14.22 Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.
Q.15.27 A bat is flitting about in a cave, navigating via ultrasonic beeps. Assume that the sound emission frequency of the bat is . During one fast swoop directly toward a flat wall surface, the bat is moving at 0.03 times the speed of sound in air. What frequency does the bat hear reflected off the wall ?
Q.15.26 Earthquakes generate sound waves inside the earth. Unlike a gas, the earth can experience both transverse (S) and longitudinal (P) sound waves. Typically the speed of S wave is about , and that of P wave is . A seismograph records P and S waves from an earthquake. The first P wave arrives 4 min before the first S wave. Assuming the waves travel in straight line, at what distance does the earthquake occur?
Q.15.25 A SONAR system fixed in a submarine operates at a frequency . An enemy submarine moves towards the SONAR with a speed of . What is the frequency of sound reflected by the submarine? Take the speed of sound in water to be .
Q.15.24 One end of a long string of linear mass density is connected to an electrically driven tuning fork of frequency . The other end passes over a pulley and is tied to a pan containing a mass of . The pulley end absorbs all the incoming energy so that reflected waves at this end have negligible amplitude. At , the left end (fork end) of the string has zero transverse displacement and is moving along positive y-direction. The amplitude of the wave is . Write down the transverse displacement y as function of x and t that describes the wave on the string.
Q.15.23 A narrow sound pulse (for example, a short pip by a whistle) is sent across a medium. (a) Does the pulse have a definite (i) frequency, (ii) wavelength, (iii) speed of propagation? (b) If the pulse rate is 1 after every , (that is the whistle is blown for a split of second after every ), is the frequency of the note produced by the whistle equal to or ?
Q.15.22 A travelling harmonic wave on a string is described by
(b) Locate the points of the string which have the same transverse displacements and velocity as the point at 5 s and 11 s.
Q.15.22 (a) A travelling harmonic wave on a string is described by
(a) what are the displacement and velocity of oscillation of a point at , and ? Is this velocity equal to the velocity of wave propagation?
Q. 14.25 A mass attached to a spring is free to oscillate, with angular velocity , in a horizontal plane without friction or damping. It is pulled to distance and pushed towards the centre with a velocity at time Determine the amplitude of the resulting oscillations in terms of the parameters , and . [Hint : Start with the equation and note that the initial velocity is negative.]
Q. 14.24 (c) A body describes simple harmonic motion with an amplitude of and a period of Find the acceleration and velocity of the body when the displacement is
Q. 14.24 (b) A body describes simple harmonic motion with an amplitude of and a period of Find the acceleration and velocity of the body when the displacement is
Q. 14.24 (a) A body describes simple harmonic motion with an amplitude of and a period of Find the acceleration and velocity of the body when the displacement is
Q .14.23 A circular disc of mass is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be The radius of the disc is . Determine the torsional spring constant of the wire. (Torsional spring constant is defined by the relation, where J is the restoring couple and θ the angle of twist).
Q. 14.21 (b) You are riding in an automobile of mass . Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by during one complete oscillation. Estimate the values of
(b) the damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports.
Q. 14.21 (a) You are riding in an automobile of mass . Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by during one complete oscillation. Estimate the values of
(a) the spring constant
Q. 14.20 An air chamber of volume V has a neck area of cross section a into which a ball of mass m just fits and can move up and down without any friction (Fig.14.33). Show that when the ball is pressed down a little and released , it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal [see Fig. 14.33].
Q13.14 Given below are densities of some solids and liquids. Give rough estimates of the size of their atoms :
|Substance||Atomic Mass (u)||Density (103 Kg m3)|
Q13.13 A gas in equilibrium has uniform density and pressure throughout its volume. This is strictly true only if there are no external influences. A gas column under gravity, for example, does not have uniform density (and pressure). As you might expect, its density decreases with height. The precise dependence is given by the so-called law of atmospheres
where n2, n1 refer to number density at heights h2 and h1 respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:
where is the density of the suspended particle, and , that of the surrounding medium. [ s Avogadro’s number, and R the universal gas constant.] [Hint: Use Archimedes principle to find the apparent weight of the suspended particle.]
Q 13.12 From a certain apparatus, the diffusion rate of hydrogen has an average value of . The diffusion of another gas under the same conditions is measured to have an average rate of . Identify the gas.
Q13.11 A metre long narrow bore held horizontally (and closed at one end) contains a long mercury thread, which traps a column of air. What happens if the tube is held vertically with the open end at the bottom ?