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- Physics
- Classical Dynamics Of Particles

Get questions and answers for Classical Dynamics Of Particles

1 Million+ Step-by-step solutions * Q:Evaluate the total energy associated with a normal mode, andEvaluate the total energy associated with a normal mode, and show that it is constant in time. Show this explicitly for the case of Example 12.3Q:Rework the problem in Example 12.7 assuming that all threeRework the problem in Example 12.7 assuming that all three particles are distanced a distance a and released from rest. Q:Consider three identical pendula instead of the two shown inConsider three identical pendula instead of the two shown in Figure 12.5 with a spring of constant 0.20 N/m between the center pendulum and each of the side ones. The mass bobs are 250g, and the pendula lengths are 47 cm. Find the normal frequencies.Q:Consider the case of a double pendulum shown in FigureConsider the case of a double pendulum shown in Figure 12-E where the top pendulum has length L1 and the bottom length is L2, and similarly, the bob masses are m1 and m2. The motion is only in the plane. Find and describe the normal modes and coordinates. Assume small oscillations.*

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Q:Find the normal modes for the coupled pendulums if in Figure 12Find the normal modes for the coupled pendulums if in Figure 12-5 when the pendulum on the left has mass bob m1 = 300 g and the right has mass bob m2 = 500g. The length of both pendula is 40cm, and the spring constant is 0.020 N/m. When the left pendulum is initially pulled back to θ1 = – 7o and released from rest when θ2 = θ2 = 0, what is the maximum angle that θ2 reaches? Use the small angle approximation.Q:Consider the line connecting (x1, y1) = (0, 0) and (x2, y2)Consider the line connecting (x1, y1) = (0, 0) and (x2, y2) = (1, 1). Show explicitly that the function y(x) = x produces a minimum path length by using the varied function y(a, x) = x + a sin π (1 – x), use the first few terms in the expansion of the resulting elliptic integral to show the equivalent of Equation 6.4.Q:Show that the shortest distance between two points on aShow that the shortest distance between two points on a plane is a straight line.

Q:Show that the shortest distance between two points in (three-dimShow that the shortest distance between two points in (three-dimensional) space is a straight line.

Q:Show that the geodesic on the surface of a rightShow that the geodesic on the surface of a right circular cylinder is a segment of a helix.

Q:Consider the surface generated by revolving a line connecting twConsider the surface generated by revolving a line connecting two fixed points (x1, y1) and (x2, y2) about an axis coplanar with the two points, Find the equation of the line connecting the points such that the surface area generated by the revolution (i.e., the area of the surface of revolution) is a minimum.

Q:Reexamine the problem of the brachistochrone (Example 6.2) andReexamine the problem of the brachistochrone (Example 6.2) and show that the time required for a particle to move (frictionlessly) to the minimum point of the cycloid is π √a/g, independent of the starting point.Q:Consider light passing from one medium with index of refractionConsider light passing from one medium with index of refraction n1 into another medium with index of refraction n2 (Figure 6-A). Use Fermatâs principle to minimize time, and derive the law of refraction: n1 sin θ1 = n2 sin θ2.

Q:Find the dimension of the parallelepiped of maximum volume circuFind the dimension of the parallelepiped of maximum volume circumscribed by

(a) A sphere of radius R;

(b) An ellipsoid with semi axes a, b, c.

Q:Find an expression involving the function Ф (x1, x2, x3) tFind an expression involving the function Ф (x1, x2, x3) that has a minimum average value of the square of its gradient within a certain volume V of space.Q:Find the ratio of the radius R to the heightFind the ratio of the radius R to the height H or a right-circular cylinder of fixed volume V that minimizes the surface area A.

Q:A disk of radius R rolls without slipping inside theA disk of radius R rolls without slipping inside the parabola y = ax2. Find the equation of constraint. Express the condition that allows the disk to roll so that I contact the parabola at one and only one point, independent of its position.

Q:Repeat Example 6.4, finding the shortest path between any twoRepeat Example 6.4, finding the shortest path between any two points on the surface of a sphere, but use the method of the Euler equations with an auxiliary condition imposed.

Q:Repeat Example 6.6 but do not use the constraint thatRepeat Example 6.6 but do not use the constraint that the y = 0 line is the bottom part of the area. Show that the plane curve of a given length, which encloses a maximum area, is a circle.

