GET Classical Dynamics Of Particles TEXTBOOK SOLUTIONS
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Find the transformation matrix that rotates the axis x3 of a rectangular coordinate system 45o toward x1 around the x2 axis.
Prove Equations 1.10 and 1.11 from trigonometric considerations.
Find the transformation matrix that rotates a rectangular coordinate system through and angle of 120o about an axis making equal angles with the original three coordinate axes.
(a) (AB)t = Bt At
(b) (AB)-1 = B–1 A–1
Show by direct expansion that |λ|2 = 1. For simplicity, take λ to be a two-dimensional orthogonal transformation matrix.
Show that Equation 1.15 can be obtained by using the requirement that the transformation leaves unchanged the length of a line segment.
Consider a unit cube with one corner at the origin and three adjacent sides lying along the three axes of a rectangular coordinate system. Find the vectors describing the diagonals of the cube. What is the angle between any pair of diagonals?
Let A be vector from the origin to point P fixed in space. Let r be a vector from the origin to a variable point Q (x1, x2, x3). Show that A · r = A2 is the equation of a plane perpendicular to A and passing through the point P.
For the two vectors A = I + 2j – k, B = – 2i + 3j + k find
(a) A – B and | A – B |
(b) Component of B along A
(c) Angle between A and B
(d) A X B
(e) (A – B) X (A + B)
A particle moves in a plane elliptical orbit described by the position vector r = 2b sin wti + b cos wtj
(a) Find v, a, and the particle speed.
(b) What is the angle between v and a at time t = π/2w?
Show that the triple scalar product (A X B) · C can be written as
Show also that the product is unaffected by an interchange of the scalar and vector product operations or by a change in the order of A, B, C, as long as they are in cyclic, order; that is, (A X B) · C = A · (B X C) = B · (C X A) = (C X A) · B, etc. why may therefore use the notation ABC to denote the triple scalar product, Finally give a geometric interpretation of ABC by computing the volume of the parallelepiped defined by the three vectors A, B, C.
Let a, b, c be three constant vectors drawn from the origin to the points A, B, C. what is the distance from the origin to the plane defined by the points A, B, C? What is the area of the triangle ABC?
X is an unknown vector satisfying the following relations involving the known vectors A and B and the scalar Ø. A x X = B, A · X = Ø. Express X in terms of A, B, Ø, and the magnitude of A.
Consider the following matrices:
Find the following
(d) AB – BtAt
Find the values of a needed to make the following transformation orthogonal.
What surface is represented by r ·a = const, that is described if a is a vector of constant magnitude and direction from the origin and r is the position vector to the point P(x1, x2, x3) on the surface?
Obtain the cosine law of plane trigonometry by interpreting the product (A – B) · (A – B) and the expansion of the product.
Obtain the sine law of plane trigonometry by interpreting the product A x B and the alternate representation (A – B) X B.
Derive the following expressions by using vector algebra.
(a) cos (a – β) = cos a cos β + sin a sin β
(b) sin (a – β) = sin a cos β – cos a sin β
Show (See also Problem 1-11) that
Use the εijk notation and derive the identity (A X B) x (C X D) = (ABC) C – (ABC) D
Let A be an arbitrary vector, and let e be a unit vector in some fixed direction. Show that A = e(A · e) + e x (A x e) what is the geometrical significance of each of the two terms of the expansion?
Find the components of the acceleration vector a in spherical coordinates.
A particle moves with v = const, along the curve r = k (1 + cos θ) (a cardioids). Find r · e, = a · e,, |a|, and θ.
If r and r = v are both explicit functions of time, show that
Find the angle between the surfaces defined by r2 = 9 and x + y + z2 = 1 at the point (2, – 2, 1).
Show that where r is the vector from the origin to the point (x1, x2, x3).
The quantities r and r are the magnitudes of the vectors r and r, respectively and a and b are constants.
Where C is a constant vector
Evaluate the integral
Show that the volume common to the intersecting cylinders defined by x2 + y2 = a2 and x2 + z2 = a2 is V = 16a3/3.
