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A 2.50-kg rock is attached at the end of a thin, very light rope 1.45 m long. You start it swinging by releasing it when the rope makes an 11 angle with the vertical. You record the observation that... A mass is oscillating with amplitude A at the end of a spring. How far (in terms of A) is this mass from the equilibrium position of the spring when the elastic potential energy equals the kinetic... A small block is attached to an ideal spring and is moving in SHM on a horizontal, friction-less surface. The amplitude of the motion is 0.250 m and the period is 3.20 s. What are the speed and... A small block is attached to an ideal spring and is moving in SHM on a horizontal, friction-less surface. The amplitude of the motion is 0.120 m. The maximum speed of the block is 3.90 m s. What is... A sinusoidally varying driving force is applied to a damped harmonic oscillator. (a) What are the units of the damping constant b? (b) Show that the quantity km has the same units as b. (c) In terms... For the oscillating object in Fig. E14.4, what are (a) Its maximum speed and (b) Its maximum acceleration? Figure E14.4: () 10.0 t (s) A0.0 15.0 5.0 -10.0 A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the amplitude of the motion is 0.090 m, it takes the block 2.70 s to travel from x =... The motion of an under-damped oscillator is described by Eq. (14.42). Let the phase angle be zero. (a) According to this equation, what is the value of x at t = 0? (b) What are the magnitude and... A mass m is attached to a spring of force constant 75 N m and allowed to oscillate. Figure E14.34 shows a graph of its velocity as a function of time t. Find (a) The period, (b) The frequency, (c)... In February 2004, scientists at Purdue University used a highly sensitive technique to measure the mass of a vaccinia virus (the kind used in smallpox vaccine). The procedure involved measuring the... An object is undergoing SHM with period 0.900 s and amplitude 0.320 m. At t = 0 the object is at x = 0.320 m and is instantaneously at rest. Calculate the time it takes the object to go (a) From x =... (a) Music. When a person sings, his or her vocal cords vibrate in a repetitive pattern that has the same frequency as the note that is sung. If someone sings the note B flat, which has a frequency of... For a string stretched between two supports, two successive standing-wave frequencies are 525 Hz and 630 Hz. There are other standing-wave frequencies lower than 525 Hz and higher than 630 Hz. If the... A 1.80-m-long uniform bar that weighs 536 N is suspended in a horizontal position by two vertical wires that are attached to the ceiling. One wire is aluminum and the other is cop-per. The aluminum... A large rock that weighs 164.0 N is suspended from the lower end of a thin wire that is 3.00 m long. The density of the rock is 3200 kg/m 3 . The mass of the wire is small enough that its effect on... A deep-sea diver is suspended beneath the surface of Loch Ness by a 100-m-long cable that is attached to a boat on the surface (Fig. P15.84). The diver and his suit have a total mass of 120 kg and a... A fan at a rock concert is 30 m from the stage, and at this point the sound intensity level is 110 dB. (a) How much energy is transferred to her eardrums each second? (b) How fast would a 2.0-mg... (a) Blue whales apparently communicate with each other using sound of frequency 17 Hz, which can be heard nearly 1000 km away in the ocean. What is the wavelength of such a sound in seawater, where... At point A, 3.0 m from a small source of sound that is emitting uniformly in all directions, the sound intensity level is 53 dB. (a) What is the intensity of the sound at A? (b) How far from the... (a) If two sounds differ by 5.00 dB, find the ratio of the intensity of the louder sound to that of the softer one. (b) If one sound is 100 times as intense as another, by how much do they differ in... You live on a busy street, but as a music lover, you want to reduce the traffic noise. (a) If you install special sound-reflecting windows that reduce the sound intensity level (in dB) by 30 dB, by... A turntable 1.50 m in diameter rotates at 75 rpm. Two speakers, each giving off sound of wavelength 31.3 cm, are attached to the rim of the table at opposite ends of a diameter. A listener stands in... (a) Show that for a wave on a string, the kinetic energy per unit length of string is where is the mass per unit length of string is (b) Calculate for a sinusoidal wave given by Eq. (15.7). (c) There... Many opera singers (and some pop singers) have a range of about 2 octaves or even greater. Sup-pose a sopranos range extends from A below middle C (frequency 220 Hz) up to -flat above high C... A person leaning over a 125-m-deep well accidentally drops a siren emitting sound of frequency 2500 Hz. Just before