Q:Find the shortest path between the (x, y, z) points (0, – 1, 0)Find the shortest path between the (x, y, z) points (0, – 1, 0) and (0, 1, 0) on the conical surface z = 1 – √x2 + y2. What is the length of the path?Q:(a) Find the curve y(x) that passes through the endpoints(a) Find the curve y(x) that passes through the endpoints (0, 0) and (1, 1) and minimizes the functional I[y] = ∫1/0 [(dy/dx) 2 – y2] dx.

(b) What is the minimum value of the integral?

(c) Evaluate I[y] for a straight line y = x between the points (0, 0) and (1, 1).

Q:(a) What curve on the surface z = x3/2 joining(a) What curve on the surface z = x3/2 joining the points (z, y, z) = (0, 0, 0) and (1, 1, 1) has the shortest are length?

(b) Use a computer to produce a plot showing the surface and the shortest curve on single plot.

Q:The corners of a rectangle lie on the ellipse (x/a)2The corners of a rectangle lie on the ellipse (x/a)2 + (y/b)2 = 1.

(a) Where should the corners be located in order to maximize the area of the rectangle?

(b) What fraction of the area of the ellipse is covered by the rectangle with maximum area?

Q:A particle of mass m is constrained to move underA particle of mass m is constrained to move under gravity with no friction on the surface xy = z. What is the trajectory of the particle if it starts from rest at (x, y, z) = (1, – 1, – 1) with the z-axis vertical?

Q:Calculate the centrifugal acceleration, due to Earth’s rotation,Calculate the centrifugal acceleration, due to Earth’s rotation, on a particle on the surface of Earth at the equator. Compare this result with the gravitational acceleration. Compute also the centrifugal acceleration due to the motion of Earth about the Sun and justify the remark made in the text that this acceleration may be neglected compared with the acceleration caused by axial rotation.

Q:An automobile drag racer drives a car with acceleration aAn automobile drag racer drives a car with acceleration a and instantaneous velocity v. The tires (of radius r0) are not slipping Find which point on the tire has the greatest acceleration relative to the ground. What is this acceleration?

Q:In Example 10.2, assume that the coefficient of static frictionIn Example 10.2, assume that the coefficient of static friction between the hockey puck and a horizontal rough surface (on the merry-go-round) is μs. How far away from the center of the merry-go-round can the hockey puck be placed without sliding?Q:In Example 10.2, for what initial velocity and direction inIn Example 10.2, for what initial velocity and direction in the rotating system will the hockey puck appear to be subsequently motionless in the fixed system? What will be the motion in the rotating system? Let the initial position be the same as in Example 10.2. You may choose to do a numerical calculation.

Q:Perform a numerical calculation using the parameters in Example Perform a numerical calculation using the parameters in Example 10.2 and Figure 10-4e, but find the initial velocity for which the path of motion passes back over the initial position in the rotating system. At what time does the puck exit the merry-go-round?

Q:A bucket of water is set spinning about its symmetryA bucket of water is set spinning about its symmetry axis. Determine the shape of the water in the bucket.

Q:Determine how much greater the gravitational field strength gDetermine how much greater the gravitational field strength g is at the pole than at the equator. Assume a spherical Earth. If the actual measured difference is ∆g = 52 mm/s2. Explain the difference. How might you calculate this difference between the measured result and your calculation?Q:If a particle is projected vertically upward to a height hIf a particle is projected vertically upward to a height h above a point on Earth’s surface at a northern latitude λ, show that it strikes the ground at a point 4/3 w cos λ ∙ √8h3/g to the west (Neglect air resistance, and consider only small vertical heights).Q:If a projectile is fired due east from a point on the surfaceIf a projectile is fired due east from a point on the surface of Earth at a northern latitude λ with a velocity of magnitude V0 and at an angle of inclination to the horizontal of a, show that the lateral deflection when the projectile strikes Earth is

Where w is the rotation frequency of Earth.

Q:In the preceding problem, if the range of the projectileIn the preceding problem, if the range of the projectile is R0 for the case w = 0, show that the change of range due to the rotation of Earth is

Q:Obtain an expression for the angular deviation of a particleObtain an expression for the angular deviation of a particle projected from the North Pole in a path that lies close to Earth. Is the deviation significant for a missile that makes a 4,800-lm flight in 10 minutes? What is the “miss distance” if the missile is aimed directly at the target? Is the miss distance greater for a 19,300-km flight at the same velocity?

