Questions and Answers of Classical Dynamics Of Particles

An almost circular orbit (i.e., ε
A particle moves in an almost circular orbit in a force field described by F(r) – k/r2) exp (– r/a). Show that the upsides advance by an amount approximately equal to πp/a in each
A communication satellite is in a circular orbit around Earth at a distance above Earth equal to Earth’s radius. Find the minimum velocity ∆v required to double the height of the satellite
Calculate the minimum ∆v required to place a satellite already in Earth’s heliocentric orbit (assumed circular) into the orbit of Venus (also assumed circular and coplanar with Earth).
Assuming a rocket engine can be fired only once from a low Earth orbit, does Mars flyby or a Venus flyby require a larger ∆v? Explain.
A spacecraft is being designed to dispose of nuclear waste either by carrying it out of the solar system or crashing into the Sun. Assume that no planetary flyby are permitted and that thrusts occur
A spacecraft is parked in a circular orbit 200 km above Earth’s surface. We want to use a Hohmann transfer to send the spacecraft to the Moon’s orbit. What are the total ∆v and the transfer
A spacecraft of mass 10,000kg is parked in a circular orbit 200km above Earth’s surface. What is the minimum energy required (neglect the fuel mass burned) to place the satellite in a synchronous
A satellite is moving in circular orbit of radius R about Earth. By what fraction must its velocity v be increased for the satellite to be in an elliptical orbit with rmin = R and rmax = 2R?
The Yukawa potential adds an exponential term to the long-range Coulomb potential, which greatly shortens the range of the Coulomb potential. It has great usefulness in atomic and nuclear
A particle of mass m moves in a central force field that has a constant magnitude F0, but always points toward the origin. (a) Find the angular velocity wФ required for the particle to move in
Two double stars of the same mass as the sun rotate about their common center of mass. Their separation is 4 light years. What is their period of revolution?
Two double stars, one having mass 1.0 M sun and the other 3.0 M sun, rotate about their common center of mass. Their separation is 6 light years. What is their period of revolution?
Calculate the moments of inertia I1, I2, and I3, for a homogeneous sphere of radius R and mass M. (Choose the origin at the center of the sphere).
Calculate the moments of inertia I1, I2, and I3, for a homogeneous cone of mass M whose height is h and whose base has a radius R. Choose the x3-axis along the axis of symmetry of the cone. Choose
Calculate the moments of inertia I1, I2, and I3, for a homogeneous ellipsoid of mass M with axes’ lengths 2a > 2b > 2c.
Consider a thin rod of length l and mass m pivoted about one end. Calculate the moment of inertia. Find the point at which, if all the mass were concentrated, the moment of inertia about the pivot
(a) Find the height at which a billiard ball should be struck so that it will roll with no initial slipping.(b) Calculate the optimum height of the rail of a billiard table. On what basis is the
Two spheres are of the same diameter and same mass, but one is solid and the other is a hollow shell, Describe in detail a non-destructive experiment to determine which is solid and which is hollow.
A homogeneous disk of radius R and mass M rolls without slipping on a horizontal surface and is attracted to a point a distance d below the plane. If the force of attraction is proportional to the
A door is construct of a thin homogenous slab of material: it has a width of 1m. If the door is opened through 90o, it is found that on release it closes itself in 2s. Assume that the hinges are
A homogeneous slab of thickness a is placed atop a fixed cylinder of radius R whose axis is horizontal. Show that the condition for stable equilibrium of the slab, assuming no slipping, is R > a/2.
A solid sphere of mass M and radius R rotates freely in space with an angular velocity w about a fixed diameter. A particle of mass m, initially at one pole, moves with a constant velocity v along a
A homogeneous cube, each edge of which has a length l, is initially in a position of unstable equilibrium with one edge is contact with a horizontal plane. The cube is then given a small displacement
Show that none of the principal moments of inertia can exceed the sum of the other two.
A three-particle system consists of masses mi and coordinates (x1, x2, x3) as follows:Find the inertia tensor, principal axes, and principal moments of inertia.
Determine the principal axes and principal moments of inertia of a uniformly solid hemisphere of radius b and mass m about its center of mass.
If a physical pendulum has the same period of oscillation when pivoted about either of two points of unequal distances from the center of mass, show that the length of the simple pendulum with the
Consider the following inertia tensor:Perform a rotation of the coordinate system by an angle θ about the x3-axis. Evaluate the transformed tensor elements, and show that the choice θ =
Consider a thin homogeneous plate that lies in the x1-x2 plane. Show that the inertia tensor takes the form
If, in the previous problem, the coordinate axes are rotated through and angle θ about the x3-axis, show that the new inertia tensor isAnd hence show that the x1- and x2-axes become principal
Consider a plane homogeneous plate of density p bounded by the logarithmic spiral r = ke aθ and the radii θ = 0 and θ = π. Obtain the inertia tensor for the origin at r = 0 if
A uniform rod of length b stands vertically upright on a rough floor and then tips over. What is the rod’s angular velocity when it hits the floor?
