2. A uniform bar of length 2a, and mass m is linked to a uniform cylinder of radius b and mass M. In addition, the two bodies are connected to a linear spring. Derive the equations of motion using x and 0 as generalized coordinates. X. k b,

2. A uniform bar of length 2a, and mass m is linked to a uniform cylinder of radius b and mass M. In addition, the two bodies are connected to a linear spring. Derive the equations of motion using x and 0 as generalized coordinates. X. k b,

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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A giant snowboard, represented by a rigid and slender rod with mass m and length L, slides on the undersized cylindrical half-pipe of radius r without friction at its ends A and B. The motion of the snowboard within the half-pipe occurs only within the vertical plane shown. Using the generalized coordinate 0, derive the equations of motion for this system using Lagrange's Equations. Note: The moment of inertia of a slender rod about its center of mass IG=1/12*m*L2 A C
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A slender, uniform, metal rod with mass M is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring with force constant k is attached to the lower end of the rod, with the other end of the spring attached to a rigid support. If the rod is displaced by a small angle 0 from the vertical and released, show that it moves in angular SHM and calculate the period. (Hint: Assume that the angle 0 is small enough for the approximations sin 0 z 0 and cos 0 z 1 to be valid. The motion is + w?0 = 0, and the period is then T = 21 ) simple harmonic if dt2 d²0 :)
Exercise 30: A uniform and homogenous rod (R) of length L=100cm and mass m=lkg is free to rotate about a fixed axis passing through its extremity O. The rod is held in equilibrium by connecting its other extremity A to a point B in a vertical wall by means of a light in- extensible string that makes an angle 0 = 60° with respect to the wall. A block (S) of mass M=lkg is connected to the rod at its extremity A by means of a light in-extensible string. Take g=10m/s. 1. Name and represent all the forces acting on the rod. 2. Determine T the magnitude of the tension force T in the string AB. 3. Determine the magnitude of the reaction force R exerted by the wall on (R). 30 120° (R) (S)
A Hartnell type spring loaded governor rotates about a vertical axis. The two rotating masses weigh 10 N each rotate at a radius of 130 mm when the speed is 600 r.p.m. At this speed the arms, which are vertical and horizontal, have effective lengths of 110 mm and 80 mm respectively. The equilibrium sped is 630 r.p.m when the rotating masses are at their maximum radius of 160 mm. Determine the stiffness rate of spring, the compression of spring at 600 r.p.m.
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5. Car doors are easy to slam shut but difficult to press shut by hand force. The door lock as two latches, the first easily engaged, and the second requiring the rubber seal around the door to be quite compressed before it can engage. The rubber seal gives a spring constant of 50,000N/m at the lock position for the door, and the mass moment of inertia for the door around its hinges is 2.5kg- m². The distance between the lock and the hinges is Im. Calculate what force is needed to press the car door shut at the lock if a speed of 0.8m/s at the lock slams it shut.
h2 L2 G2 G1 L1 www x2 www m2, k2 02 m1, k1 Fig Qa Fig Qb Two compound pendulums of lengths L1 and L2 are pivoted at a support and connected together through a horizontal spring of stiffness k as in Fig Qa. The masses and radii of gyration of the pendulums about the centres of mass (G1 and G2) are m1 & m2 and k1 & k2 respectively. The centres of mass are at distances hl and h2 respectively whilst the spring is distance 1 from the pivots as shown. The pendulums are set into motion by small displacements xl and x2 opening up small angles 01 and 02 respectively as in Fig Qb. 4.1 Find the equations of motion of the two pendulums. 4.2 Also determine the natural frequencies of the vibration of the pendulums if mı = 10 kg, m2 = 8 kg, kı = 0.433 m, k2. = 0.3464 m, L1 = 1.2 m, L2. = 1.0 m, hi = 0.6 m, h2. = 0.5 m, 1= 0.8 m, and k = 2000 N/m.
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