Q:Show that the small angular deviation ε of a plumb line froShow that the small angular deviation ε of a plumb line from the true vertical (i.e., toward the center of Earth) at a point on Earthâs surface at a latitude λ is where R is the radius of Earth.

What is the value (in seconds of arc) of the maximum deviation? Note that the entire denominator in the answer is actually the effective g, and g0 denotes the pure gravitational component.

Q:Refer Example 10.3 concerning the deflection from the plumbRefer Example 10.3 concerning the deflection from the plumb line of a particle falling in Earthâs gravitational field. Take g to be defined at ground level and use the zeroth order result for the time-of-fall, T = √2h/g. Perform a calculation in second approximation (i.e., retain terms in w2) and calculate the southerly deflection. There are three components to consider;

(a) Coriolis force to second order (C1),

(b) Variation of centrifugal force with height (C2), and

(c) Variation of gravitational force with height (C3).

Show that each of these components gives a result equal to with C1 = 2/3, C2 = 5/6, and C3 = 5/2. The total southerly deflection is therefore (4h2w2 sin λ cos λ)/g.

Q:Refer to Example 10.3 and the previous problem, but dropRefer to Example 10.3 and the previous problem, but drop the particle at Earthâs surface down a mineshaft to a depth h. Show that in this case there is no southerly deflection due to the variation of gravity and that the total southerly deflection is only

Q:Consider a particle moving in a potential U(r). Rewrite theConsider a particle moving in a potential U(r). Rewrite the Lagrangian in terms of a coordinate system in uniform rotation with respect to an inertial frame. Calculate the Hamiltonian and determine whether H = E. Is H a constant of the motion? If E is not a constant of motion, why isn’t it? The expression for the Hamiltonian thus obtained is the standard formula ½ mv2 + U plus and additional term. Show that the extra term is the centrifugal potential energy. Use the Lagrangian you obtained to reproduce the equations of motion given in Equation 10.25 (without the second and third terms).

Q:Consider Problem 9-63 but include the effects of the CoriolisConsider Problem 9-63 but include the effects of the Coriolis force on the probe. The probe is launched at a latitude of 45o straight up. Determine the horizontal deflection in the probe at its maximum height for each part of Problem 9-63.

Q:Approximate Lake Superior by a circle of radius 162 kmApproximate Lake Superior by a circle of radius 162 km at a latitude of 47o. Assume the water is at rest with respect to Earth and find the depth that the center is depressed with respect to the shore due to the centrifugal force.

Q:A British warship fires a projectile due south near theA British warship fires a projectile due south near the Falkland Islands during Word War I at latitude 50oS. If the shells are fired at 37o elevation with a speed of 800 m/s, by how much do the shells miss their target and in what direction? Ignore air resistance.

Q:Find the Coriolis force on an automobile of mass 1300Find the Coriolis force on an automobile of mass 1300 kg driving north near Fairbanks, Alaska (latitude 65oN) at a speed of 100 km/h.

Q:Calculate the effective gravitational field vector g at Earth’sCalculate the effective gravitational field vector g at Earth’s surface at the poles and the equator. Take account of the difference in the equatorial (6378km) and polar (6357km) radius as well as the centrifugal force. How well does the result agree with difference calculated with the result g = 9.780356[1 + 0.0052885 sin2λ -0.0000059 sin2 (2λ) m/s2 where λ is the latitude?Q:Water being diverted during a flood in Helsinki, Finland (latituWater being diverted during a flood in Helsinki, Finland (latitude 60oN) flows along a diversion channel of width 47 m in the south direction at a speed of 3.4m/s. On which side is the water the highest (from the standpoint of non-inertial systems) and by how much?

Q:Shot towers were popular in the eighteenth and nineteenth centurShot towers were popular in the eighteenth and nineteenth centuries for dropping melted lead down tall towers to form spheres for bullets. The lead solidified while falling and often landed in water to cool the lead bullets. Many such shot towers were built in New York State. Assume a shot tower was constructed at latitude 42oN, and the lead fell a distance of 27m. In what direction and how far did the lead bullets land from the direct vertical?

Q:Discuss the motion of a continuous string when the initialDiscuss the motion of a continuous string when the initial condition are q(x, 0) = 0 q(x, 0) = A sin (3πx/L). Resolve the solution into normal modes.Q:Rework the problem in Example 13.1 in the event thatRework the problem in Example 13.1 in the event that the plucked point is a distance L/3 from one end. Comment on the nature of the allowed modes.

