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All Textbook Solutions for Classical Dynamics of Particles and Systems
1.1P1.2. Prove Equations 1.10 and 1.11 from trigonometric considerations.
1.3PShow
(a) (AB)t = BtAt (b) (AB)−1 = B−1 A−1
1.5P1.6PConsider 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?
1.8PFor the two vectors
find
A − B and |A – B|
component of B along A
angle between A and B
A × B
(A − B) × (A + B)
A particle moves in a plane elliptical orbit described by the position vector r=2bsinti+bcostj (a) Find v, a, and the particle speed. (b) What is the angle between v and a at time t = /2?1.11PLet 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 ϕ,
A × X = B, A · X = ϕ.
Express X in terms of A, B, ϕ, and the magnitude of A.
1.14P1.15PWhat 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 × B and the alternate representation (A − B) × B.
1.19P1-20. Show that
Show (see also Problem 1–11) that
1.22PUse the εijk notation and derive the identity
(A × B) × (C × D) = (ABD)C − (ABC)D
1.24PFind the components of the acceleration vector a in spherical coordinates.1.26PIf r and are both explicit functions of time, show that
Show that
1.29P1.30PShow that
(a)
(b)
(c)
Show that (2arr+2brr)dt=ar2+br2+const. 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.Show that (rrrrr2)dt=rr+C where C is a constant vector.1.34P1.35P1.36P1.37P1.38PA 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.
For what values of a are the vectors A = 2ai − 2j + ak and B = ai + 2aj + 2k perpendicular?
2.1P2.2PIf a projectile is fired from the origin of the coordinate system with an initial velocity υ0 and in a direction making an angle α with the horizontal, calculate the time required for the projectile to cross a line passing through the origin and making an angle β < α 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 80 m above the flat range, (a) She wanted the bales of hay to land 30 m 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.
2.7PA projectile is fired with a velocity 0 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 d=0g024ghConsider a projectile fired vertically in a constant gravitauonal field. For the same initial velocities, compare the times required 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.2.11PA particle is projected vertically upward in a constant gravitational field with an initial speed 0. 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 0t02+t2 where t, is the terminal speed.
A particle moves in a medium under the influence of a retarding force equal to mk(υ3+ a2υ), where k and a are constants. Show that for any value of the initial speed the particle will never move a distance greater than π/2kaand that the particle comes to rest only for t → ∞.
A projectile is fired with initial speed 0 at an elevation angle of up a hill of slope ( ). (a) How far up the hill will the projectile land? (b) At what angle will the range be a maximum? (c) What is the maximum range?
A particle of mass m slides down an inclined plane under the influence of gravity. If the motion is resisted by a force f = kmυ2, 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 0 up a slope that makes an angle with the horizontal. Assume frictionless motion and find the time required for the particle to return to its starting position. Find the time for 0 = 2.4 m/s and = 26.A strong softball player smacks the ball at a height of 0.7 m above home plate. The ball leaves the players bat at an elevation angle of 35 and travels toward a fence 2 m high and 60 m away in center field. What must the initial speed of the softball be to clear the center field fence? Ignore air resistance.2.19PA gun fires a projectile of mass 10 kg of the type to which the curves of Figure 2-3 apply. The muzzle velocity is 140 m/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.
2.21P2.22PA skier weighing 90 kg starts from rest down a hill inclined at 17°. He skis 100 m down the hill and then coasts for 70 m 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?
A block of mass m = 1.62 kg slides down a frictionless incline (Figure 2-A). The block is released a height h = 3.91 m above the bottom of the loop.
What is the force of the inclined track on the block at the bottom (point A)?
What is the force of the track on the block at point B?
At what speed does the block leave the track?
How far away from point A does the block land on level ground?
Sketch the potential energy U(x) of the block. Indicate the total energy on the sketch.
FIGURE 2-A Problem 2-25.
A child slides a block of mass 2 kg along a slick kitchen floor. If the initial speed is 4 m/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 2 m of a rough floor that has μk = 0.2?
