4. A cuckoo clock pendulum consists of two pieces attached together:5. A thin rod with mass m₁ and length 7.6. A circular disk with mass m2 and radius r, centered at the midpoint of the thin rod.The pendulum is attached at one end to a fixed pivot O, as shown below, where a time-varyingtorque (t) is also applied. Note that gravity acts on the system.Instructions:a) Find the expression for the mass moment of inertia of the pendulum Izz with respect to O.b) Find an expression for the angular momentum of the pendulum with respect to O.c) Draw a free-body diagram for the system.Y₁₁(t)T(t)OXθm119m2

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Answered: 4. A cuckoo clock pendulum consists of…

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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1. In lecture I focused on a horizontal mass-spring system so we could ignore gravity, but in practice it's much easier to build a vertical mass-spring system. In the figure below, we can see a suspended spring with spring constant k both before and after a mass m is hooked to it. Y Yo yo is the equilibrium height of the spring (without the mass) and in the figure y = 0 is set at the ground. a) Using the coordinate system in the figure, show that Newton's second law for the hanging mass-spring system leads to the following differential equation: d²y dt² k (y - Yo) - g m b) Now define a new coordinate, y', related to y by a constant shift: y = y' + C, where C = constant. Show that if you choose the right value of C, Newton's second law is identical in form to a mass-spring system without gravity. What is the period of the system? c) Effectively, gravity just shifts the equilibrium position of the system, but the system still undergoes simple harmonic motion. What is the net force on the…
igure 2. Assume that the rod is massless, perfectly rigid, and pivoted at point P. When the rod is perfectly horizontal, the angle 0 = 0, the displacement y 0, and the springs are in neither tension nor compression. Gravity acts on the system (e.g. on mass M). We assume that y is a small displacement. A mass M is attached at the end of the rod. DE only 225 Your tasks: X k 0 3k a F a M y A Derive an equation of motion for the system in terms of the angular displacement 0, and its derivatives (you should not have y or its derivatives in this equation.) B Derive an equation of motion for the system in terms of the displacement y, and its derivatives (you should not have or its derivatives in this equation. C Assuming there is no external actuator force F acting on the system, write down the total energy H of the system in terms of 0,0 and element constants. Derive an expression for the time derivative H of the total energy. D Transform the equation from part B, which is in y, to another…
For this problem, take a look at Figure 2 below. A disk with uniformly distributed mass m, radius R, and center of mass at point O is connected to a combination of springs at point P, which are then connected to a fixed wall. The disk rolls without slipping at point Q along an inclined plane that is at an angle a from the horizontal. Gravity acts in the vertical direction (towards the bottom of the page). ₁ is the linear coordinate of the point O along the inclined plane. The positive direction of ₁ is as shown. When the springs are undeflected, *₁ = 0. An angle , about the instant center of rotation, is shown. You may assume that the motion (and therefore angle ) is small. puny m Massless structure between springs R Figure 2: System schematic. Your tasks: A Draw the FBD for the disk. Don't forget the forces at point Q B Derive the equation of motion with as the dynamic variable. Be sure to put it in input-output standard form (inputs and constant forces on the right, things related to…
Questions A and B
M=2kg m=2kg w=4 e=2 omega=4t please solve with angular motion
Q2
Block String Rod Disk Mech 3. As shown above, a uniform disk is mounted to an axle and is free to rotate without friction. A thin uniform rod is rigidly attached to the disk so that it will rotate with the disk. A block is attached to the end of the rod. Properties of the disk, rod, and block are as follows. Disk: Зт, radius R, moment of inertia about center ID = mR² mass = %3D Rod: m, length = 2R, moment of inertia about one end IR mR? mass = Block: mass = 2m The system is held in equilibrium with the rod at an angle 0, to the vertical, as shown above, by a horizontal string of negligible mass with one end attached to the disk and the other to a wall. Express your answers to the following in terms of m, R, 00, and g. (a) Determine the tension in the string.
Dynamics
The radius, r, of the pulley shown in figure Q1 is 100mm and has a mass moment of inertia of I = 0.1 kg m². The mass, m, is 5kg and the spring constant is k = 135N/m. The system is released from rest with the spring unstretched. At the instant the mass has fallen 0.2m, determine the: A) Angular acceleration of the pulley. B) Tension in the rope between the mass and the pulley. www m X Figure Q1 - Mass pulley system
* F2 # 3 E Problem 4. EOM Derivation Derive the equation of motion for the following system: We want to measure the spin rate w of the rotating shaft of a turbine. To do it we push a small mass m against the end of the shaft. The shaft makes friction contact with the drag cup mounted on the mass. The frictional torque is b (w). Assume that the rotational spring torque is ke and that the moment of inertia of mass m is J. J Rotating Shaft F3 @ $ 4 R b J, m Drag Cup F4 Q Search % 5 F5 T 0 Wall * A 6 F6 Y 2 * F7 & 7 PrtScn U F8 4 8 Home 6. F9 9 End F10 Po 0
3. Quarter car suspension model The figure below a model of the suspension of a car. r(t)is the position of the mass of the car from its equilibrium position while z is the profile of the road. Note that z is a function of the distance y along the road, but you need ż(t) which you can obtain from z(y) using the chain rule. The car is traveling to the right on the road at a speed v. Assume the mass of the car is 250 kg, the spring constant is 16 kN/m, and damping constant is 1000 Ns/m. (a) Derive the differential equation for the position of the car r(t). (b) If z(t) = Zoejot, determine the amplitude and the phase of the particular solution. (c) (MATLAB) If the car is travelling with constant velocity along a road with rolling hills that can be represented as a cosine function with amplitude Zo and wavelength A (wavelength is the distance between adjacent crests or adjacent troughs, determine the vertical motion of the car as a function of time. X(t) Z m 1 C Figure 1: Quarter car…
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