Q:Refer to Example 13.1. Show by a numerical calculation thatRefer to Example 13.1. Show by a numerical calculation that the initial displacement of the string is well represented by the first three terms of the series in Equation 13.13. Sketch the shape of the string at intervals of time of 1/8 of a period.

Q:Discuss the motion of a string when the initial conditionsDiscuss the motion of a string when the initial conditions are q(x, 0) = 4x (L – x)/L2, q(x, 0) = 0. Find the characteristic frequencies and calculated the amplitude of the nth mode.

Q:A string with no initial displacement is set into motionA string with no initial displacement is set into motion by being struck over a length 2s about its center. This center section is given an initial velocity v0. Describe the subsequent motion.

Q:A string is set into motion by being struck atA string is set into motion by being struck at a point L/4 from one end by triangular hammer. The initial velocity is greatest at x = L/4 and decreases linearly to zero at x = 0 and x = L/2. The region L/2 Q:A string is pulled aside a distance h at aA string is pulled aside a distance h at a point 3L/7 from one end. At a point 3L/7 from the other end, the string is pulled aside a distance h in the opposite direction. Discuss the vibrations in terms of normal modes.

Q:Compare, by plotting a graph, the characteristic frequencies wTCompare, by plotting a graph, the characteristic frequencies wT as a function of the mode number r for a loaded string consisting of 3, 5, and 10 particles and for a continuous string with the same values fo τ and m/d = p. Comment on the results.Q:In Example 13.2, the complementary solution (transient part) wasIn Example 13.2, the complementary solution (transient part) was omitted. If transient effects are included, what are the appropriate conditions for over-damped, critically damped, and under damped motion? Find the displacement q(x, t) that results when under damped motion is included in Example 13.2 (assume that the motion is under damped for all normal modes).

Q:Consider the string of Example 13.1. Show that if theConsider the string of Example 13.1. Show that if the string is driven at an arbitrary point, none of the normal modes with nodes at the driving point will be excited.

Q:When a particular driving force is applied to a string,When a particular driving force is applied to a string, it is observed that the string vibration is purely of the nth harmonic. Find the driving force.

Q:Determine the complementary solution for Example 13.2Determine the complementary solution for Example 13.2

Q:Consider the simplified wave function assume that w and vConsider the simplified wave function

Assume that w and v are complex quantities and that k is real: w = a + iβ v = u + iw

Show that the wave is damped in time. Use the fact that k2 = w2/v2 to obtain expressions for a and β in terms of u and w. Find the phase velocity for this case.

Q:Consider and electrical transmission line that has a uniformConsider and electrical transmission line that has a uniform inductance per unit length L and a uniform capacitance per unit length C. Show that an alternating current I in such a line obeys the wave equation

So that the wave velocity is v = 1√LC.

Q:Consider the superposition of two infinitely long wave trains wiConsider the superposition of two infinitely long wave trains with almost the same frequencies but with different amplitudes. Show that the phenomenon of beats occurs but that the waves never beat to zero amplitude.

Q:Treat the problem of wave propagation along a string loadedTreat the problem of wave propagation along a string loaded with particles of two different masses, mâ and mââ, which alternate in placement; that is,

Show that the w â k curve has two branches in this case, and show that there is attenuation for frequencies between the branches as well as for frequencies above the upper branch.

Q:Sketch the phase velocity V (k) and the group velocity U (k)Sketch the phase velocity V (k) and the group velocity U (k) for the propagation of waves along a loaded string in the range of wave numbers 0 Q:Consider an infinitely long continuous string with linear mass dConsider an infinitely long continuous string with linear mass density p1 for x L, but density p2 > p1 for 0 Q:Consider an infinitely long continuous string with tension τConsider an infinitely long continuous string with tension τ. A mass M is attached to the string at x = 0. If a wave train with velocity w/k is incident from the left, show that reflection and transmission occur at x = 0 and that the coefficients R and T are given by R = sin2 θ, T = cos2 θ where tan θ = Mw2/2kt Consider carefully the boundary condition on the derivatives of the wave function at x = 0. What are the phase changes for the reflected and transmitted waves?Q:Consider a wave packet in which the amplitude distribution isConsider a wave packet in which the amplitude distribution is given by

Show that the wave function is ψ (x, t) = 2sin [w0t â x) ∆k]/wot â x e i (w0t â kox) Sketch the shape of the wave packet (choose t = 0 for simplicity).

