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- An object of mass 2 kg is attached to a spring of stiffness 200 N / m on a horizontal frictionless surface . Then it is extend by 10 cm and then let free to oscillate . 1- How long will it take to make one complete oscillation ? 2- Determine the amplitude and give an expression for the displacement as a function of time . 3- Calculate its displacement , velocity , acceleration and force acting on it after sec . 4- Calculate its velocity by the time it is extended by 0.5 cmquestion 2 A 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 of the system yields: 1 2 !91 + 1 2 6,1 + 1 2 7;1 − !); = =, ?ℎABA , = 9 5 , = = CDE&/, ; = FG&H'5CA!AE/ IBD! AJ(G'GKBG(! Find the natural circular frequency of the system in terms of !, 5, 7,6, and ).A reciprocating engine runs at 110 rpm with a stroke of 1.6 m and a connecting rod 2.4 m in length. It imparts simple harmonic motion to a 250 mm diameter piston. When the crank pin is 32º beyond top dead centre the pressure on the piston is 0.8 MN/m². Calculate EACH of the following: (a) the instantaneous velocity of the piston; (b) the instantaneous torque at the crank pin; (c) the instantaneous acceleration of the piston.
- What is the period for small oscillations of the half cylinder shown in the figure. Mass of the cylinder is M; radius of the cylinder is "r" and the cylinder rolls back and forth without slipping. 11 rl21 x(1) y(1) k m The roller cam in Fig. 3 imparts a periodic motion to the mass-spring system, as shown. Use the principle of superposition and (a)find the possible values of t, (b) the response of the system m =1 kg, k =100 N/m, Y = 0.2 m, and m=1kg, k =100 N/m, Y = 0.2m, µ=0.1. Assume no initial conditions. The roller cam in Fig. 3 imparts a periodic motion to the mass-spring system, as shown. Use the principle of superposition and (a)find the possible values of E, (b) the response of the system m =1kg, k =100N m, Y = 0.2m,and m =1kg, k =100N m, Y = 0.2m, E = 0.1. %3D Assume no initial conditions.Three cylindrical rollers of equal length and material are arranged as shown in end view below. Cylinders 2A and 2B have radiusr and cylinder 1 has radius 4r. Their axes are parallel to each other and to that of the fixed concave cylindrical surface, of radius 6r, that supports them. They roll without slip. Assuming no damping and following the steps below, determine the frequency of small free rolling oscillations. (a) For rotation 6, of the cylinder 1, determine the resulting rotation of the lower cylinders. [Hint: Does it look like an epicyclic gear train?] When the upper roller rotates with velocity 6, determine the translational and rotational velocities of the centre of mass of the lower rollers. [Hint: In an epicyclic gear train the centres of these would be supported on the planet carrier.] (b) (c) Write an expression for the kinetic energy of the whole system at some instant during the oscillation when the rotational velocity of cylinder 1 is , . (d) Write an expression for…
- The complete slide- back and forth motion of a single reciprocating compressor piston completes in a 2-sec period. The volume displacement of the piston though the piston cylinder is 8,902.63 in3. The stiffness of crank is said to be equal to 65.5N/m and the mass of the said piston is at 10kgs. A.)Draw out your conclusion of the vibration found during the motion of the piston through its piston cylinder.For the system shown below, the disc of mass M rolls without slipping. Assume that the mass M = 681.8627 kg and the stiffnesses K = 30338487.4491 N/m, K2 = 14293643.7357 N/m, the radius of disc R = 0.3164 m and moment of inertia of disc about its centre / = MR?/2. The natural frequency of the free vibration of this system is: K1 K2 * No Slip14. For the system shown to the right, the disk of mass m rolls without slip and z measures the displaccment of the disk from the unstretchcd position of the spring. 3k a) find the cquations of motion; b) if the system is inderdamped, whiat is the frequency of the frec vibrations of this system in terms of the parancters k, с, and m;
- Problem 6: An object of mass 2 kg is attached to a spring of stiffness 200 N/m on a horizontal frictionless surface. Then it is extending by 10 cm and then let free to oscillate. 1- How long will it take to make one complete oscillation? 2- Determine the amplitude and give an expression for the displacement as a function of time. 3- Calculate its displacement, velocity, acceleration and force acting on it after sec. 4- Calculate its velocity by the time it is extended by 0.5 cm.7.7. Obtain the equation of torsional oscillation of a uniform shaft with a torsional spring of stiffness k attached to each end.In an oscillatory motion of a simple pendulum, the ratio of the maximum angular acceleration, O"max, to the maximum angular velocity, O'max, is 2t s^(-1). What is the time needed for the pendulum to complete two ocillations?