Find the value of the integral ∫s A · da, where A = xi – yj + zk and S is the closed surface defined by the cylinder c2 = x2 + y2. The top and bottom of the cylinder are at z = d and 0, respectively.
Find the value of the integral ∫s A · da, where A = (x2 + y2 + z2) (xi + yi + zk) and the surface S is the defined by the sphere R2 = x2 + y2 + z2. Do the integral directly and also by using Gauss’s theorem.
Find the value of the integral ∫s (Δ x A) · da if the vector A = ai + zj + xk and S is the surface defined by the paraboloid z = 1 – x2 – y2, where x > 0.
A Plane passes through the three points (x, y, z) = (1, 0, 0), (0, 2, 0), (0, 0, 3).
(a) Find a unit vector perpendicular to the plane.
(b) Find the distance from the point (1, 1, 1) to the closest point of the plane and the coordinates of the closest point.
The height of a hill in meters is given by z = 2xy – 3x2 – 4y2 – 18x + 28y + 12, where x is the distance east and y is the distance north of the origin.
(a) Where is the top of the hill and how high is it?
(b) How steep is the hill at x = y 1, that is, what is the angle between a vector perpendicular to the hill and the z axis?
(c) In which compass direction is the slope at x = y = 1 steepest?
Sketch the equipotential surface and the lines of force for two point masses separated by a certain distance. Next, consider one of the masses to have a fictitious negative mass – M. Sketch the equipotential surfaces and lines of force for this case. To what kind of physical situation does this set of equipotentials and field lines apply?
If the field vector is independent of the radial distance within a sphere, find the function describing the density p = p(T) of the sphere.
Assuming that air resistance is unimportant, calculate the minimum velocity a particle must have at the surface of Earth to escape from Earth’s gravitational field. Obtain a numerical value for the result. (This velocity is called the escape velocity).
A particle at rest is attracted toward a center of force according to the relation F = – mk2/x3. Show that the time required for the particle to reach force center from a distance d is d2/k.
A particle falls to Earth starting from rest at a great height (many times Earth’s radius). Neglect air resistance and show that the particle requires approximately 9/11 of the total time of fall to traverse the first half of the distance.
Computer directly the gravitational force on a unit mass at a point exterior to a homogeneous sphere of matter
Calculate the gravitational potential due to a thin rod of length l and mass M at a distance R from the center of the rod and in a direction perpendicular to the rod.
Calculate the gravitational field vector due to a homogeneous cylinder at exterior points on the axis of the cylinder. Perform the calculation
(a) By computing the force directly and
(b) By computing the potential first.
Calculate the potential due to a thin circular ring of radius a and mass M for points lying in the plane of the ring and exterior to it. The result can be expressed as an elliptic integral. Assume that the distance from the center of the ring to the field point is large compared with the radius of the ring. Expand the expression for the potential and find the first correction term.
Find the potential at off-axis points due to a thin circular ring of radius a and mass M. Let R be the distance from the center of the ring to the field point, and let θ be the angle between the line connecting the center of the ring with the field point and the axis of the ring. Assume R >> a so that terms of order (a/R) 3 and higher may be neglected.
Consider a massive body of arbitrary shape and a spherical surface that is exterior to and does not contain the body. Show that the average value of the potential due to the body taken over the spherical surfaces is equal to the value of the potential at the center of the sphere.
In the previous problem, let the massive body be inside the spherical surface. Now show that the average value of the potential over the surface of the sphere is equal to the value of the potential that would exist on the surface of the sphere if all the mass of the body were concentrated at the center of the sphere.
A planet of density p1 (spherical core, radius R1) with a thick spherical cloud of dust (density p2, radius R2) is discovered. What is the force on a particle of mass m placed within the dust cloud?
Show that the gravitational self-energy (energy of assembly piecewise from infinity) of a uniform sphere of mass M and radius R is
A particle is dropped into a hole drilled straight through the center of Earth. Neglecting rotational effects show that the particle’s motion is simple harmonic if you assume Earth has uniform density. Show that the period of the oscillation is about 84 min.