A rope having a total mass of 0.4 kg 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?A superball of mass M and a marble of mass m are dropped from a height h with the marble just on top of the superball. A superball has a coefficient of restitution of nearly 1 (i.e., its collision is essentially elastic). Ignore the sizes of the superball and marble. The superball 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 superball go?An automobile driver traveling down an 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 k = 0.45. Is the lawyer correct to accuse the driver of exceeding the 25-MPH speed limit? Explain.A student drops a water-filled balloon from the roof of the tallest building in town trying to hit her roommate on the ground (who is too quick). The first student ducks back but hears 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.
2.31PTwo blocks of unequal mass are connected by a string over a smooth pulley (Figure 2 B). If the coefficient of kinetic friction is μk, what angle θ of the incline allows the masses to move at a constant speed?
FIGURE 2-B Problem 2-32.
A particle is released from rest (y = 0) and falls under the influence of gravity and air resistance. Find the relationship between and the distance of falling y when the air resistance is equal to (a) and (b) 2.Perform the numerical calculations of Example 2.7 for the values given in Figure 2-8. Plot both Figures 2-8 and 2-9. Do not duplicate the solution in Appendix H; compose your own solution.
2.36PA particle of mass m has speed υ = α/x, where x is its displacement. Find the force F(x) responsible.
The speed of a particle of mass m varies with the distance x as υ(x) = αx−n. Assume υ(x = 0) = 0 at t = 0. (a) Find the force F(x) responsible. (b) Determine x(t) and (c) F(t).
A boat with initial speed υ0 is launched on a lake. The boat is slowed by the water by a force F = −αeβυ. (a) Find an expression for the speed υ(t). (b) Find the time and (c) distance for the boat to stop.
A train moves along the tracks at a constant speed u. A woman on the train throws a ball of mass m straight ahead with a speed υ 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?
2.42P2.45P2.46PConsider a particle moving in the region x > 0 under the influence of the potential
where U0 = 1 J and α = 2 m. Plot the potential, find the equilibrium points, and determine whether they are maxima or minima.
2.48P2.49PAccording 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 << 1. (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/m0 = 10 m/s2 ( ≈ g on Earth). How much time is required for the particle to reach half the speed of light and of 99% the speed of light?
Let us make the (unrealistic) assumption that a boat of mass m gliding with initial velocity v0 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/l000? (b) Find x(t). How far does the boat travel in this time? Let m = 200 kg, v0 = 2 m/s, and b = 0.2 Nm-2s2.
A 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 constants. (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 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.A 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 it is initially shot directly upward with a student-made potato gun with an initial velocity of 120 m/s.
2.55P3.1PAllow the motion in the preceding problem to take place in a resisting medium. After oscillating for 10 s, the maximum amplitude decreases to half the initial value. Calculate (a) the damping parameter β, (b) the frequency υ1 (compare with the undamped frequency υ0), and (c) the decrement of the motion.
3.3P3.4PObtain an 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.
Two masses m1 = 100 g and m2 = 200 g 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.3.7P3.8PA particle of mass m is at rest 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 a 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
If 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.
3.11P3.12P3.13P3.14PReproduce Figures 3-10b and c for the same values given in Example 3.2, but instead let = 0.1 s1 and = rad. How many times does the system cross the x = 0 line before the amplitude finally falls below 102 of its maximum value? Which plot, b or c, is more useful for determining this number? Explain.3.16PFor 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.
Show that, if a driven oscillator is only lightly damped and driven near resonance, the Q of the system is approximately Q2(TotalenergyEnergylossduringoneperiod)3.19PPlot 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 is approximately equal to ω0/6.