Q:Consider a wave packet with a Gaussian amplitude distributionConsider a wave packet with a Gaussian amplitude distribution A (k) = B exp [â σ (k â k0)2] where 2/√σ is equal to the 1/e width* of the packet. Using this function for A (k), show that

Sketch the shape of this wave packet. Next, expand w (k) in a Taylor series, retain the first two terms, and integrate the wave packet equation to obtain the general result ψ(x, t) = B √π/σ exp [â(w0t â x) 2/4σ] exp [i (w0t â k0x)] Finally, take one additional term in the Taylor series expression w (k) and show that σ is now replaced by a complex quantity. Find the expression for the 1/e width of the packet as a function of time for this case and show that the packet moves with the same group velocity as before but spreads in width as it moves. Illustrate this result with a sketch.

Q:In section 8.2, we showed that the motion of twoIn section 8.2, we showed that the motion of two bodies interacting only with each other by central forces could be reduced to an equivalent one-boy problem. Show by explicit calculation that such a reduction is also possible for bodies moving in an external uniform gravitational field.

Q:Perform the integration of Equation 8.38 to obtain Equation 8.39Perform the integration of Equation 8.38 to obtain Equation 8.39.

Q:A particle moves in a circular orbit in a forceA particle moves in a circular orbit in a force field given by

Show that, if k suddenly decreases to half its original value, the particleâs orbit becomes parabolic.

Q:Perform an explicit calculation of the time average (i.e., thePerform an explicit calculation of the time average (i.e., the average over one complete period) of the potential energy for a particle moving in an elliptical orbit in a central inverse-square-law force field. Express the result in terms of the force constant of the field and the semi major axis of the ellipse. Perform a similar calculation for the kinetic energy. Compare the results and thereby verify the virial theorem for this case.

Q:Two particles moving under the influence of their mutualTwo particles moving under the influence of their mutual gravitational force describe circular orbits about one another with a period τ. If they are suddenly stopped in their orbits and allowed to gravitate toward each other, show that they will collide after a time τ/4√2.Q:Two gravitating masses m1 and m2 (m1 + m2 =Two gravitating masses m1 and m2 (m1 + m2 = M) are separated by a distance r0 and released from rest. Show that when the separation is r (

Q:Show that the a real velocity is constant for aShow that the a real velocity is constant for a particle moving under the influence of an attractive force given by F(r) = – kr. Calculate the time averages of the kinetic and potential energies and compare with the results of the virial theorem.

Q:Investigate the motion of a particle repelled by a forceInvestigate the motion of a particle repelled by a force center according to the law F(r) = kr. Show that the orbit can only by hyperbolic.

Q:A communications satellite is in a circular orbit around EarthA communications satellite is in a circular orbit around Earth at radius R and velocity v. A rocket accidentally fires quite suddenly, giving the rocket an outward radial velocity v in addition to its original velocity.

(a) Calculate the ratio of the new energy and angular momentum to the old.

(b) Describe the subsequent motion of the satellite and plot T(r), V(r), U(r), and E(r) after the rocket fires.

Q:Assume Earth’s orbit to be circular and that the Sun’sAssume Earth’s orbit to be circular and that the Sun’s mass suddenly decreases by half. What orbit does Earth then have? Will Earth escape the solar system?

Q:A particle moves under the influence of a central forceA particle moves under the influence of a central force given by F(r) = – k/rn, If the particle’s orbit is circular and passes through the force center, show that n = 5.

Q:Consider a comet moving in a parabolic orbit in the plane ofConsider a comet moving in a parabolic orbit in the plane of Earthâs orbit. If the distance of closest approach of the comet to the Sun is βrE, where rE is the radius of Earthâs (assumed) circular orbit and where β

If the comet approaches the Sun to the distance of the perihelion of Mercury, how many days is it within Earthâs orbit?

