[{"problem_text": " An automobile with a mass of $1000 \\mathrm{~kg}$, including passengers, settles $1.0 \\mathrm{~cm}$ closer to the road for every additional $100 \\mathrm{~kg}$ of passengers. It is driven with a constant horizontal component of speed $20 \\mathrm{~km} / \\mathrm{h}$ over a washboard road with sinusoidal bumps. The amplitude and wavelength of the sine curve are $5.0 \\mathrm{~cm}$ and $20 \\mathrm{~cm}$, respectively. The distance between the front and back wheels is $2.4 \\mathrm{~m}$. 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.\r\n", "answer_latex": " -0.16", "answer_number": "-0.16", "unit": " $ \\mathrm{~mm}$", "source": "class", "problemid": " Problem 3.40", "comment": " "}, {"problem_text": "Find the shortest path between the $(x, y, z)$ points $(0,-1,0)$ and $(0,1,0)$ on the conical surface $z=1-\\sqrt{x^2+y^2}$. What is the length of the path? Note: this is the shortest mountain path around a volcano.", "answer_latex": " $2 \\sqrt{2} \\sin \\frac{\\pi}{2 \\sqrt{2}}$", "answer_number": "2.534324263", "unit": "", "source": "class", "problemid": " Problem 6.14", "comment": " "}, {"problem_text": "A simple pendulum of length $b$ and bob with mass $m$ is attached to a massless support moving vertically upward with constant acceleration $a$. Determine the period for small oscillations.\r\n", "answer_latex": " $2 \\pi$", "answer_number": "6.283185307", "unit": " $\\sqrt{\\frac{b}{a+g}}$", "source": "class", "problemid": " Problem 7.14", "comment": " "}, {"problem_text": "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 $160 \\mathrm{~km} / \\mathrm{hr}$ and dropped the bales from a height of $80 \\mathrm{~m}$ above the flat range. She wanted the bales of hay to land $30 \\mathrm{~m}$ behind the cattle so as to not hit them. How far behind the cattle should she push the bales out of the airplane?", "answer_latex": " 210", "answer_number": "210", "unit": "$\\mathrm{~m}$ ", "source": "class", "problemid": "Problem 2.6 ", "comment": " "}, {"problem_text": "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.\r\n", "answer_latex": " $\\frac{8 \\pi}{\\sqrt{64 \\pi^2+1}}$", "answer_number": "0.9992093669", "unit": "", "source": "class", "problemid": " Problem 3.44", "comment": " "}, {"problem_text": "What is the minimum escape velocity of a spacecraft from the moon?", "answer_latex": " 2380", "answer_number": "2380", "unit": "$\\mathrm{~m} / \\mathrm{s}$ ", "source": "class", "problemid": " Problem 8.28", "comment": " "}, {"problem_text": "Find the value of the integral $\\int_S \\mathbf{A} \\cdot d \\mathbf{a}$, where $\\mathbf{A}=x \\mathbf{i}-y \\mathbf{j}+z \\mathbf{k}$ and $S$ is the closed surface defined by the cylinder $c^2=x^2+y^2$. The top and bottom of the cylinder are at $z=d$ and 0 , respectively.", "answer_latex": " $\\pi$", "answer_number": "3.141592", "unit": " $c^2 d$", "source": "class", "problemid": " Question 1.36", "comment": " "}, {"problem_text": "A rocket has an initial mass of $7 \\times 10^4 \\mathrm{~kg}$ and on firing burns its fuel at a rate of 250 $\\mathrm{kg} / \\mathrm{s}$. The exhaust velocity is $2500 \\mathrm{~m} / \\mathrm{s}$. If the rocket has a vertical ascent from resting on the earth, how long after the rocket engines fire will the rocket lift off? What is wrong with the design of this rocket?\r\n", "answer_latex": "25", "answer_number": "25", "unit": "$\\mathrm{~s}$ ", "source": "class", "problemid": " Problem 9.60", "comment": " "}, {"problem_text": "A spacecraft of mass $10,000 \\mathrm{~kg}$ is parked in a circular orbit $200 \\mathrm{~km}$ above Earth's surface. What is the minimum energy required (neglect the fuel mass burned) to place the satellite in a synchronous orbit (i.e., $\\tau=24 \\mathrm{hr}$ )?", "answer_latex": " 2.57", "answer_number": "2.57", "unit": "$10^{11} \\mathrm{~J}$ ", "source": "class", "problemid": " Problem 8.42", "comment": " "}, {"problem_text": "A uniformly solid sphere of mass $M$ and radius $R$ is fixed a distance $h$ above a thin infinite sheet of mass density $\\rho_5$ (mass/area). With what force does the sphere attract the sheet?", "answer_latex": " $2\\pi$", "answer_number": "6.283185307", "unit": " $\\rho_s G M$", "source": "class", "problemid": " Problem 5.16", "comment": " "}, {"problem_text": "Consider a thin rod of length $l$ and mass $m$ pivoted about one end. Calculate the moment of inertia. ", "answer_latex": " $\\frac{1}{3}$", "answer_number": "0.33333333", "unit": " $m l^2$", "source": "class", "problemid": " Problem 11.4", "comment": " "}, {"problem_text": "A clown is juggling four balls simultaneously. Students use a video tape to determine that it takes the clown $0.9 \\mathrm{~s}$ 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?