Obtain the Lagrangian and equations of motion for the double pendulum.
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Obtain the Lagrangian and equations of motion for the double pendulum.
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- Determine the theoretical equation for the dependence of Period of the Physical Pendulum to location of the pivot point for a solid bar of length L. This will be T(x) - Period as a function of pivot point position. Mathematically determine minima of the function T(x).A block of mass m = 2 kg is attached a spring of force constant k = 500 N/m as shown in the figure below.The block is initially at the equilibrium point at x = 0 m where the spring is at its natural length. Then theblock is set into a simple harmonic oscillation with an initial velocity 2.5 m/s at x = 0 cm towards right. Thehorizontal surface is frictionless. a) What is the period of block’s oscillation? b) Find the amplitude A of the oscillation, which is the farthest length spring is stretched to. c) Please represent block’s motion with the displacement vs. time function x(t) and draw the motion graphx(t) for at least one periodic cycle. Note, please mark the amplitude and period in the motion graph.Assume the clock starts from when the block is just released. d) Please find out the block’s acceleration when it is at position x = 5cm. e) On the motion graph you draw for part c), please mark with diamonds ♦ where the kinetic energy of theblock is totally transferred to the spring…Consider the simple pendulum: a ball hanging at the end of a string. Derive the expression for the period of this physical pendulum, taking into account the finite size ball (i.e. the ball is not a point mass). Assume that the string is massless. Start with the expression for the period T'of a physical pendulum with small amplitude oscillati T = 2π The moment of inertia of the ball about an axis through the center of the ball is Here, I, is the moment of inertia about an axis through the pivot (fixed point at the top of the string, m is the mass of the ball, g is the Earth's gravitational constant of acceleration, and h is the distance from the pivot at the top of the string to the center of mass of the ball. Note, this pre-lab asks you to do some algebra, and may be a bit tricky. I mgh Iball = / mr² T
- A mass M is free to slide along a frictionless rail. A pendulum of length L and mass m hangs from M. Find the equations of motion. Find the total energy.An object is attached to a coiled spring. The object is pulled down (negative direction from the rest position) 9 centimeters, and then released. Write an equation for the distance d of the object from its rest position, after t seconds if the amplitude is 9 centimeters and the period is 6 seconds. The equation for the distance d of the object from its rest position is (Type an exact answer, using z as needed. Use integers or fractions for any numbers in the equation.) (? Enter your answer in the answer box. Save for Later 3:18 PM O Type here to search O 11/15/2020 PgUp PgDn F12 DII PrtScn Home F9 End F10 F11 Ins F4 F5 F6 F7 F8 F1 F2 F3 2$ & ) %23 %3D 3. 4. 5 6 7 8 E R Y U | [ TConsider a simple pendulum of length L with a mass m. Derive the angular frequency of oscillation for the pendulum when it is displaced by a small angle. Assume it is in a gravitational field with the magnitude of acceleration due to gravity equal to g. (That is, you need to find MOI and d for a simple pendulum and then use the equation for ω for a simple pendulum.)
- Consider a hollow sphere (I = 2/3 M R2 when rotated about its center) of radius 0.49 m. The sphere is pinned at its north pole (this is not its center) at allowed to undergo small oscillations about this point. Calculate the period of the oscillation, is s, using g = 10 m/s2. (Please answer to the fourth decimal place - i.e 14.3225)A simple pendulum of length L and mass m is pivoted to the mass M which slides without friction on a horizontal plane as shown in the fig. below. Use Lagrange’s equation to write the equation of motion of the system.Develop a Lagrangian for the double pendulum. You may need to make some assumptions to simplify the problem. You may also need to introduce some new variables to make the problem work. Make sure that is explained.