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Additional info for Analytical dynamics : theory and applications
The energy method is simple when appropriate. The advantages are: (i) Unit vectors, free-body diagrams, and coordinate systems are not needed, (ii) It gives an integral of the motion; that is, it gives a relation of speeds, not accelerations. The disadvantages are: (i) It gives only one equation, and thus possibly doesn't give the information desired, (ii) It is cumbersome when forces do work and are nonconservative. These observations motivate the search for new approachs to dynamics. Purposes of Analytical Dynamics.
12 m - - j Problem 1/41 Review of Newtonian 1/42. Dynamics 37 A 4000 lb car travels up a hill as shown. The car starts from rest at A and the engine exerts a constant force in the direction of travel of 1000 lb until position B is reached, at which time the engine is shut off. How far does the car roll up the hill before stopping? Neglect all friction and air resistance. 30\ 400 ft . ^ . A k = 20 kN/m BVProblem 1/42 Problem 1/43 1/43. The small 4 kg collar is released from rest at A and slides down the circular rod in the vertical plane.
10): ^^Fx = mx =^ —Tsm6 = 'mx 22 Py — "^y =^ T cos 9 — mg = my eliminate T to get: cos 9.. "« in normal-tangential components (Eqn. 16): 22 Ft = mil =^ —mgsm9 = mi9 2_^Fn = m,— =^ T —mg cos 9 = rriY The first equation provides the equation of motion: 9 + ^sm9 = 0 and the second gives the force T. (c) The work-energy relation (Eqn. 46): Since the chord tension T is perpendicular to v_, U = I T_- v_dt = 0 and T does no work. The Analytical 24 Dynamics only force doing work is weight, mg, and this force is conservative; therefore energy is conserved and Eqn.