A uniformly solid sphere of mass M and radius R is fixed a distance h above a thin infinite sheet of mass density ps (mass/area). With what force does the sphere attract the sheet?
Newton’s model of the tidal height, using the two water wells dug to the center of Earth, used the fact that the pressure at the bottom of the two wells should be the same. Assume water is incompressible and find the tidal height difference h, Equation 5.55, due to the Moon using this model.
Show that the ratio of maximum tidal heights due to the Moon and Sun is given by and that this value is
2.2. REs is the distance between the Sun and Earth, and Ms is the Sun’s mass.
The orbital revolution of the Moon about Earth takes about 27.3 days and is in the same direction as Earth’s rotation (24h). Use this information to show that high tides occur everywhere on Earth every 12 h and 26 min.
A thin disk of mass M and radius R lies in the (x, y) plane with the z-axis passing through the center of the disk. Calculate the gravitational potential Ф (z) and the gravitational field g(z) - ∆ Ф (z) = – kd Ф (z)/dz on the z-axis.
A point mass m is located a distance D from the nearest end of a thin rod of mass M and length L along the axis of the rod. Find the gravitational force exerted on the point mass by the rod.
Assume that x (t) = b cos (w0t) = u (t) is a solution of the van der Pol Equation 4.19. Assume that the damping parameter is small and keep terms in u (t) to first order in μ. Show that b = 2a and u (t) – (μa3/4w0) sin (3w0t) is a solution. Produce a phase diagram of x versus x and produce plots of x (t) and x (t) for values of a = 1, w0 = 1, and μ = 0.5.
Use numerical calculations to find a solution for the van der Pol oscillator of Equation 4.19. Let x0 and w0 equal 1 for simplicity. Plot the phase diagram, x(t) and x(t) for the following conditions.
(a) μ = 0.07, x0 = 1.0, x0 = 0 at t = 0;
(b) μ = 0.07, x0 = 3.0, x0 = 0 at t = 0.Discuss the motion; does the motion appear to approach a limit cycle?
Repeat the previous problem with μ = 0.5. Discuss also the appearance of the limit cycle, x(t), and x(t).
Make a plot of the Henon map, this time starting from the initial values x0 = 0.63135448, y0 = 0.18940634. Computer the shape of this plot with that obtained in the previous problem is the shape of the curves independent of the initial conditions?
Refer to Example 4.1. If each of the springs must be stretched a distance d to attach the particle at the equilibrium position (i.e., in its equilibrium position, the particle is subject to two equal and oppositely directed forces of magnitude kd). Then show that the potential in which the particle moves is approximately. U(x) ≡ (kd/l) x2 + [k (l – d) dl3] x4
Construct a phase diagram for the potential in Figure 4-1.
Construct a phase diagram for the potential U(x) = – (λ/3) x3.
Lord Rayleigh used the equation x – (a – bx2) x + w2/0 x = 0 in his discussion of nonlinear effects in acoustic phenomena. * Show that differentiating this equation with respect to time and making the substitution y = y0√3b/ax results in van der Poll’s equation;
Solve by a successive approximation procedure, and obtain a result accurate to four significant figures;
(a) x + x2 + 1 – tan x, 0 < x < π/2
(b) x (x + 3) = 10 sin x, x > 0
(c) 1 _ x _ cos x = 3x, x > 0
It may be profitable to make a crude graph to choose a reasonable first approximation.
Derive the expression for the phase paths of the plane pendulum if the total energy is E > 2mgl. Note that this is just the case of a particle moving in a periodic potential U (θ) = mgl (1 - cos θ).
Consider the free motion of a plane pendulum whose amplitude is not small. Show that the horizontal component of the motion may be represented by the approximate expression (components through the third order are included).
Where w2/0 = g/l and ε = 3g/2l3, with l equal to the length of the suspension.
