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1 Million+ Step-by-step solutions * Q:If a projectile moves such that its distance from theIf a projectile moves such that its distance from the point of projection is always in creasing, find the maximum angle above the horizontal with which the particle could have been projected. (Assume no air resistance)Q:A gun fires a projectile of mass 10kg of theA gun fires a projectile of mass 10kg of the type to which the curves of Figure 2-3 apply. The muzzle velocity is 140m/s. Through what angle must the barrel be elevated to hit a target on the same horizontal plane as the gun and 1000 m away? Compare the results with those for the case of no retardation.Q:Show directly that the time rate of change of theShow directly that the time rate of change of the angular momentum about the origin for a projectile fired from the origin constant g is equal to the moment force or torque about the origin.Q:The motion of a charged particle in an electromagnetic fieldThe motion of a charged particle in an electromagnetic field can be obtained from the Lorentz equation for the force on a particle in such a field, if the electric field vector is E and the magnetic field vector is B, the force on a particle of mass m that carries a charge q and has a velocity v is given by F = qE + qv x B where we assume that v (a) If there is no electric field and if the particle enters the magnetic field in a direction perpendicular to the lines of magnetic flux, show that the trajectory is a circle with radius r = mv/qB = v/wc where wc = qB/m is the cyclotron frequency.(b) Choose the z-axis to lie in the direction of B and let the plane containing E and B be the yz-plane. Thus B = Bk, E = E, j + Ezk Show that the z component of the motion is given by z(t) = z0 + z0t qEz/2mt2 where z(0) = z0 and z(0) = z0(c) Continue the calculation and obtain expressions for x(t) and y(t). Show that the time averages of these velocity components are (x) Ey/Bâ (y) = 0 (Show that the motion is periodic and then average over one complete period).(d) Integrate the velocity equations found in Â© and show (with the initial condition x(0) = â A A/w, x(0) = Ey/B, y(0) = 0, y(0) â A) that*

These are the parametric equations of a trochoid. Sketch the projection of the trajectory on the xy â plane for the cases (i) A > |Ey/B|, (ii) A

Q:A particle of mass m = 1 kg is subjectedA particle of mass m = 1 kg is subjected to a one dimensional force F(t) – kte–at, where k = 1 N/s and a = 0.5 s–1. If the particle is initially at rest, calculate and plot with the aid of a computer the position, speed, and acceleration of the particle as a function of time.

Q:A skier weighing 90kg starts from rest down a hillA skier weighing 90kg starts from rest down a hill inclined at 17o. He skis 100 m down the hill and then coasts for 70 along level snow until he stops. Find the coefficient of kinetic friction between the skis and the snow. What velocity does the skier have at the bottom of the hill?

Q:A block of mass m = 1.62kg slides down aA block of mass m = 1.62kg slides down a frictionless incline (Figure 2-A). The block is released a height h = 3.91, above the bottom of the loop.

(a) What is the force of the inclined track on the block at the bottom (point A)?

(b) What is the force of the track on the block at point B?

(c) At what speed does the block leave the track?

(d) How far away from point A does the block land level ground?

(e) Sketch the potential energy U(x) of the block. Indicate the total energy on the sketch.

Q:A child slides a block of mass 2 kg along a slick kitchen floorA child slides a block of mass 2 kg along a slick kitchen floor. If the initial speed is 4m/s and the block hits a spring with spring constant 6 N/m, what is the maximum compression of the spring? What is the result if the block slides across 22m of a rough floor that ha μh = 0.2?Q:A rope having a total mass of 0.4kg and totalA rope having a total mass of 0.4kg and total length 4 m has 0.6 m of the rope hanging vertically down off a work bench. How much work must be done to place all the rope on the bench?

Q:A super ball of mass M and a marble ofA super ball of mass M and a marble of mass m are dropped from a height h with the marble just on top of the super ball. A super ball has a coefficient of restitution of nearly 1 (i.e., its collision is essentially elastic). Ignore the sizes of the super ball and marble. The super ball collides with the floor, rebounds, and smacks the marble, which moves back up. How high does the marble go if all the motion is vertical? How high does the super ball go?

