A Rod with mass m and length L can rotate freely about one of its ends. A pointmass with the same mass as the rod (m) can move freely along the length of the rod(ignore friction). The point mass is attached to a spring with a spring constant Kand a negligible initial length. Gravity acts. Use Newton's Second Law (thisincludes the torque equation if you need it) and derive the Equation(s) of motionfor this system.LmuKmm

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Answered: A Rod with mass m and length L can…

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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Figure shows a mechanical system. The connecting link has moment of inertia J about its pivot point, and rotation angle is positive clockwise. Position of mass m is positive to the right. Both the angular and translational displacements are measured from the equilibrium position where all springs are undeflected. Derive the mathematical model of this system assuming small rotation angle 8. Link, moment. of inertia J L k₁ www m O k₂ www
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QUESTION 14 The collar of mass m = 77 kg in Figure Q14 slides outwards with an initial radial velocity vo = 0.58 m-s-1 from zero initial radial position (see dashed box) along a smooth rotating shaft (see unit vector ur). The shaft rotates with constant angular speed of w = 2 rad-s-1. Determine the magnitude of the constant force F (N) that needs to be applied in the radial direction (pointing towards the centre of the shaft) in order to decelerate the collar to zero radial velocity the moment when the collar reaches a radial position r = 0.19 m. Consider motion in a horizontal plane, such that gravity can be ignored. Provide only the numerical value (in Newtons) to zero decimal places and do not include the units in the answer box. Ur F m Vo Figure Q14: Rotating collar-shaft system
The structure of a football goal is shown in the figure below. The bending motion of the structure can be represented by a flexible cantilever beam with tip mass M¡ as shown in the figure. The tip mass M, represents the mass of the horizontal crossbar which is assumed to be rigid in this motion. A single vertical straight beam having mass of MH, shown in the figure below, is the equivalent model for the two symmetrical vertical posts. A soccer ball of mass M, hits the horizontal crossbar with the velocity of Vp. x(t) м, VINGLI SOLAR The response of the horizontal crossbar, mass M, is shown by x(t) in the figure. The vertical beam has a circular cross section of radius r, Young's modulus E and length L. For numerical calculations, use M, = 23.2130 kg, MH = 28.6617 kg. M, = 0.5497 kg, Vý = 8.8974 m/s, L= 2.7757 m, E = 133.5685 GPa andr= 0.0437 m. For circular cross- section I = Tr/4.Considering zero damping and the numerical data given above, obtain the maximum amplitude of response x(t)…
Obtain the Lagrange's equations of motion.
Consider a square plate at rest in the horizontal plane with mass m, sides of length 2L, and center of mass at point G (located at the center of the square). Two forces are applied to the plate: FA is applied at corner A with some angle o relative to the edge of the plate, and FB is applied at point B parallel with the edge of the plate, as shown. The resulting motion is a pure acceleration (translation) to the right of magnitude a. The plate does not rotate. Determine the magnitude of the force applied at B, ||FB||. (Hint: Use Euler's 2nd Law about an arbitrary point (in this case about point A). FAS 2L 2L F B B a
P.4) You are tossing the baton up into the air while it is turning, resulting in a motion that is translating as well as rotating. The instance when the baton was tossed into the air is illustrated in a figure below. The baton is modeled as two point masses m attached at the end of a massless rod of length 2r. The center of the rod is thrown directly upward at a speed vo. The rod rotates at an angular speed of è. Derive the expression for the kinetie energy for the baton's motion (translation and rotation at this instance) using the given variables. /B
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For each of the systems shown in Figure P4.52, the input is the force f andthe outputs are the displacements x1 and x2 of the masses. The equilibriumpositions with f = 0 correspond to x1 = x2 = 0. Neglect any friction betweenthe masses and the surface. Derive the equations of motion of the systems.
4.14. The system illustrated in Figure 4.26 is comprised of a board of length / with uniform mass per unit length of p. It sits on top of two counter-rotating cylinders, which are rotating with an angular velocity of @ in the directions indicated and are separated by a distance 1/2. You may assume that is large enough that the points of contact between the board and cylinders are always slipping and that the coefficient of dynamic friction between the board and cylinders is μ. If the board is not centered on the two cylinders, then the normal force will be greater on the side to which it is displaced because more of the weight of the board will be supported on that side. For example, in the figure, the normal force between the board and cylinder will be greater at the left cylinder, and this will cause a net force to the right. Let x denote the distance that the center of the board is displaced from the center of the two cylinders, as is illustrated in the figure, and determine the…
A mechanical system consists of a ("long") rigid rod of length d and mass M, and another ("short") rigid rod of length I and mass m. The rods are attached at a fixed 90° angle such that the endpoint of the long rod touches the middle of the short rod; see the figure below. The mass per unit length of the rods is constant. The system can move in a vertical plane (x, y) with the endpoint O of the long rod fixed, so that the system is free to oscillate in the vertical plane about the point O. Gravity acts along the vertical y direction. 0 d a) How many degrees of freedom does the system have? [Expect to use a line to answer this question.] â b) Calculate the moment of inertia of the system constituted by the rods about an axis orthog- onal to the plane (x, y) and passing through O. Assume that the widths of the rods are negligible, so that you can effectively treat them as one-dimensional objects. [Expect to use about half a page to answer this question.] c) Determine the distance of the…
A sphere of mass M and radius R is not necessarily solid or hollow. It has moment of inertia I= cMR2. As shown in the figure, the sphere starts from rest and rolls without slipping down a ramp from height H. It then moves back up the other side, but now with no friction at all between the sphere and the ramp. What height does the sphere reach?

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