this siren hits the bottom of the well, find the frequency and wavelength of the... The siren of a fire engine that is driving northward at 30.0 m/s emits a sound of frequency 2000 Hz. A truck in front of this fire engine is moving northward at 20.0 m/s. (a) What is the frequency of... Two swift canaries fly toward each other, each moving at 15.0 m/s relative to the ground, each warbling a note of frequency 1750 Hz. (a) What frequency note does each bird hear from the other one?... Two small stereo speakers are driven in step by the same variable-frequency oscillator. Their sound is picked up by a microphone arranged as shown in Fig. E16.38. For what frequencies does their... Two loudspeakers, A and B (see Fig. E16.33), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. The frequency of the sound waves... A uniform 165-N bar is supported horizontally by two identical wires A and B (Fig. P16.62). A small 185-N cube of lead is placed three-fourths of the way from A to B. The wires are each 75.0 cm long... The motors that drive airplane propellers are, in some cases, tuned by using beats. The whirring motor produces a sound wave having the same frequency as the propeller. (a) If one single-bladed... You are exploring a newly discovered planet. The radius of the planet is 7.20 10 7 m. You suspend a lead weight from the lower end of a light string that is 4.00 m long and has mass 0.0280 kg. You... The lower end of a uniform bar of mass 45.0 kg is attached to a wall by a friction-less hinge. The bar is held by a horizontal wire attached at its upper end so that the bar makes an angle of 30.0 o... You are designing a two-string instrument with metal strings 35.0 cm long, as shown in Fig. P15.58. Both strings are under the same tension. String S 1 has a mass of 8.00 g and produces the note... A 1750-N irregular beam is hanging horizontally by its ends from the ceiling by two vertical wires (A and B), each 1.25 m long and weighing 0.360 N. The center of gravity of this beam is one-third of... A light wire is tightly stretched with tension F. Trans-verse traveling waves of amplitude A and wavelength 1 carry average power P av,1 = 0.400 W. If the wavelength of the waves is doubled, so 2 =... A horizontal wire is stretched with a tension of 94.0 N, and the speed of transverse waves for the wire is 492 m s. What must the amplitude of a traveling wave of frequency 69.0 Hz be in order for... One string of a certain musical instrument is 75.0 cm long and has a mass of 8.75 g. It is being played in a room where the speed of sound is 344 m/s. (a) To what tension must you adjust the string... The upper end of a 3.80-m-long steel wire is fastened to the ceiling, and a 54.0-kg object is suspended from the lower end of the wire. You observe that it takes a transverse pulse 0.0492 s to travel... A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes 2.5 s for the boat to travel from its highest point to its lowest, a total... (a) The range of audible frequencies is from about 20 Hz to 20,000 Hz. What is the range of the wavelengths of audible sound in air? (b) The range of visible light extends from 400 nm to 700 nm. What... A circular saw blade with radius 0.120 m starts from rest and turns in a vertical plane with a constant angular acceleration of 3.00 rev/s 2 . After the blade has turned through 155 rev, a small... A 15.0-g bullet traveling horizontally at 865 m/s passes through a tank containing 13.5 kg of water and emerges with a speed of 534 m/s. What is the maximum temperature increase that the water could... If the air temperature is the same as the temperature of your skin (about 30C), your body cannot get rid of heat by transferring it to the air. In that case, it gets rid of the heat by evaporating... An insulated beaker with negligible mass contains 0.250 kg of water at a temperature of 75.0 o C. How many kilograms of ice at a temperature of 20.0 o C must be dropped into the water to make the... The blood plays an important role in removing heat from the body by bringing this heat directly to the surface where it can radiate away. Never the-less, this heat must still travel through the skin... You are making pesto for your pasta and have a cylindrical measuring cup 10.0 cm high made of ordinary glass [ = 2.7 10 5 (C o ) 1 ] that is filled with olive oil [ = 6.8 10 4 (C o ) 1 ] to a... A surveyors 30.0-m steel tape is correct at a temperature of 20.0 o C. The distance between two points, as measured by this tape on a day when its temperature is 5.00 o C, is 25.970 m. What is the... A Styrofoam bucket of negligible mass contains 1.75 kg of water and 0.450 kg of ice. More ice, from a refrigerator at -15.0 o C, is added to the mixture in the bucket, and when thermal equilibrium... (a) If the jogger in the preceding problem were not able to get rid of the excess heat, by how much would his