Q:An automobile driver traveling down and 8% grade slams on hisAn automobile driver traveling down and 8% grade slams on his brakes and skids 30 m before hitting a parked car. A lawyer hires an expert who measures the coefficient of kinetic friction between the tires and road to be μk = 0.45. Is the lawyer correct to accuse the driver of exceeding the 25-MPH speed limit? Explain.Q:A student drops a water-filled balloon from the roof ofA student drops a water-filled balloon from the roof of the tallest building in two trying to hit her roommate on the ground (who is too quick). The first student ducks back but hear the water splash 4.021 s after dropping the balloon. If the speed of sound is 331 m/s, find the height of the building, neglecting air resistance.

Q:In Example 2.10, the initial velocity of the incoming chargedIn Example 2.10, the initial velocity of the incoming charged particle had no component along the x-axis. Show that, even if it had and x component, the subsequent motion of the particle would be the same that only the radius of the helix would be altered.

Q:Two blocks of unequal mass are connected by a string over aTwo blocks of unequal mass are connected by a string over a smooth pulley (Figure 2-B). If the coefficient of kinetic friction is μv what angle θ of the incline allows the masses to move at a constant speed?

Q:A particle is released from rest (y = 0) and falls under theA particle is released from rest (y = 0) and falls under the influence of gravity and air resistance. Find the relationship between v and the distance of falling y when the air resistance is equal to.

(a) av and

(b) βv2.

Q:Perform the numerical calculations of Example 2.7 for the valuesPerform the numerical calculations of Example 2.7 for the values given are Figure 2-8. Plot Figures 2-8 and 2-9. Do not duplicate the solution in Appendix H; compose your own solution.

Q:A gun is located on a bluff of height hA gun is located on a bluff of height h overlooking a river valley. If the muzzle velocity is v0 find the expression for the range as a function of the elevation angle of the gun. Solve numerically for the maximum range out into the valley for a given h and v0.

Q:A particle of mass m has speed v = a/x,A particle of mass m has speed v = a/x, where x is its displacement. Find the force F(x) responsible.

Q:The speed of a particle of mass m varies withThe speed of a particle of mass m varies with the distance x as v(x) = ax –n. Assume v(x = 0) = 0 at t = 0.

(a) Find the force F(x) responsible.

(b) Determine x (t) and

(c) F (t)

Q:A boat with initial speed v0 is launched on aA boat with initial speed v0 is launched on a lake. The boat is slowed by the water by a force F = – ae.

(a) Find an expression for the speed v (t).

(b) Find the time and

(c) Distance for the boat to stop.

Q:A particle moves in a two-dimensional orbit defined by (a)A particle moves in a two-dimensional orbit defined by

(a) Find the tangential acceleration at and normal acceleration an as a function of time where the tangential and normal components are taken with respect to the velocity.

(b) Determine at what times in the orbit an has a maximum.

Q:A train moves along the tracks at a constant speedA train moves along the tracks at a constant speed v. A woman on the train throws a ball of mass m straight ahead with a speed v with respect to herself.

(a) What is the kinetic energy gain of the ball as measured by a person on the train?

(b) by a person standing by the railroad track?

(c) How much work is done by the woman throwing he ball and

(d) By the train?

Q:A solid cube of uniform density and sides of bA solid cube of uniform density and sides of b is in equilibrium on top of a cylinder of radius R (Figure 2-C). The planes of four sides of the cube are parallel to the axis of the cylinder. The contact between cube and sphere is perfectly rough. Under what conditions is the equilibrium stable or not stable?

Q:A particle is under the influence of a force FA particle is under the influence of a force F – kx + kx3/a2, where k and a are constants and k is positive. Determine U(x) and discuss the motion. What happens when E = (1/4) ka2?