Q:Discuss the motion of a particle in a central inverse-squareDiscuss the motion of a particle in a central inverse-square-law force field for a superimposed force whose magnitude is inversely proportional to the cube of the distance from the particle to the force center, that is,

Show that the motion is described by a precessing ellipse. Consider the cases λ l2/μ

Q:Find the force law for a central-force field that allows aFind the force law for a central-force field that allows a particle to move in a spiral orbit given by r = kθ2, where k is a constant.Q:A particle of unit mass moves from infinity along a straightA particle of unit mass moves from infinity along a straight line that, if continued would allow it to pass a distance b √2 from a point P. If the particle is attracted toward P with a force varying as k/r5, and if the angular momentum about the point P is √k/b, show that the trajectory is given by

Q:A particle executes elliptical (but almost circular) motionA particle executes elliptical (but almost circular) motion about a force center. At some point in the orbit a tangential impulse is applied to the particle, changing the velocity from v to v + δv. Show that the resulting relative change in the major and minor axes of the orbit is twice the relative change in the velocity and that the axes are increased if δv > 0.Q:A particle moves in an elliptical orbit in an inverse-square-lawA particle moves in an elliptical orbit in an inverse-square-law central-force field. If the ratio of the maximum angular velocity to the minimum angular velocity of the particle in its orbit is n, then show that the eccentricity of the orbit is

Q:Use Kepler’s results (i.e., his first and second laws) toUse Kepler’s results (i.e., his first and second laws) to show that the gravitational force must be central and that the radial dependence musty be 1/r2. Thus, perform an inductive derivation of the gravitational force law.

Q:Calculate the missing entries denoted by c in Table 8-1.Calculate the missing entries denoted by c in Table 8-1.

Q:For a particle moving in an elliptical orbit with semi majorFor a particle moving in an elliptical orbit with semi major axis a and eccentricity ε, show that

Where the angular brackets denote a time average over one complete period

Q:Consider the family of orbits in a central potential forConsider the family of orbits in a central potential for which the total energy is a constant. Show that if a stable circular orbit exists, the angular momentum associated with this orbit is larger than that for any other orbit of the family.

Q:Discuss the motion of a particle moving in an attractiveDiscuss the motion of a particle moving in an attractive central-force field described by F(r) = – kr3. Sketch some of the orbits for different values of the total energy. Can a circular orbit be stable in such a force field?

Q:An Earth satellite moves in an elliptical orbit with a periodAn Earth satellite moves in an elliptical orbit with a period τ, eccentricity ε, and semi major axis a. Show that the maximum radial velocity of the satellite is 2πaε/π√1 – ε2).Q:An Earth satellite has a perigee of 300 km andAn Earth satellite has a perigee of 300 km and an apogee of 3,500 km above Earth’s surface. How far is the satellite above Earth when

(a) It has rotated 90o around Earth from perigee and

(b) It has moved halfway from perigee to apogee?

Q:An Earth satellite has a speed of 28,070 km/hr whenAn Earth satellite has a speed of 28,070 km/hr when it is at its perigee of 220 km above Earth’s surface. Find the apogee distance, its speed at apogee, and its period of revolution.

Q:Show that the most efficient way to change the energyShow that the most efficient way to change the energy of an elliptical orbit for a single short engine thrust is by firing the rocket along the direction of travel at perigee.

Q:A spacecraft in an orbit about Earth has the speedA spacecraft in an orbit about Earth has the speed of 10,160m/s at a perigee of 6,680 km from Earth’s center. What speed does the spacecraft have at apogee of 42,200 km?

Q:What is the minimum escape velocity of a spacecraft fromWhat is the minimum escape velocity of a spacecraft from the moon?

Q:The minimum and maximum velocities of a moon rotating aroundThe minimum and maximum velocities of a moon rotating around Uranus are vmin = v – v0 and vmax v + v0. Find the eccentricity in terms of v and v0.

Q:A spacecraft is placed in orbit 200km above Earth inA spacecraft is placed in orbit 200km above Earth in a circular orbit. Calculate the minimum escape speed from Earth. Sketch the escape trajectory, showing Earth and the circular orbit. What is the spacecraft’s trajectory with respect to Earth?

Q:Consider a force law of the form Show that if p2kConsider a force law of the form

Show that if p2k > kâ, then a particle can move in stable circular orbit at r = p.

Q:Consider the form stability of circular orbits this force field.Consider a force law of the form (F(r) = – (k/r2) exp (– r/a). Investigate the stability of circular orbits in this force field.

Q:Consider a particle of mass m constrained to move on theConsider a particle of mass m constrained to move on the surface of a paraboloid whose equation (in cylindrical coordinates) is r2 = 4az. If the particle is subject to a gravitational force, show that the frequency of small oscillations about a circular orbit with radius p = √4az0 is

Q:Consider the problem of the particle moving on the surfaceConsider the problem of the particle moving on the surface of a cone, as discussed in Examples 7.4 and 8.7. Show that the effective potential is

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