body temperature increase above the normal 37C in a half hour of jogging? The specific... (a) By how much would the body temperature of the bicyclist in the preceding problem increase in an hour if he were unable to get rid of the excess heat? (b) Is this temperature increase large enough... On a cool (4.0 o C) Saturday morning, a pilot fills the fuel tanks of her Pitts S-2C (a two-seat aerobatic airplane) to their full capacity of 106.0 L. Before flying on Sunday morning, when the... (a) Equation (17.12) gives the stress required to keep the length of a rod constant as its temperature changes. Show that if the length is permitted to change by an amount ÎL when its... Animals in cold climates often depend on two layers of insulation: a layer of body fat (of thermal conductivity 0.20 W/m K) surrounded by a layer of air trapped inside fur or down. We can model a... The basal metabolic rate is the rate at which energy is produced in the body when a person is at rest. A 75-kg (165-lb) person of height 1.83 m (6 ft) has a body surface area of approximately 2.0 m 2... You have probably seen people jogging in extremely hot weather and wondered Why? As we shall see, there are good reasons not to do this! When jogging strenuously, an average runner of mass 68 kg and... A violinist is tuning her instrument to concert A (440 Hz). She plays the note while listening to an electronically generated tone of exactly that frequency and hears a beat of frequency 3 Hz, which... In the northern hemisphere, June 21 (the summer solstice) is both the longest day of the year and the day on which the suns rays strike the earth most vertically, hence delivering the greatest amount... (a) The average normal body temperature measured in the mouth is 310 K. What would Celsius and Fahrenheit thermometers read for this temperature? (b) During very vigorous exercise, the bodys... Perform a degrees of freedom analysis for the model in Eqs. 2-64 through 2-68. Identify parameters, output variables, and inputs (manipulated and disturbance variables). Plot the level response for a tank with constant cross-sectional area of 4 ft 2 as a function of time for the following sequence of events; assume an initial level of 1.0 ft with the drain open, and... Consider the PCM Furnace module of Appendix E, which is used to preheat a high-molecular-weight hydro-carbon feed (C 16 C 26 ) to a cracking unit at a petroleum refinery. Do the following sequence... Consider the PCM distillation column module of Appendix E, in which a 50%50% mixture of methanol (MeOH) and ethanol is separated. Do the following sequence of simulations: (a) Change the Vapor Flow... Show that when K = 1, the ramp response of the third-order system G(s)= K / (s+1) 3 lags behind the input signal (u = at) by three time constants, once the output is changing linearly in time. Assume that step response data obtained from an FOPTD system with K = = 1 are available. Determine the accuracy of the FOPTD approximate model derived from these data using the Sundaresan and... Consider a standard feedback control system where each component is functioning properly. Briefly indicate whether you agree or disagree with the following statements: (a) For proportional-only... The transfer function relating the blood pressure of a patient to the infusion rate of a blood pressure drug is given by where θ 1 = 30 s, θ 2 = 45 s, and Ï 1 = 40 s.... composition analyzer is used to measure the concentration of a pollutant in a wastewater stream. The relationship between the measured composition Cm and the actual composition C is given by the... Consider a third-order process with G m = G v = 2 and G p = G d = 2 / (s + 2) 3 . (a) Find the roots of the characteristic equation for a proportional controller G c = K c for K c = 1, 8, and 27.... Suppose a control system is modeled by G v G p = 0.25 / (s + 1) 3 , G m = 4, and G c = K c . Find the maximum value of K c for a proportional controller for which the system is stable, using direct... Repeat Exercise 10.7 for the case where the flow meters are triply redundant; that is, there are three identical flow meters with two in the backup mode. How much would the overall system reliability... The system equations for two liquid surge tanks in series are Using state-space notation, determine the matrices A, B, C, and E, assuming that the level deviations is h² 1 and h² 2 are the... A two-input / two-output process involving simultaneous heating and liquid-level changes is illustrated in Fig. E6.20. Find the transfer function models and expressions for the gains and the time... Consider the following cascade connection of isothermal reactors. A first-order reaction A B occurs in both reactors, with reaction rate constant k. The volumes of liquid in the reactors,V, are... The probability of a particular type of sensor functioning properly is 