Q:A potato of mass 0.5 kg moves under Earth’s gravityA potato of mass 0.5 kg moves under Earth’s gravity with an air resistive force of –kmv.

(a) Find the terminal velocity if the potato is released from rest and k = 0.01 s–1.

(b) Find the maximum height of the potato if it has the same value of k, but is initially shot directly upward with a student-made potato gun with an initial velocity of 120m/s.

Q:A pumpkin of mass 5 kg shot out of aA pumpkin of mass 5 kg shot out of a student-made cannon under air pressure at an elevation angle of 45o fell at a distance of 142 m from the cannon. The students used light beams and photocells to measure the initial velocity of 54m/s. If the air resistive force was F = – kmv, what was the value of k?

Q:Write the criteria for determining whether an equilibrium is staWrite the criteria for determining whether an equilibrium is stable or unstable when all derivatives up through order n, (dnU/dxn) 0 = 0.

Q:Consider a particle moving in the region x > 0Consider a particle moving in the region x > 0 under the influence of the potential

Where U0 = 1 J and a = 2m. Plot the potential, find the equilibrium points, and determine whether they are maxima or minima.

Q:Two gravitationally bound stars with equal masses m, separated bTwo gravitationally bound stars with equal masses m, separated by a distance d, revolve about their center of mass in circular orbits. Show that the period r is proportional to d3/2 (Kepler’s Third Law) and find the proportionality constant.

Q:Two gravitationally bound stars with unequal masses m1 and m2,Two gravitationally bound stars with unequal masses m1 and m2, separated by a distance d, revolve about their center of mass in circular orbits. Show that the period r is proportional to d3/2 (Kepler’s Third Law) and find the proportionality constant.

Q:According to special relativity, a particle of rest mass m0According to special relativity, a particle of rest mass m0 accelerated in one dimension by a force F obeys the equation of motion dp/dt = F. Here p = m0v/(1 – v2/c2)1/2 is the relativistic momentum, which reduces to m0v for v2/c2 (a) For the case of constant F and initial conditions x (0) = 0 – v (0), find x (t) and v(t).

(b) Sketch your result for v (t).

(c) Suppose that F/mo = 10 m/s2 (≈ g on Earth). How much time is required for the particle ro reach half the speed of light and of 99% the speed of light?

Q:Let us make the (unrealistic) assumption that a boat ofLet us make the (unrealistic) assumption that a boat of mass m gliding with initial velocity vo in water is slowed by a viscous retarding force of magnitude bv2, where b is a constant.

(a) Find and sketch v(t). How long does it take the boat to reach a speed of v0/1000?

(b) Find x (t). How far does that boat travel in this time? Let m = 200kg, vo = 2m/s, and b = 0.2 Nm– 2s2.

Q:A particle of mass m moving in one dimension hasA particle of mass m moving in one dimension has potential energy U(x) = U0 [2(x/a) 2 – (x/a) 4], where U0 and a are positive constant.

(a) Find the force F(x), which acts on the particle.

(b) Sketch U(x). Find the positions of stable and unstable equilibrium.

(c) What is the angular frequency w of oscillations about the point of stable equilibrium?

(d) What is the minimum speed the particle must have at the origin to escape to infinity?

(e) At t = 0 the particle is at the origin and its velocity is positive and equal in magnitude to the escape speed of part (d). Find x(t) and sketch the result.

Q:Which of the following forces are conservative? If conservative,Which of the following forces are conservative? If conservative, find the potential energy U(r).

(a) Fx = ayz + bx + c, Fy = axz + bz, Fz = axy + by.

(b) Fx = – ze-x, F, = In z, Fx = e–x +y/z.

(c) F = era/r(a, b, c are constants).

Q:A simple harmonic oscillator consists of a 100-g mass attachedA simple harmonic oscillator consists of a 100-g mass attached to a spring whose force constant is 104 dyne/cm. The mass is displaced 3cm and released from rest. Calculate

(a) The natural frequency v0 and the period T0.

