SIMPLE HARMONIC MOTION 20 The motion of an undamped linear spring mass system spring constant k 25 N m m 0 25 kg is set into motion with an initial 0 60 J potential energy and an initial 0 20 J kinetic energy a x0 b c displacement Determine the initial position x0 and the initial velocity V0 of the system assuming and V0 The motion in general is described by the equation of motion x t A cos t assign numerical values to the amplitude A the angular frequency and the initial phase Determine the potential energy and the kinetic energy when the system d e x A 2 Determine the maximum speed of the mass during the motion At what displacement s x is the potential energy 75 of the total energy STANDING WAVES 20 Consider a fixed end L 1 50 m taut string under tension T vibrating in the standing wave pattern displayed below at t 0 the maximum displacement of the string from equilibrium is 0 01 m Additionally for this fixed end string the frequency of the fundamental standing wave mode is 60 Hz a b c d e The standing wave pattern is described by the wave function y x t A sinkx cos t Assign values to the amplitude A the wave number k and the angular frequency From the wave function y x t A sinkx cos t demonstrate prove that the antinodes of the standing wave pattern are located at x L 6 L 2 and 5L 6 Determine the speed V of transverse disturbances on this string Determine the maximum transverse velocity of the string at x L 6 Radiation damping The string when vibrating disturbs the surrounding air and produces sound waves which radiate from the string source at 344 m s Determine the frequency and wavelength of the sound radiation produced and describe the subsequent motion of the string Explain QUENCHING 20 A metric tonne of 450oC steel is cooled to 20oC by dousing the sample with a constant spray of 20oC water During the process the water is converted to and removed from the quenching chamber as 150oC steam DATA Water cliquid 1 00 cal goC csteam 0 48 cal goC LF 80 cal g LV 540 cal g steel cs 0 122 cal g oC 1 metric tonne 1000 kg 1 cal 4 186 J a process b c d final e steel Determine the change in thermal internal energy of the steel during the quenching Determine the change in thermal internal energy of M grams of water heated from 20oC to 100oC Determine the change in thermal internal energy of M grams of water converted from liquid to steam at the boiling temperature Determine the change in thermal internal energy of M grams of steam heated to a 150oC Determine the minimum mass M of water required to quench the metric tonne of Explain reasoning logic A THERMODYNAMIC CYCLE 20 An ideal diatomic gas undergoes a four stage thermal engine cycle from an initial state p1 V1 T1 2 00x105 Pa 3x10 3 m3 300 K During process 1 2 the gas is heated a constant pressure to 500 K then cooled at constant volume to state 3 at 250 K The system is further cooled at constant pressure to T4 150 K and finally heated at constant volume to complete the cycle Draw the cycle on the accompanying p V diagram labeling all states 1 2 3 4 and the processes Complete a STATE ANALYSIS P Pa V m3 ETH J T K Determine the HEAT INPUT to the system during the cycle Determine the NET WORK OUTPUT associated with the cycle Determine the efficiency of the cycle for the conversion of HEAT INPUT to NET WORK OUTPUT Compare the cycle efficiency to that of a CARNOT cycle operating between the temperature extremes of this cycle a b c d e STATE 1 2 3 4 GAUSS LAW 20 Consider a uniformly charged spherical insulator radius r total charge Q surrounded by a metallic spherical shell inner radius a outer radius b total charge 2Q a b c d e Determine as a function of R the ELECTRIC FIELD magnitude and direction within the charged insulator R r Determine as a function of R the ELECTRIC FIELD magnitude and direction in the region between the insulator and the inside of the conducting shell r R a Determine as a function of R the ELECTRIC FIELD magnitude and direction within the charged conducting shell a R b on the inner surface of the shell R a and on the outer surface of the shell R b Determine the magnitude and sign or of the charge lying on the inside surface R a and on the outside surface R b of the conducting shell Determine as a function of R the ELECTRIC FIELD magnitude and direction for R b CHARGE INTERACTIONS 20 Consider the SOURCE CHARGE DISTRIBUTION Q1 Q 0 a 0 Q2 Q 0 0 0 Q3 Q 0 a 0 where Q 1 0 C a 0 01 m a b charge c charge d charge e to Sketch the source charge distribution Determine the NET ELECTROSTATIC FORCE direction and magnitude on the test q0 0 10 C located at position s 0 0 where s 2a Determine the NET ELECTROSTATIC FORCE direction and magnitude on the test q0 0 10 C located at position 0 0 s where s 2a Determine the NET ELECTROSTATIC FORCE direction and magnitude on the test q0 0 10 C located at position 0 s 0 where s 2a Determine the minimum energy required to transfer q0 from the initial position s 0 0 the final position 0 s 0 where s 2a EXPLAIN
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