mass ! hangs on the end of a cord around a pulley of radius 5 and moment of inertia 6, rotating with an angular velocity ,, as shown in the figure below. The rim of the pulley is attached to a spring (with constant 7). Assume small oscillations so that the spring remains essentially horizontal and neglect friction so that the conservation of energy

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

5. The pendulum shown below consists of a 30 kg point mass at the end of a uniform bar with mass m = 45 kg and length L = 2 m. The pendulum moves in a vertical plane while acted upon by a 500 N-m torque as shown. If the pendulum's angular velocity is 4 rad/s when 0 = 90 degrees, how much force is exerted on the pendulum by the pin at point A when 0 = 330 degrees? 500 N-m B F = 2470 N Z -52 degrees

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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

11 A cylinder of mass m and mass moment of inertia J is free to roll without slipping, but is restrained by the spring k, as shown in Fig. P2-11. Determine the natural frequency of oscillation. Use:force balance method m. Jo k

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k 0.5m 1m The figure shows a slender rod with a mass of 6 kg. Attached to the rod is a spring of stiffness k = 250 N/m.Amoveable point mass m1 = 1 kg is position at a distance X from the pivot point. You can assume small angle oscillations from the equilibrium position shown in the figure. Calculate the required position, X, of the moveable point mass m1 needed to give the system a natural frequency of 5.3 rad/s. Give your answer to two decimal places.

Consider a mass-and -spring system containing two masses m1=1 and m2=1 whose displacement functions x(t) and y(t) satisfy the differential equations : x'' = -10x + 2y y'' = 3x -15y Assume that the two masses  start in motion  with the initial  conditions x(0) =32, x'(0)=6, and y(0)=36, y'(0)=3 and are acted on by the same force F1(t)=F2(t)=-1320cos(7t). Describe the resulting motion as a superposition of oscillations at the three different frequencies.  The three frequencies consist of lower, higher, and the forcing frequency.  For each one please compare the magnitude of m2 to m1.

4.20. Assuming small angular oscillations, derive the differential equation of motion of the vibratory system shown in Fig. P4.5. If m1 = m2 = 0.5 kg, 4 = 12 = 0.5 m, k = 10* N/m, c = 50N s/m, and F = 10 sin 5t N, determine the complete solution as a function of time for the initial conditions 60 = 0 and 6o = 2 rad/s. F- m2 12 Fig. P4.5