown by the broken line as shown. Because of an approaching lightning storm, the semicircular ring and the rod become charged. The ring and the rod each attain a charge Q. What is the electric potential at point P, which is at a position x along the midline of the field, measured from the center of the rod, due to the letter D? Think quickly and use all resources available to you, which include your physics textbook: you are in competition!Calculate the commutators [^px, ^x2].
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- Find the arc length of the space curve given by the vector function r (t) t3 i + t2 j + (2t + 1) k from the point P(0 , 0 , 1) to the point Q (9 , 9 , 7) on the curve. Enter an integer or a fully reduced fraction such as 14 , 7/9 , etc.Consider the Initial Value Problem: X₁ 01 x²₁ x₂ = (a) Find the eigenvalues and eigenvectors for the coefficient matrix. 181 An ellipse with clockwise orientation the trajectory. = - 3x1 + 3x₂ -6x₁ + 3x₂² (b) Solve the initial value problem. Give your solution in real form. I1 = 1-8 x2 = *1(0) x2(0) and X2 = = = ໜຶ່ງ : = 27 18 1. Use the phase plotter pplane9.m in MATLAB to describeShow a[1 2 1] direction on a simple cubic unit cell and give the location (in correct notation) for each position where the direction vector pierces the unit cell.
- Let vectors A=(2,1,−4), B=(−3,0,1), and C=(−1,−1,2).Calculate the following: What is the angle θAB between A and B?To illustrate that the length of a smooth space curve does not depend on the parameterization used to compute it, calculate the length of one turn of the helix with the following parameterizations. a. r(t) = (cos 4t)i + (sin 4t)j + 4tk, 0sts, 1 = 27 b. r(t) = cos i+ sin 2 이, 0sts 4T %3D + c. (t) = (cos t)i - (sin t)j – tk, - 2nsts0 Note that the helix shown to the right is just one example of such a helix, and does not exactly correspond to the parametrizations in parts a, b, or c. (1, 0,0) 1 = 0For this problem, I know the answer is B but I do not know how to get there. Seeing the steps and equations would be much appreciated. Thank you