(b) The total energy, and

(c) The maximum speed.

Q:Allow the motion in the preceding problem to take place inAllow the motion in the preceding problem to take place in a resisting medium. After oscillating for 10s, the maximum amplitude decreases to half the initial value. Calculate

(a) The damping parameter β.

(b) The frequency v1 (compare with the un-damped frequency v0), and

(c) The decrement of the motion.

Q:The oscillator of Problem 3-1 is set into motion byThe oscillator of Problem 3-1 is set into motion by giving it an initial velocity of 1 cm/s at its equilibrium position. Calculate

(a) The maximum displacement and

(b) The maximum potential energy.

Q:Consider a simple harmonic oscillator. Calculate the time averagConsider a simple harmonic oscillator. Calculate the time averages of the kinetic and potential energies over one cycle, and show that these quantities are equal. Why is this a reasonable result? Next calculate the space averages of the kinetic and potential energies. Discuss the results.

Q:Obtain and expression for the fraction of a complete periodObtain and expression for the fraction of a complete period that a simple harmonic oscillator spends within a small interval Δx at a position x. Sketch curves of this function versus x for several different amplitudes. Discuss the physical significance of the results. Comment on the areas under the various curves.Q:Two masses m1 = 100 g and m2 = 200gTwo masses m1 = 100 g and m2 = 200g slide freely in a horizontal frictionless track and are connected by a spring whose force constant is k = 0.5 N/m. Find the frequency of oscillatory motion for this system.

Q:Where g is the gravitational field strength, determine the valueWhere g is the gravitational field strength, determine the value of T.

Q:A pendulum is suspended from the cusp of a cycloid*A pendulum is suspended from the cusp of a cycloid* cut in a rigid support (Figure 3-A). The path described by the pendulum bob is cycloidal and is given by x = a (Ã – sin Ã), y = a (cos Ã â 1)

Where the length of the pendulum is l = 4a, and where Ã is the angle of rotation of the circle generating the cycloid. Show that the oscillations are exactly isochronous with frequency w0 = √g / l, independent of the amplitude.

Q:A particle of mass m is at r4est at theA particle of mass m is at r4est at the end of a spring (force constant = k) hanging from a fixed support. At t = 0, a constant downward force F is applied to the mass and acts for time t0. Show that, after the force is removed, the displacement of the mass from its equilibrium position (x = x0, where x is down) is where w2/0 = k/m.

Q:If the amplitude of a damped oscillator decreases to 1/e of itsIf the amplitude of a damped oscillator decreases to 1/e of its initial value after n periods, show that the frequency of the oscillator must be approximately [1 = (8π2n2)–1] times the frequency of the corresponding undamped oscillator.Q:Derive the expressions for the energy and energy-loss curves shoDerive the expressions for the energy and energy-loss curves shown in Figure 3-8 for the damped oscillator. For a lightly damped oscillator, calculate the average rate at which the damped oscillator loses energy (i.e., compute a time average over one cycle).

Q:A simple pendulum consists of a mass m suspended from a fixedA simple pendulum consists of a mass m suspended from a fixed point by a weight-less, extension less rod of length l. Obtain the equation of motion and, in the approximation that sin θ ≡ θ, show that the natural frequency is w0 = √g/l, where g is the gravitational field strength. Discuss the motion in the event that the motion takes place in a viscous medium with retarding force 2m√gl θ.Q:Show that Equation 3.43 is indeed the solution for criticalShow that Equation 3.43 is indeed the solution for critical damping by assuming a solution of the form x(t) = y(t) exp( – βt) and determining the function y(t).Q:Express the displacement x(t) and the velocity x(t) for theExpress the displacement x(t) and the velocity x(t) for the over damped oscillator in terms of hyperbolic functions.

