Questions and Answers of Classical Dynamics Of Particles

Evaluate the total energy associated with a normal mode, and show that it is constant in time. Show this explicitly for the case of Example 12.3
Rework the problem in Example 12.7 assuming that all three particles are distanced a distance a and released from rest.
Consider 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
Consider 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
Find 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
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
Show that the shortest distance between two points on a plane is a straight line.
Show that the shortest distance between two points in (three-dimensional) space is a straight line.
Show that the geodesic on the surface of a right circular cylinder is a segment of a helix.
Consider 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
Reexamine 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,
Consider light passing from one medium with index of refraction n1 into another medium with index of refraction n2 (Figure 6-A). Use Fermat€™s principle to minimize time, and derive the law
Find 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.
Find 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.
Find 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.
A 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
Repeat 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.
Repeat 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.
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?
(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)
(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
The 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
A 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
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
An 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
In 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
In 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
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
A bucket of water is set spinning about its symmetry axis. Determine the shape of the water in the bucket.
Determine 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
If 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 ∙
If 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
In 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
Obtain 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
Show 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 Earth€™s surface at a latitude λ is where R is
Refer Example 10.3 concerning the deflection from the plumb line of a particle falling in Earth€™s gravitational field. Take g to be defined at ground level and use the zeroth order result
Refer to Example 10.3 and the previous problem, but drop the particle at Earth€™s surface down a mineshaft to a depth h. Show that in this case there is no southerly deflection due to the
Consider 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
Consider 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
Approximate 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
A 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
Find the Coriolis force on an automobile of mass 1300 kg driving north near Fairbanks, Alaska (latitude 65oN) at a speed of 100 km/h.
Calculate 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
Water 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
Shot 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
Discuss 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.
Rework 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.
Refer 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
Discuss 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.
A 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.
A 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
A 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
Compare, 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
In 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
Consider 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.
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.
Determine the complementary solution for Example 13.2
Consider the simplified wave functionAssume that w and v are complex quantities and that k is real: w = a + iβ v = u + iwShow that the wave is damped in time. Use the fact that k2 = w2/v2 to
Consider 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
Consider 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
Treat the problem of wave propagation along a string loaded with particles of two different masses, m€™ and m€™€™, which alternate in placement; that is,Show that the w
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 < k < π/d. Show that U (π/d) = 0, whereas
Consider an infinitely long continuous string with linear mass density p1 for x < 0 and for x > L, but density p2 > p1 for 0 < x < L. If a wave train oscillating with an angular frequency w is
Consider 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
Consider a wave packet in which the amplitude distribution is given byShow that the wave function is ψ (x, t) = 2sin [w0t €“ x) ∆k]/wot €“ x e i (w0t €“ kox)
Consider a wave packet with a Gaussian amplitude distribution A (k) = B exp [€“ σ (k €“ k0)2] where 2/√σ is equal to the 1/e width* of the packet. Using this
In 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
Perform the integration of Equation 8.38 to obtain Equation 8.39.
A particle moves in a circular orbit in a force field given byShow that, if k suddenly decreases to half its original value, the particle€™s orbit becomes parabolic.
Perform 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
Two 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
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 (
Show 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
Investigate 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.
A 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
Assume 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?
A 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.
Consider a comet moving in a parabolic orbit in the plane of Earth€™s orbit. If the distance of closest approach of the comet to the Sun is βrE, where rE is the radius of
Discuss 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
Find 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.
A 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
A 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 +
A 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,
Use 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
Calculate the missing entries denoted by c in Table 8-1.
For a particle moving in an elliptical orbit with semi major axis a and eccentricity ε, show thatWhere the angular brackets denote a time average over one complete period
Consider 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
Discuss 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
An 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
An 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
An 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.
Show 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.
A 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?
What is the minimum escape velocity of a spacecraft from the moon?
The 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.
A 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
Consider a force law of the formShow that if p2k > k€™, then a particle can move in stable circular orbit at r = p.
Consider a force law of the form (F(r) = – (k/r2) exp (– r/a). Investigate the stability of circular orbits in this force field.
Consider 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
Consider 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

Showing 200 - 300 of 1329