Let the initial position and speed of an overdamped, nondriven 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=2x0+v021 and A2=1x0+v021 where 1 = 2 and 2 = + 2. (b) Show that when A1 = 0, the phase paths of Figure 3-11 must be along the dashed curve given by x=2x, otherwise the asymptotic paths are along the other dashed curve given by x=1x. Hint: Note that 2 1 and find the asymptotic paths when t .3.26P3.27P3.28P3.29P3.30P3.31PObtain the response of a linear oscillator to a step function and to an impulse function (in the limit τ → 0) for overdamping. Sketch the response functions.
Calculate the maximum values of the amplitudes of the response functions shown in Figures 3-22 and 3-24. Obtain numerical values for β = 0.2ω0 when a = 2 m/s2, ω0 = 1 rad/s, and t0 = 0.
Consider an undamped linear oscillator with a natural frequency ω0 = 0.5 rad/s and the step function a = 1 m/s2. Calculate and sketch the response function for an impulse forcing function acting for a time τ = 2π/ω0. Give a physical interpretation of the results.
3.35P3.36P3.37P3.38P3.39PAn automobile with a mass of 1000 kg, including passengers, settles 1.0 cm closer to the road for every additional 100 kg of passengers. It is driven with a constant horizontal component of speed 20 km/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 2.4 m. 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.
3.41PAn undamped driven harmonic oscillator satisfies the equation of motion m(d2x/dt2+ 02x) = F(t). The driving force F(t) = F0 sin(t) 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 = 0 by taking the limit 0 in your result for part (a). Sketch your result for x(t). Hint: In part (a) look for a particular solution of the differential equation of the form x = A sin(t) and determine A. Add the solution of the homogeneous equation to this to obtain the general solution of the inhomogeneous equation.Consider 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.A grandfather clock has a pendulum length of 0.7 m and mass bob of 0.4 kg. A mass of 2 kg falls 0.8 m 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?
4.1P4.2P4.3P4.4P4.5P4.6P4.7P4.8P4.9P4.10P4.11P4.12P4.13P4.14P4.15P4.16P4.17P4.18P4.19P4.20P4.21P4.22P4.23P4.24P4.25P4.26P5.1P5.2P5.3P5.4P5.5PCompute directly the gravitational force on a unit mass at a point exterior to a homogeneous sphere of matter.5.7P5.8P5.9P5.10P5.11P5.12PA planet of density 1 (spherical core, radius R1) with a thick spherical cloud of dust (density 2, radius R2) is discovered. What is the force on a particle of mass m placed within the dust cloud?
5.14P5.15P5.16P5.17P5.18P5.19PA 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 (z) and the gravitational field 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.6.1P6.2P6.3P6.4P6.5P6.6P6.7P6.8P6.9P6.10P6.11P6.12P6.13P6.14P6.15P6.16P6.17P6.18P7.1P7.2P7.3P7.4P7.5P7.6P7.7P7.8P7.9P7.10P7.11P7.12P7.13P7.14P7.15P7.16P7.17P7.18P7.19P7.20P7.21P7.22P7.23P7.24P7.25P7.26P7.27P7.28P7.29P7.30P7.31P7.32P7.33P7.34P7.35P7.36P7.37P7.38P7.39P7.40P7.41P8.1P8.2P8.3P8.4P8.5P8.6P8.7P8.8P8.9P8.10P8.11P8.12P8.13P8.14P8.15P8.16P8.17P8.18P8.19P8.20P8.21P8.22P8.23P8.24P8.25P8.26P8.27P8.28P8.29P8.30P8.31P8.32P8.33P8.34P8.35P8.36P8.37P8.38P8.39P8.40P8.41P8.42P8.43P8.44P8.45P8.46PTwo double stars, one having mass 1.0 Msun and the other 3.0 Msun, rotate about their common center of mass. Their separation is 6 light years. What is their period of revolution?
Find the center of mass of a hemispherical shell of constant density and inner radius r1 and outer radius r2.
9.2P9.3P9.4P9.5P9.6P9.7P9.8P9.9P9.10P9.11P9.12P9.13P9.14P9.15P9.16P9.17P9.18P9.19P9.20P