# C write an expression for the position of the mass as

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c. Write an expression for the position of the mass as a real function, x ( t ). d. Write an expression for the position of the mass as a complex function, z ( t ), in the irreducible form. Problem 1.3 (20 pts) A particle of mass m moves on the x axis with potential energy V ( x ) = E 0 a 4 ( x 4 + 4 ax 3 - 8 a 2 x 2 ) a. Find the positions at which the particle is in stable equilibrium. b. Find the angular frequency of small oscillations about each stable equilibrium position. c. What do you mean by small oscillations? Be quantitative and give a separate answer for each point of stable equlibrium. Problem 1.4 (20 pts) Consider a simple pendulum consisting of a point-like mass m attached to a massless string of length L hanging from a fixed support and constrained to move in a vertical plane (see Figure 2). Assume gravitational acceleration to be g . a. Parametrize the motion of the pendulum in terms of the angle θ , its deviation from the vertical. Find the exact equation of motion ( ~ τ = I~α ) for the pendulum as a function of θ . b. Assume that the angle θ is small and find the approximate simple harmonic equation of motion. c. Justify your approximations. Find the range of θ that the pendulum can be considered a SHM. What is the period of oscillations of this SHM? d. Calculate the exact potential energy of the pendulum as a function of θ . Then, show that the Taylor expansion leads to the same result as in part (b). e. Parametrize the motion of the pendulum in terms of the cartesian coordinate x in the coordinate sys- tem with origin at the pendulum equilibrium position and x -axis horizontal in the plane of pendulum. Find the exact equation of motion ( ~ F = m~a ) of the pendulum in terms of x . 2

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x y θ Figure 2: Pendulum f. Assume that x is small and find the approximate simple harmonic equation of motion. g. Justify your approximations. Find the range of x such that the pendulum can be considered a SHM. 3
Problem 1.5 (20 pts) m k b cos t ) Figure 3: Vibration Free Table Many precision scientific and commercial instruments require vibration free supports. In this problem you will evaluate a design of a vibration free table.

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