0.95. Consequently, a triply redundant sensor system has been proposed for a critical measurement. Thus, three independent... Consider the PCM furnace module of Appendix E. Assume that hydrocarbon temperature T HC is the output variable and that air flow rate F A is the input variable. (a) Develop an FOPTD model from... In the introduction to Section 16.4, it is stated that based on a steady-state degrees-of-freedom analysis of a control system, it is not possible to eliminate offset in the controlled variables for... The analog exponential flter in Eq. 17-4 is used to flter the output signal from a transfer function G = 1/(2s + 1). The input signal is d(t) = 1 + 0.2 sin t. Compare the responses for no fltering (... Section 18.1 states that ordinarily, pressure, volume, and temperature cannot change individually without one affecting the others. Yet when a liquid evaporates, its volume changes, even though its... In the ideal-gas equation, could an equivalent Celsius temperature be used instead of the Kelvin one if an appropriate numerical value of the constant R is used? Why or why not? On a chilly morning you can see your breath. Can you really? What are you actually seeing? Does this phenomenon depend on the temperature of the air, the humidity, or both? Explain. When a car is driven some distance, the air pressure in the tires increases. Why? Should you let out some air to reduce the pressure? Why or why not? The coolant in an automobile radiator is kept at a pressure higher than atmospheric pressure. Why is this desirable? The radiator cap will release coolant when the gauge pressure of the coolant... Unwrapped food placed in a freezer experiences dehydration, known as freezer burn. Why? If the collector doping density in a transistor is 6 10 15 atoms/cm 3 , and is much less than the base doping, find BV CEO for F = 200 and n = 4. Use crit = 3 10 5 V/cm. A person who has skin of surface area and temperature 30.0C is resting in an insulated room where the ambient air temperature is 20.0C. In this state, a person gets rid of excess heat by radiation.... A 20.0-L tank contains 4.86 10 4 of helium at 18.0 o C. The molar mass of helium is 4.00 g/mol. (a) How many moles of helium are in the tank? (b) What is the pressure in the tank, in pascals and in... A process is described by the transfer function Find the range of controller settings that yield stable closed-loop systems for: (a) A proportional-only controller. (b) A proportional-integral... A PI controller is to be used in a temperature control system for a bioreactor. For nominal conditions, it has been determined that the closed-loop system is stable when I = 10 min and 10 < K c < 0.... Helium gas with a volume of 2.60 L, under a pressure of 0.180 atm and at a temperature of 41.0 o C, is warmed until both pressure and volume are doubled. a. What is the final temperature? b. How many... A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains 0.110 m 3 of air at a pressure of 0.355 atm. The piston is slowly pulled... A large cylindrical tank contains 0.750 m 3 of nitrogen gas at 27 o C and 7.50 10 3 (absolute pressure). The tank has a tight-fitting piston that allows the volume to be changed. What will be the... Using the failure rate data in Table 10.1, evaluate the reliability and mean time between failures for the high-pressure interlock in Fig. 10.4. Assume that the failure rate for the solenoid switch... A certain heat engine operating on a Carnot cycle absorbs 150 J of heat per cycle at its hot reservoir at 135C and has a thermal efficiency of 22.0%. (a) How much work does this engine do per cycle?... A 10.0-L gas tank containing 3.20 moles of ideal He gas at 20.0C is placed inside a completely evacuated, insulated bell jar of volume 35.0 L. A small hole in the tank allows the He to leak out into... Air samples from a process area are continuously drawn through a 14 -in diameter tube to an analytical instrument that is located 60m away. The tubing has an outside diameter of 6.35 mm and a wall... It has been suggested that the liquid flow rate in a large diameter pipeline could be better regulated by using two control valves instead of one. Suppose that one control valve has a large C v... A 4.50-kg block of ice at 0.00C falls into the ocean and melts. The average temperature of the ocean is 3.50C, including all the deep water. By how much does the melting of this ice change the... A refrigerator has a coefficient of performance of 2.25, runs on an input of 95 W of electrical power, and keeps its inside compartment at 5C. If you put a dozen 1.0-L plastic bottles of water at 31C... The coefficient of performance K = H/P is a dimension-less quantity. Its value is independent of the units used for H and P, as long as the same units, such as watts, are used for both quantities....
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