Q:Reproduce Figures 3-10b and c for the same values given inReproduce Figures 3-10b and c for the same values given in Example 3.2, but instead let β = 0.1 s–1 and δ = π rad. How many times does the system cross the x = 0 line before the amplitude finally falls below 10-2 of its maximum value? Which plot, b or c, is more useful for determining this number? Explain.Q:Discuss the motion of a particle described by Equation 3.34Discuss the motion of a particle described by Equation 3.34 in the event that b Q:For a damped, driven oscillator, show that the average kineticFor a damped, driven oscillator, show that the average kinetic energy is the same at a frequency of a given number of octaves* above the kinetic energy resonance as at a frequency of the same number of octaves below resonance.

Q:Show that, if a driven oscillator is only lightly dampedShow that, if a driven oscillator is only lightly damped and driven near resonance, the Q of the system is approximately.

Q:For a lightly damped oscillator, show that Q ≡ w0/Δw For a lightly damped oscillator, show that Q ≡ w0/Δw (Equation 3.65).Q:Plot a velocity resonance curve for a driven, damped oscillatorPlot a velocity resonance curve for a driven, damped oscillator with Q = 6, and show that the full width of the curve between the points corresponding to xmax/√2 is approximately equal to w0/6.Q:Use a computer to produce a phase space diagram similar toUse a computer to produce a phase space diagram similar to Figure 3-11 for the case of critical damping. Show analytically that the equation of the line that the phase paths approach asymptotically is x = – βx. Show the phase paths for at least three initial positions above and below the line.Q:Let the initial position and speed of an over damped,Let the initial position and speed of an over damped, non-driven oscillator be x0 and v0, respectively.

(a) Show that the values of the amplitudes A1 and A2 in Equation 3.44 have the values A1 = β2×0 + v0/ β2 – β1 and A2 = β1×0 + v0/ β2 – β1 where β1 β – w2 and β2 = β + w2.

(b) Show that when A1 = 0, the phase paths of Figure 3-11 must be along the dashed curve given by x = – β2x, otherwise the asymptotic paths are along the other dashed curve given by x = – β1x.

Q:To better understand under damped motion, use a computer toTo better understand under damped motion, use a computer to plot x(t) of Equation 3.40 (with A = 1m) and its two components [e –βt and cos (w1t – δ)] and comparisons (with β = 0) on the same plot as in Figure 3-6. Let w0 = 1 rad/s and make separate plots for β2/w20 = 0.1, 0.5, and 0.9 and for δ (in radians) = 0, π/2, and π. Have only one value of δ and β on each plot (i.e., nine plots). Discuss the results.Q:For β = 0.2 s–1, produce computer plots like those shown inFor β = 0.2 s–1, produce computer plots like those shown in Figure 3-15 for a sinusoidal driven, damped oscillator where xp(t), xc(t), and the sum x(t) are shown . Let k = 1kg/s2 and m = 1kg. Do this for values of w/w1 of 1/9, 1/3, 1.1,3, and 6. For the xc (t) solution (Equation 3.40), let the phase angle δ = 0 and the amplitude A = – 1m. For the xp (t) solution (Equation 3.60), let A = 1 m/s2 but calculate δ. What do you observe about the relative amplitudes of the two solutions as w increases? Why does this occur? For w/w1 = 6, A 20 m/s2 for xp (t) and produce the plot again.Q:Figure 3-B illustrates a mass m1 driven by a sinusoidalFigure 3-B illustrates a mass m1 driven by a sinusoidal force whose frequency is w. The mass m1 is attached to a rigid support by a spring of force constant k and slides on a second mass m2. The frictional force between m1 and m2 is represented by the damping parameter b1, and the frictional force between m2 and the support is represented by b2. Construct the electrical analog of this system and calculate the impedance.

Q:Show that the Fourier series of Equation 3.89 can beShow that the Fourier series of Equation 3.89 can be expresses as

Relate the coefficients cn to the an and bn of Equation 3.90.

Q:Obtain the Fourier expansion of the function In the intervalObtain the Fourier expansion of the function

In the interval â π/wQ:Obtain the Fourier series representing the function SOLUTION: Obtain the Fourier series representing the function

Q:Obtain the Fourier representation of the output of a full-waveObtain the Fourier representation of the output of a full-wave rectifier. Plot the first three terms of the expansion and compare with the exact function.