A mass m moves in one dimension and is subject to a constant force + F0 when x < 0 and to a constant force – F0 when x > 0. Describe the motion by constructing a phase diagram. Calculate the period of the motion in terms of m, F0, and the amplitude A (Disregard damping).
Investigate the motion of an un-damped particle subject to a force of the form
Where k and δ are positive constants
The parameters F = 0.7 and c = 0.05 are fixed for Equation 4.43 describing the driven, damped pendulum. Determine which of the values for w (0.1, 0.2, 0.3.., 1.5) produce chaotic motion. Produce a phase plot for w = 0.3. Do this problem numerically.
A really interesting situation occurs for the logistic equation, Equation 4.46 when a = 3.82831 and x1 = 0.51. Show that a three cycle occurs with the approximate x values 0.16, 0.52, and 0.96 for the first 80 cycles before the behavior apparently terns chaotic. Find for what iteration the next apparently periodic cycle occurs and for how many cycles it stays periodic.
Let the value of a in the logistic equation, Equation 4.46, be equal to 0.9, Make a map like that in Figure 4-21 when x1 = 0.4. Make the plot for three other values of x1 for which 0 < x1 < 1.
Perform the numerical calculation done in Example 4.3 and show that the two calculations clearly diverge by n = 39. Next, let the second initial value agree to within another factor of 10 (i.e., 0.700 00 000 1), and confirm the statement in the text that only four more iterations are gained in the agreement between the two initial values.
Use the function described in Example 4.3, xn+1 = axan (1 – xn2) where a = 2.5. Consider two starting values of x1 that are similar, 0.900 000 0 and 0.900 000. Make a plot of xn versus n for the two starting values and determine the lowest value of n for which the two values diverge by more than 30%.
Use direct numerical calculation to show that the map f(x) = a sin π x also leads to the Feigenbaum constant, where x and a are limited to the interval (0, 1)
The curve xn+1 = f (xn) intersects the curve xn+1 = xn at x0. The expansion of xn+1 about x0 is xn+1 – x0 = β (xn – x0) where β = (df/dx) at x = x0.
(a) Describe the geometrical sequence that the successive values of xn+1 – x 0 forms.
(b) Show that the intersection is stable when |β| < 1 and unstable when |β| > 1.
The tent map is represented by the following iterations;
Where 0 < a < 1 make a map up to 20 iterations for a = 0.4 and 0.7 with x1 = 0.2. Does it appear that either of the maps represents chaotic behavior?
Plot the bifurcation diagram for the tent map of the previous problem. Discuss the results for the various regions.
Show analytically that the Lyapunov exponent for the tent maps is λ = in (2a). This indicates that chaotic behavior occurs for a > ½.
Consider the Henon map described by Let a = 1.4 and b = 0.3, and use a computer to plot the first 10,000 points (xw yn) starting from the initial values x0 = 0, y0 =0. Choose the plot region as – 1.5 < x < 1.5 and – 0.45 < y < 0.45.
A circuit with a nonlinear inductor can be modeled by the first-order differential equations.
Chaotic oscillations for this situation have been extensively studied. Use a computer to construct the Poincare section plot for the case k = 0.1 and 9.8 < B < 13.4. Describe the map.
The motion of a bouncing ball, on successive bounces, when the floor oscillates sinusoidally can be described by the Chirikov map:
Where – π < p < and – π < q < π. Construct two-dimensional maps for K = 0.8, 3.2 and 6.4 starting with random values of p and q and iterating them. Use periodic boundary conditions, which means that if the iterated values of p or q exceed π, a value of 2π is subtracted and whenever they are less than –π, a value of 2π is added. Examine the maps after thousands of iterations and discuss the differences.
Suppose that the force acting on a particle is factorable into one of the following forms;
(a) F(xi, t) = f(xi) g(t)
(b) F(xi, t) = f(xi) g(t)
(c) F(xi, xi) = f(xi) g(xi) for which cases are the equations of motion integral?
A particle of mass m is constrained to move on the surface of a sphere of radius R by ab applied force F (θ, Ø). Write the equation of motion.
