This question describes a simplified model of a device used to de-spin a satellite. A uniform circular disc of mass 12m and radius a lies on a smooth horizontal table and is free to rotate about a fixed vertical axis through its centre. A light wire is attached to a point on the rim of the disc and is wound round this rim. A particle of mass m is attached to the free end of the wire and is initially attached to the rim. When the disc is rotating with angular speed in the opposite sense to that in which the wire is wound the particle is released so that the wire unwinds and remains taut. The length of the wire is chosen so that it is completely….
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In a diatomic molecule, one atom is very much heavier than the other and the lighter atom is free to move on a straight line through the centre of the heavier atom. The potential energy for the force between two atoms in a diatomic molecule can be represented approximately by where x is the atomic separation and a and b are positive constants. i) Locate any positions of stable equilibrium of the lighter atom. ii) Find the period of small oscillations about such positions, assuming the mass of the lighter atom is m. iii) Sketch the potential energy as a function of x. How does the force vary as the atoms approach each other?
A uniform circular cylinder of mass M and radius a lies at rest inside a fixed hollow cylinder of radius 3a. The axes of both cylinders are horizontal. A particle of mass m is fixed to a point at the top of the inner cylinder which can roll without slipping inside the hollow cylinder. The system is subject to a vertical gravitational field g. as shown in the diagram. ii) Find an expression for the gravitational potential energy of the system as a function of . iii) Locate the positions of equilibrium and determine the relationship between M and m for the initial position to be one of stable equilibrium. Show that in this case it is the only equilibrium position.
A light plane uniform square lamina of side 2a is mounted in a vertical plane and is free to rotate about an axis through its centre and perpendicular to its plane. Initially two of the edges of the lamina are horizontal. A particle of mass km, k > 0, is attached to the centre of the upper edge and particles of mass m are attached to each of the two bottom corners. i) By considering a rotation through an angle of the lamina show that the potential energy of the system relative to the centre of the lamina is mga(k – 2) cos . ii) Locate any positions of equilibrium. iii) Discuss the dependence of the nature of the stability of any equilibrium positions on the value of k…..
Stock Valuation at Ragan, Inc.
Ragan, Inc., was founded nine years ago by brother and sister Carrington and Genevieve Ragan. The company manufactures and installs commercial heating, ventilation, and cooling (HVAC) units. Ragan, Inc., has experienced rapid growth because of a proprietary technology that increases the energy efficiency of its units. The company is equally owned by Carrington and Genevieve. The original partnership agreement between the siblings gave each 50,000 shares of stock. In the event either wished to sell stock, the shares first had to be offered to the other at a discounted price.
Although neither sibling wants to sell, they have decided they should value their holdings in the company. To get started, they have gathered the information about their main competitors in the table below.
MODERN LITERATURE – II (1700 – 1850) MAXIMUM MARKS : 30 ANSWER ALL QUESTIONS 1. Annotate of the following: (a) What mad pursuit? What struggle to escape? What pipes and timbrels? What wild estacy? (b) A vital breeze which travelled gently on o’ver Things which it hath made, and is become. (c) He has long outlived his century, the term commonly fixed as the test of literary merit. (d) A quibble was to him the fatal Cleopatra for which he lost the world, and was content to lose it. (e) Where beauty cannot keep her lustrous eyes, Or new Love pine at them beyond to-morrow. (f) Wisdom and spirit of the universe! Thou soul that art the eternity of thought! (g) Nothing can please many, and please long,….
In yet another similar case, the vibration levels despite controlling the oil-inlet temperature and pressure did not come down to safe limits. Obviously, the problem was arising of too-less clearance between the bearing and the shaft. As mentioned, the oil-whirl problem arises when the load to be supported by the oil film is less than the vertical component of the pressure of the hydrodynamic film of lubrication.
Since the problem could not be sorted out by manipulating the oil-inlet temperature and pressure, a deliberate unbalance was put on the shaft so that the bearing load increases. With this, though the vibration levels in the other non-whirling bearings increased to a certain extent, all the bearing-vibration levels came within the acceptable range.
Many a times, a deliberate misalignment in….
The case study pertains to 4 x 80 hydro station (Fig. 7.49) having high vibrations and intense noise problem. The machines, 428.6 rpm, operated at a net head of 251 m,. the number of runner vanes were 11 and the number of guide vanes were 24. The vibrations and noise occurred predominantly at 2 x blade-passing frequency. Interference calculations (Equation 7.26) showed that the calculated speed factor S = 0.070 and (KZr – Zg)/KZr = 0.09 were in the near neighbourhood of each other. Changing the number of runner vanes to 13 was suggested.
There are certain problems of vibration / noise in the components of hydro unit which arise out of poor tail-race design. In one particular case where four units of 80 MW were operating, the tailrace boundary….
A large utility turbine comprises of HP turbine, IP turbine, and LP turbine on a common shaft driving an AC generator, experienced repeated failures of one of the stages of IP turbine. The stage consists of blades provided with a lacing wire to increase the natural frequency from 147 HZ to 153HZ of the free-standing blade to 160 HZ to 165 HZ so that the third-harmonic excitation (around 150 HZ) does not cause high vibrations of the blades and consequent failures. At a particular site, a few of these blades failed during the service causing damages to the diaphragm blades. The failed blades were replaced with new blades and the lacing wire was rebraced. The damaged diaphragms were repaired at site by grinding (dressing) and the machine was….
A motor-driven pump assembly shows a high vibration behaviour at motor drive endbearing as well as the front-bearing of the pump. Figure below shows the schematic of the system. The salient features of the vibration data are as follows.
Motor drive end-bearing horizontal vibration 95 µ pk–pk, vertical 20 µ pk–pk, axial 10 µ pk–pk.
Pump bearing 1 horizontal 85 µ pk–pk, vertical 15 µ pk–pk, and axial µ 20 pk–pk.
What inference can you draw from the above data? Suggest suitable further investigations. The pump is driven by a 3000 rpm AC induction motor.
Another important aspect of vibration analysis is vibration isolation. In these exercises, we shall include some problems on vibration isolation although the basics of the same have been covered elsewhere in this….