The proof represented by Equation 11-54-11-61 is expressed entirely in the summation convention. Rewrite this proof in matrix notation.
The trace of tensor is defined as the sum of the diagonal elements;Show, by performing a similarity transformation, that the trace is and invariant quantity, in other words, show that tr {l} = tr
Show by the method used in the previous problem that the determinant of the elements of a tensor is an invariant quantity under a similarity transformation. Verify this result also for the case of
Find the frequency of small oscillations for a thin homogeneous plate if the motion takes place in the plane of the plate and if the plate has the shape of an equilateral triangle and is suspended(a)
Consider a thin disk composed of two homogeneous halves connected along a diameter of the disk. If one half has density p and the other has density 2p, find the expression for the Lagrangian when the
Obtain the components of the angular velocity vector w(see Equation 11.02) directly from the transformation matrix λ(Equation 11.09).
A symmetric body moves without the influence of force or torques. Let x3 be the symmetry axis of the body and L be along x3. The angle between w and x3 is a. Let w and L initially be in the x2-x3
Show from Figure 11-9C that the components of w along the fixed (xi) axes are
Investigate the motion of the symmetric top discussed in Section 11.11 for the case in which the axis of rotation is vertical (i.e., the x3- and x3 axes coincide). Show that the motion is either
Refer to the discussion of the symmetric top is Section 11.11. Investigate the equation for the turning points of the notational motion by setting θ = 0 in Equation 11.162. Show that the
Consider a thin homogeneous plate with principal moment a of inertiaLet the origins of the xi and xi systems coincide and be located at the center of mass O of the plate. At time t = 0, the plate is
Solve Example 11.2 for the case when the physical pendulum does not undergo small oscillations. The pendulum is released from rest at 67o at time t = 0. Find the angular velocity when the pendulum
Do a literature search and explain how a cat can always land on its feet when dropped from a position at rest with its feet pointing upward. Estimate the minimum height a cat needs to fall in order
Consider a symmetrical rigid body rotating freely about its center of mass. A frictional torque (Nf = bw) acts to slow down the rotation. Find the components of the angular velocity along the
A disk rolls without slipping across a horizontal plane. The plane of the disk remains vertical, but it is free to rotate about a vertical axis. What generalized coordinates may be used to describe
Work out Example 7.6 showing all the steps, in particular those leading to Equation 7.36 and 7.41. Explain why the sign of the acceleration a cannot affect the frequency w. Give an argument why the
A sphere of radius P is constrained to roll without slipping on the lower half of the inner surface of a hollow cylinder of inside radius R. Determine the Lagrangian function, the equation of
A particle moves in a plane under the influence of a force f = – Ara-1 directed toward the origin; A and a (> 0) are constants. Choose appropriate generalized coordinates, and let the potential
Consider a vertical plane in a constant gravitational field. Let the origin of a coordinate system be located at some point in this plane. A particle of mass m moves in the vertical plane under the
A hoop of mass m and radius R rolls without slipping down and inclined plane of mass M, which makes an angle a with the horizontal. Find the Lagrange equations and the integrals of the motion if the
A double pendulum consists of two simple pendula, with on pendulum suspended from the bob of the other. If the two pendula have equal lengths and have bobs of equal mass and if both pendula are
Consider a region of space divided by a plane. The potential energy of a particle in region 1 is U1 and in region 2 it is U2. If a particle of mass m and with speed v1 in region 1 passes from region
A disk of mass M and radius R rolls without slipping down a plane inclined from the horizontal by an angle a. The disk has short weightless axle of negligible radius. From this axis is suspended a
Two blocks, each of mass M, are connected by an extension less, uniform string of length l. One block is placed on a smooth horizontal surface, and the other block hangs over the side, the string
A particle of mass m is constrained to move on a circle of radius R. The circle rotates in space about one point on the circle, which is fixed. The rotation takes place in the plane of the circle and
A particle of mass m rests of a smooth plane. The plane is raised to an inclination angle θ at a constant rate a (θ = 0 at t = 0), causing the particle to move down the plane. Determine the
A simple pendulum of length b and bob with mass m is attached to a mass less support moving horizontally with constant acceleration a. Determine(a) The equations of motion and(b) The period for small
A simple pendulum of length b and bob with mass m is attached to a mass less support moving vertically upward with constant acceleration a. Determine(a) The equations of motion.(b) The period for
A pendulum consists of a mass m suspended by a mass less spring with unexpended length b and spring constant k. Find Lagrange’s equations of motion.
The point of support of a simple pendulum of mass m and length b is driven horizontally by x = a sin wt. Find the pendulum’s equation of motion.
A particle of mass m can slide freely along a wire AB whose perpendicular distance to the origin O is h. The line OC rotates about the origin at a constant angular velocity θ = w. The position
A pendulum is constructed by attaching a mass m to an extension less string of length l. the upper end of the string is connected to the uppermost point on a vertical disk of radius R (R
Two masses m1 and m2 (m1 ≠ m2) are connected by a rigid rod of length d and of negligible mass. An extension less string of length l1 is attached to m1 and connected to a fixed point of support
A circular hoop is suspended in a horizontal plane by three strings, each of length l, which are attached symmetrically to the hoop and are connected to fixed points lying in a plane above the hoop.