Q:A damped linear oscillator, originally at rest in itsA damped linear oscillator, originally at rest in its equilibrium position, is subjected to a forcing function given by

Find the response function. Allow T → 0 and show that the solution becomes that fore a step function.

Q:Obtain the response of a linear oscillator to a step functionObtain the response of a linear oscillator to a step function and to an impulse function (in the limit T → 0) for over damping. Sketch the response functions.Q:Calculate the maximum values of the amplitudes of the responseCalculate the maximum values of the amplitudes of the response functions shown in Figures 3-22 and 3-24. Obtain numerical values for β = 0.2w0 when a = 2m/s2, w0 = 1 rad/s, and t0 = 0.Q:Consider an undamped linear oscillator with a naturalConsider an undamped linear oscillator with a natural frequency w0 = 0.5 rad/s and the step function a = 1 m/s2. Calculate the sketch the response function for an impulse forcing function acting for time T = 2π/w0. Give a physical interpretation of the results.Q:Obtain the response of a linear oscillator to the forcingObtain the response of a linear oscillator to the forcing function

Q:Derive and expression for the displacement of a linear oscillatoDerive and expression for the displacement of a linear oscillator analogous to Equation 3.110 but for the initial conditions x(t0) = x0 and x(t0) = x0.

Q:Derive the Green’s method solution for the response caused byDerive the Green’s method solution for the response caused by an arbitrary forcing function. Consider the function to consist of a series of step functions-that is, start from Equation 3.105 rather than from Equation 3.110.

Q:Use Green’s method to obtain the response of a dampedUse Greenâs method to obtain the response of a damped oscillator to a forcing function of the form

Q:Consider the periodic function represents the positive portions Consider the periodic function

Which represents the positive portions of a sine function (Such a function represents, for example, the output of a half-wave rectifying circuit)? Find the Fourier representation and plot the sum of the first four terms.

Q:An automobile with a mass of 1000 kg, including passengers,An automobile with a mass of 1000 kg, including passengers, settles 1.0 cm closer to the road for every additional 100kg of passengers. It is driven with a constant horizontal component of speed 20km/h over a washboard road with sinusoidal bumps. The amplitude and wavelength of the sine curve are 5.0 cm and 20 cm, respectively. The distance between the front and back wheels is 2.4m. Find the amplitude of oscillation of the automobile, assuming it moves vertically as an undamped driven harmonic oscillator. Neglect the mass of the wheels and springs and assume that the wheels are always in contact with the road.

Q:(a) Use the general solutions x(t) to the differential equation(a) Use the general solutions x(t) to the differential equation d2 x/dt2 + 2βdx/dt + w2/0 x = 0 for under damped, critically damped, and over damped motion and choose the constants of integration to satisfy the initial conditions x = x0 and v = v0 = 0 at t = 0.;

(b) Use a computer to plot the results for x (t) x0 as a function of w0t in the three cases β = (1/2) w0, β = w0, and β = 2w0. Show all three curves on a single plot.

Q:An un-damped driven harmonic oscillator satisfies the equation oAn un-damped driven harmonic oscillator satisfies the equation of motion m (d2x/dt2 + w2/0 = F(t). The driving force F (t) = F- sin (wt) is switched on at t = 0.

(a) Find x (t) for t > 0 for the initial conditions x = 0 and v = 0 at t = 0.

(b) Find x (t) for w = w0 by taking the limit w → w0 in your result for part (a). Sketch your result for x (t).

Q:A point mass m slides without friction on a horizontalA point mass m slides without friction on a horizontal table at one end of a mass less spring of natural length a and spring constant k as shown in Figure 3-C. The spring is attached to the table so it can rotate freely without friction. The net force on the mass is the central force F(r) = â k (r â a).

(a) Find and sketch both the potential energy U(r) and the effective potential Ueff(r).

