The SPRING and MASS system The motion of an undamped linear spring mass system spring constant k 6 40 N m is described by the equation of motion x t 0 16 cos 4t 3 m a Briefly what are the distinguishing characteristics of SIMPLE HARMONIC MOTION b Determine the amplitude and frequency associated with the motion the initial position initial velocity and initial acceleration c Determine the mass of the system d Determine the force Fx magnitude and direction on the mass at t seconds e At some position s during the motion the kinetic energy and the elastic potential energy of this system are equal Determine the position s and determine the TOTAL ENERGY of the system TUNNEL THROUGH the MOON In this problem the motion of a 0 50 kg mass released from REST at one end of a tunnel drilled straight through the center of the moon is investigated BACKGROUND Recall from PHYS 161 that the magnitude of the gravitational force inside a spherical mass distribution of constant density is Fg mgr R where r is the distance from the center of the spherical body R the radius of the body g the surface acceleration of gravity DATA For the Moon gM GMM RM 2 1 63 m s2 MM 7 35x1022 kg RM 1 723x106 m a b c d e Physically describe the motion of the 0 50 kg mass ANS SIMPLE HARMONIC MOTION Note Fg kr where k mg R If r t is the position of the mass from the center of the Moon what is the general form of the equation of motion r t ANS r t A cos t generic equation of motion for one dimension undamped SIMPLE HARMONIC MOTION Given the INITIAL CONDITIONS specify r t completely ANS INITIAL CONDITION r 0 RM and v 0 0 From equation of motion r 0 Acos and v t dr dt A sin t so v 0 A sin yielding 0 A RM and of course k m 1 2 gM RM 1 2 What is the angular frequency T ANS k m 1 2 gM RM 1 2 T 2 associated with motion of the mass the period What is v t the instantaneous particle velocity What is the maximum speed and where does maximum speed occur ANS v t dr dt A sin t where A also vmax occurs at r 0 at the center of the Moon and vmax A gMRM 1 2 STANDING WAVES on a VIOLIN STRING The length of the vibrating portion of the violin A string is 29 0 cm and is tuned to 440 Hz in its fundamental mode a Determine the speed of transverse waves on this string b If the speed of sound in the surrounding air is 344 m s determine the wavelength of the sound waves produced by the vibration of this violin string c What is the highest harmonic of this vibrating A string that a person with a 100Hz to 14 kHz acoustic range can hear d The tension in the A string of a second violin initially tuned to 440 Hz is increased by 2 0 What is the frequency of the fundamental mode of this second violin e What is the beat frequency heard when the second and first violin are played simultaneously ANSWERS a 255 2 m s b 0 78 m c 31st harmonic d 444 4 Hz e 4 4 Hz The TWO STRING GUITAR A simple two string guitar design is displayed below the unfretted length of the metal strings L 350 m and both strings are under the same tension String Sa has a linear mass density a 2 30x10 2 kg m and when plucked vibrates at a fundamental frequency of 262 Hz middle C To extend the range of this instrument a fret is located just under but not touching the strings a Determine the tension T in the strings b Determine the linear mass density of string Sb with a fundamental frequency A fb 466 Hz c Where x should the fret be placed so when pressed and held tightly against Sa that string would vibrate at a shifted fundamental frequency C 277 Hz d If string Sb were pressed and held against the fret predict the shifted fundamental frequency fb A THERMODYNAMIC CYCLE A HEAT ENGINE operates on a four stage cycle Stage 1 2 is an adiabatic compression from V1 to V2 Stage 2 3 a constant volume heating from T2 to T3 Stage 3 4 an adiabatic expansion from V3 to V4 and Stage 4 1 constant volume cooling from T4 to T1 The working substance is n moles of an IDEAL DIATOMIC GAS Denote the initial STATE 1 by the thermodynamic variables p1 V1 and T1 the compression ratio r V1 V2 and the pressure ratio s p3 p2 a Draw the CYCLE on the accompanying p V Diagram labeling each STATE and each STAGE process clearly For each STAGE of the CYCLE determine expressing all answers in terms of p1 V1 T1 r s and b c d the WORK TRANSFER associated with each STAGE process the HEAT TRANSFER associated with each process the efficiency of the cycle for converting INPUT HEAT ENERGY into NET MECHANICAL ENERGY WORK OUTPUT SOLUTION STATE and FIRST LAW ANALYSIS Diatomic IDEAL GAS Cv 5 2 R Cp Cv R Cp Cv 7 5 1 40 STATE ANALYSIS STATE 1 2 3 4 P p1 r p1 s r p1 sp1 V V1 r 1 V1 r 1 V1 V1 T T1 r 1 T1 s r 1 T1 sT1 ETH nCvT1 r 1 nCvT1 s r 1 nCvT1 S nCvT1 IMPORTANT CONNECTIONS and CONSTRAINTS C1 C2 C3 C4 C5 C6 C7 C6 1 1 T2V2 pV nRT r V1 V2 1 s p3 p2 1 adiabatic process 1 2 p1V1 p2V2 T1V1 adiabatic process 3 4 p3V3 p4V4 T3V3 iscochoric process 2 3 V2 V3 iscochoric process 4 1 V4 V1 ETH nCv T 1 T4V4 1 GIVEN p1 V1 T1 then from C4 and C2 p2 p1 V1 V2 r p1 and T2 T1 V1 V2 1 T1 r 1 and from C2 V2 V1 r r 1 V1 For the isochoric process 2 3 V3 V2 r 1 V1 from C3 p3 sp2 s r p1 and from C1 the Equation of State T3 p3V3 nR s r 1 p1V1 nR s r 1 T1 For the adiabatic process 3 4 and the isochoric process 4 1 V4 V1 so p4 p3 V3 V4 s r p1 r 1 V1 V1 sp1 and by the Equation of State C1 T4 sT1 FIRST LAW ANALYSIS PROCESS 1 2 2 3 3 4 4 1 CYCLE Qi f 0 s 1 r 1 nCvT1 0 1 s nCvT1 s 1 r 1 1 nCvT1 Wi f r 1 1 nCvT1 0 s 1 r 1 nCvT1 0 s 1 r 1 1 nCvT1 ETH r 1 1 nCvT1 s 1 r 1 nCvT1 s 1 r 1 nCvT1 1 s nCvT1 0 NOTE The FIRST LAW Qi f Wi f ETH For adiabatic processes Qi …
View Full Document
Unlocking...