A particle is constrained to move (without friction) on a circular wire rotating with constant angular speed w about a vertical diameter. Find the equilibrium position of the particle, and calculate
A particle of mass m moves in one dimension under the influence of a force where k and T are positive constants. Compute the Lagrangian and Hamiltonian functions. Compare the Hamiltonian and the
Consider a particle of mass m moving freely in a conservative force field whose potential function is U. Find the Hamiltonian function, and show that the canonical equations of motion reduce to
Consider a simple plane pendulum consisting of a mass m attached to a string of length l. After the pendulum is set motion, the length of the string is shortened at a constant rateThe suspension
A particle of mass m moves under the influence of gravity along the helix z = kθ, r = constant, where k is a constant and z is vertical. Obtain the Hamiltonian equations of motion.
Determine the Hamiltonian and Hamilton’s equations of motion for (a) A simple pendulum and(b) A simple Atwood machine (single pulley).
A mass less spring of length b and spring constant k connects two particles of masses m1 and m2. The system rests on a smooth table and may oscillate and rotate. (a) Determine Lagrange’s equations
A particle of mass m is attracted to a force center with the force of magnitude k/r2. Use plane polar coordinated and find Hamilton’s equation of motion.
Consider the pendulum described in Problem 7-15. The pendulum’s point of support rises vertically with constant acceleration a.(a) Use the Lagrangian method to find the equations of motion.(b)
A spherical pendulum consists of a bob of mass m attached to a weightless, extension less rod of length l. The end of the rod opposite the bob pivots freely (in all directions) about some fixed
A particle moves in a spherically symmetric force field with potential energy given by U(r) = – k/r. Calculate the Hamiltonian function in spherical coordinates, and obtain the canonical equations
Determine the Hamiltonian an Hamilton’s equations of motion for the double Atwood machine of Example 7.8.
A particle of mass m slides down a smooth circular wedge of mass M as shown in Figure 7-C. The wedge rests on a smooth horizontal table. Find(a) The equation of motion of m and M and(b) The reaction
Four particles are directed upward in a uniform gravitational field with the following initial conditions.Show by direct calculation that the representative points corresponding to these particles
Discuss the implications of Liouville’s theorem on the focusing of beams of charged particles by considering the following simple case. An electron beam of circular cross section (radius R0) is
Use the method of Lagrange undetermined multipliers to find the tensions in both strings of the double Atwood machine of Example 7.8.
The potential for an anharmonic oscillator is U = kx2/2 + bx4/4 where k and b are constants. Find Hamilton’s equation of motion.
An extremely limber rope of uniform mass density, mass m and total length b lies on a table with a length z hanging over the edge of the table. Only gravity acts on the rope. Find Lagrange’s
A double pendulum is attached to a cart of mass 2m that moves without friction on a horizontal surface. See Figure 7-D each pendulum has length b and mass bob m. Find the equations of motion.
A pendulum of length b and mass bob m is oscillating at small angles when the length of the pendulum string is shortened at a velocity of a (db/dt = – a). Find Lagrange’s equations of motion.
Prove Equation 14.13 by using Equations 14.9-14.12.
Show that the transformation equations connecting the K€™ and K systems (Equations 14.14) can be expressed as where tanh a = v/c.Show that the Lorentz transformation corresponds to a
Show that the equation Is invariant under a Lorentz transformation but not under a Galilean transformation. (This is the wave equation that describes the propagation of light waves in free space.)
Show that the expression for the Fitzgerald-Lorentz contraction (Equation 14.19) can also be obtained if the observer in the K’ system measures the time necessary for the rod to pass a fixed point
What is the apparent shape of a cube moving with a uniform velocity directly toward or away from an observer?
Consider two events that take place at different points in the K system at the same instant t. If these two points are separated by ad distance ∆x, show that in the K’ system the events are
Two clocks located at the origins of the K and K’ systems (which have a relative speed v) are synchronized when the origins coincide. After a time t, an observer at the origin of the K system
In his 1905 paper (see the translation in Lo23), Einstein states; “WE conclude that a balance-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock
Consider a relativistic rocket whose velocity with respect to a certain inertial frame is v and whose exhaust gases are emitted with a constant velocity V with respect to the rocket. Show that the
Show by algebraic methods that Equations 14.15 follow from Equations 14.14.
A stick of length l is fixed at an angle θ from its x1-axis in its own rest system K. What is the length and orientation of the stick as measured by an observer moving along x1 with speed v?
A racer attempting to break the land speed record rockets by two markers spaced 100 m apart on the ground in a time of 0.4 μs, as measured by an observer on the ground. How far apart do the two
A muon is moving with speed v = 0.999c vertically down through the atmosphere. If its half-life in its own rest frame is 1.5 μs, what is its half-life as measured by an observer on Earth?

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