If projectile is fired from the origin of the coordinate system with an initial velocity vo and in direction making and angle a with the horizontal, calculate the time required for the projectile to cross a line passing through the origin and making an angle a with the horizontal, calculate the time required for the projectile to cross a line passing through the origin and making an angle β < a with the horizontal.
A clown is juggling four balls simultaneously. Students use a video tape to determine that it takes the clown 0.9 s to cycle each ball through his hands (including catching, transferring, and throwing) and to be ready to catch the next ball. What is the minimum vertical speed the clown must throw up each ball?
A jet fighter pilot knows he is able to withstand an acceleration of 9g before blacking out. The pilot points his plane vertically down while traveling at Mach 3 speed and intends to pull up in a circular maneuver before crashing into the ground.
(a) Where does the maximum acceleration occur in the maneuver?
(b) What is the minimum radius the pilot can take?
In the blizzard of ’88, a rancher was forced to drop hay bales from an airplane to feed her cattle. The plane flew horizontally at 160 km/hr and dropped the bales from a height of 80m above the flat range.
(a) She wanted the bales of hay to land 30m behind the cattle so as to not hit them. Where should she push the bales out of the airplane?
(b) To not hit the cattle, what is the largest time error she could make while pushing the bales out of the airplane? Ignore air resistance.
Include air resistance for the bales of hay in the previous problem. A bale of hay has a mass of about 30kg and an average area of about 0.2m2. Let the resistance be proportional to the square of the speed and let cw = 0.80. Plot the trajectories with a computer if the hay bales land 30m behind the cattle for both including air resistance and not. If the bales of hay were released at the same time in the two cases, what is the distance between landing positions of the bales?
A projectile is fired with a velocity v0 such that it passes through two points both a distance h above the horizontal. Show that if the gun is adjusted for maximum range, the separation of the points is
Consider a projectile fired vertically in a constant gravitational field. For the same initial velocities, compare the times required for the projectile to reach its maximum height
(a) For zero resisting force,
(b) For a resisting force proportional to the instantaneous velocity of the projectile.
Consider a particle of mass m whose motion starts from rest in a constant gravitation field. If a resisting force proportional to the square of the velocity (i.e., kmv2) is encountered, show that the distance s the particle falls in accelerating from v0 to v2 is given by
A particle is projected vertically upward in a constant gravitational field with an initial speed v0. Show that if there is a retarding force proportional to the square of the instantaneous speed, the speed of the particle when it returns to the initial position is
Where vt is the terminal speed.
A particle moves in a medium under the influence of a retarding force equal to mk(v3 + a2v), where k and a are constants. Show that for any value of the initial speed the particle will never move a distance greater that π/2ka and that the particle comes to rest only for t → ∞.
A projectile is fired with initial speed v0 at an elevation angle of a up a hill of slope β(a > β).
(a) How far up the hill will the projectile land?
(b) At what angle a will the range be a maximum?
(c) What is the maximum range?
A particle of mass m slides down and inclined plane under the influence of gravity. If the motion is resisted by a force ∫ = kmv2, show that the time required to move a distance d after starting from rest is
Where θ is the angle of inclination of the plane
A particle is projected with an initial velocity v0 up a slope that makes an angle a with the horizontal. Assume frictionless motion and find the time required for the particle to return to its starting position. Find the time for v0 – 2.4 m/s and a = 26o.
A strong softball player smacks the ball at a height of 0.7 m above home plate. The ball leaves the player’s bat at an elevation angle of 35o and travels toward a fence 2 m high and 60 m away in center field. What must the initial speed of the softball to clear the center field fence? Ignore air resistance.
Include air resistance proportional to the square of the ball’s speed in the previous problem. Let the drag coefficient be cw = 0.5, the softball radius be 5 cm and the mass be 200 g.
(a) Find the initial speed of the softball needed not to clear the fence.
(b) For this speed, find the initial elevation angle that allows the ball to most easily clear the fence. By how much does the ball now vertically clear the fence?