(b) What angular velocity w0 is required for a circular orbit with radius r0?

(c) Derive the frequency of small oscillations w about the circular orbit with radius r0. Express your answers for (b) and (c) in terms of k, m, r0, and a.

Q:Consider a damped harmonic oscillator. After four cycles the ampConsider a damped harmonic oscillator. After four cycles the amplitude of the oscillator has dropped to 1/e of its initial value. Find the ratio of the frequency of the damped oscillator to its natural frequency.

Q:A grandfather clock has a pendulum length of 0.7m andA grandfather clock has a pendulum length of 0.7m and mass bob of 0.4kg. A mass of 2kg falls 0.8m in seven days to keep the amplitude (from equilibrium) of the pendulum oscillation steady at 0.03 rad. What is the Q of the system?

Q:Reconsider the problem of two coupled oscillators discussed in SReconsider the problem of two coupled oscillators discussed in Section 12.2 in the event that the three springs all have different force constants. Find the two characteristic frequencies, and compare the magnitudes with the natural frequencies of the two oscillators in the absence of coupling.

Q:Continue Problem 12-1, and investigate the case of weak couplingContinue Problem 12-1, and investigate the case of weak coupling: k12 Q:Two identical harmonic oscillators (with masses M and natural frTwo identical harmonic oscillators (with masses M and natural frequencies w0) are coupled such that by adding to the system a mass m common to both oscillators the equations of motion become

Solve this pair of coupled equation, and obtain the frequencies of the normal modes of the system.

Q:Refer to the problem of the two coupled oscillators discussedRefer to the problem of the two coupled oscillators discussed in Section 12.2. Show that the total energy of the system is constant. (Calculate the kinetic energy of each of the particles and the potential energy stored in each of the three springs, and sum the results). Notice that the kinetic and potential energy terms that have k12 as a coefficient depend on C1 and w1 but not on C2 or w2. Why it such a result to be expected?

Q:Find the normal coordinates for the problem discussed inFind the normal coordinates for the problem discussed in Section 12.2 in Example 12.1 if the two masses are different, m1 ≠ m2. You may again assume all the k are equal.Q:Two identical harmonic oscillators are placed such that the twoTwo identical harmonic oscillators are placed such that the two masses slide against one another, as in Figure 12-A. The frictional force provides a coupling of the motions proportional to the instantaneous relative velocity. Discuss the coupled oscillations of the system.

Q:A particle of mass m is attached to a rigidA particle of mass m is attached to a rigid support by a spring with force constant k. At equilibrium, the spring hangs vertically downward. To this mass-spring combination is attached an identical oscillator, the spring of the latter being connected to the mass of the former. Calculate the characteristic frequencies for one-dimensional vertical oscillations, and compare with the frequencies when one or the other of the particle is held fixed while the other oscillates. Describe the normal modes of motion for the system.

Q:A simple pendulum consists of a bob of mass m suspended by anA simple pendulum consists of a bob of mass m suspended by an inextensible (and mass less) string of length l. From the bob of this pendulum is suspended a second, identical pendulum. Consider the case of small oscillations (so that sin θ ≡ θ), and calculate the characteristic frequencies. Describe also the normal modes of the system.Q:The motion of a pair of coupled oscillators may beThe motion of a pair of coupled oscillators may be described by using a method similar to that used in constructing a phase diagram for a single oscillator (Section 3.4). For coupled oscillators, the two position x1(t) and x2(t) may be represented by a point (the system point) in the two-dimensional configuration space x1-x2. As t increases, the locus of all such points defines a certain curve. The loci of the projection of the system points onto the x1- and x2-axes represent the motions of m1 and m2, respectively. In the general case, x1(t) and x2(t) are complicated functions, and so the curve is also complicated. But it is always possible to rotate the x1-x2 axes to a new set x1-x2 in such a way that the projection of the system point onto each of the new axes is simple harmonic. The projected motions along the new axes take places

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