Starting from D'Alembert's principle, derive the differential equation of motion (in plane polar coordinates) for a simple pendulum (with inextensible string) exhibiting planar oscillations. Treat the oscillations as small oscillations.
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Starting from D'Alembert's principle, derive the differential equation of motion (in plane polar coordinates) for a simple pendulum (with inextensible string) exhibiting planar oscillations. Treat the oscillations as small oscillations.
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- A mass weighing 5 lb stretches a spring 4in. If the mass is pushed upward, contracting the spring a distance of 5 in and then set in motion with a downward velocity of 3 ft/s, and if there is no damping and no other external force on the system, find the position u of the mass at any time t. Determine the frequency (wo), period (T), amplitude (R), and phase (8) of the motion. NOTE: Enter exact answers. Use t as the independent variable. u(t) wo rad/s T = R = ft rad || ||A disk of radius 0.25 meters is attached at its edge to a light (massless) wire of length 0.50 meters to form a physical pendulum. Assuming small amplitude motion, calculate its period of oscillation. For a disk, I = (1/2)MR2 about its center of mass.Quartic oscillations Consider a point particle of mass m (e.g., marble whose radius is insignificant com- pared to any other length in the system) located at the equilibrium points of a curve whose shape is described by the quartic function: x4 y(x) = A ¹ Bx² + B² B²), (1) Where x represents the distance along the horizontal axis and y the height in the vertical direction. The direction of Earth's constant gravitational field in this system of coordinates is g = −gŷ, with ŷ a unit vector along the y direction. This is just a precise way to say with math that gravity points downwards and greater values of y point upwards. A, B > 0. (a) Find the local extrema of y(x). Which ones are minima and which ones are maxima? (b) Sketch the function y(x). (c) What are the units of A and B? Provide the answer either in terms of L(ength) or in SI units. (d) If we put the point particle at any of the stationary points found in (a) and we displace it by a small quantity³. Which stationary locations…
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