\r\n", "answer_latex": "13.2", "answer_number": "13.2", "unit": "$\\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ ", "source": "class", "problemid": " Problem 2.4", "comment": " "}, {"problem_text": "A billiard ball of initial velocity $u_1$ collides with another billiard ball (same mass) initially at rest. The first ball moves off at $\\psi=45^{\\circ}$. For an elastic collision, what are the velocities of both balls after the collision? ", "answer_latex": " $\\frac{1}{\\sqrt{2}}$", "answer_number": "0.7071067812", "unit": " $u_1$", "source": "class", "problemid": " Problem 9.34", "comment": " "}, {"problem_text": "A deuteron (nucleus of deuterium atom consisting of a proton and a neutron) with speed $14.9 \\mathrm{~km} / \\mathrm{s}$ collides elastically with a neutron at rest. Use the approximation that the deuteron is twice the mass of the neutron. If the deuteron is scattered through a $\\mathrm{LAB}$ angle $\\psi=10^{\\circ}$, what is the final speed of the deuteron?", "answer_latex": "14.44", "answer_number": "14.44", "unit": "$\\mathrm{~km} / \\mathrm{s}$", "source": "class", "problemid": " Problem 9.22", "comment": " "}, {"problem_text": "A mass $m$ moves in one dimension and is subject to a constant force $+F_0$ when $x<0$ and to a constant force $-F_0$ when $x>0$. Describe the motion by constructing a phase diagram. Calculate the period of the motion in terms of $m, F_0$, and the amplitude $A$ (disregard damping) .", "answer_latex": " 4", "answer_number": "4", "unit": " $\\sqrt{\\frac{2 m A}{F_0}}$", "source": "class", "problemid": " Problem 3.8", "comment": " "}, {"problem_text": "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 $4.021 \\mathrm{~s}$ after dropping the balloon. If the speed of sound is $331 \\mathrm{~m} / \\mathrm{s}$, find the height of the building, neglecting air resistance.", "answer_latex": " 71", "answer_number": "71", "unit": "$\\mathrm{~m}$ ", "source": "class", "problemid": " Problem 2.30", "comment": " "}, {"problem_text": "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 $\\Phi(z)$.", "answer_latex": " -2", "answer_number": "-2", "unit": " $\\frac{G M}{R^2}\\left(\\sqrt{z^2+R^2}-z\\right)$", "source": "class", "problemid": " Problem 5.20", "comment": " "}, {"problem_text": "A steel ball of velocity $5 \\mathrm{~m} / \\mathrm{s}$ strikes a smooth, heavy steel plate at an angle of $30^{\\circ}$ from the normal. If the coefficient of restitution is 0.8 , at what velocity does the steel ball bounce off the plate?", "answer_latex": " $4.3$", "answer_number": "4.3", "unit": "$\\mathrm{~m} / \\mathrm{s}$ ", "source": "class", "problemid": " Problem 9.42", "comment": " "}, {"problem_text": "Include air resistance proportional to the square of the ball's speed in the previous problem. Let the drag coefficient be $c_W=0.5$, the softball radius be $5 \\mathrm{~cm}$ and the mass be $200 \\mathrm{~g}$. Find the initial speed of the softball needed now to clear the fence. ", "answer_latex": " 35.2", "answer_number": "35.2", "unit": "$\\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ ", "source": "class", "problemid": " Problem 2.18", "comment": " "}, {"problem_text": "A child slides a block of mass $2 \\mathrm{~kg}$ along a slick kitchen floor. If the initial speed is 4 $\\mathrm{m} / \\mathrm{s}$ and the block hits a spring with spring constant $6 \\mathrm{~N} / \\mathrm{m}$, what is the maximum compression of the spring? ", "answer_latex": " 2.3", "answer_number": "2.3", "unit": "$\\mathrm{~m}$ ", "source": "class", "problemid": " Problem 2.26", "comment": " "}, {"problem_text": "If the field vector is independent of the radial distance within a sphere, find the function describing the density $\\rho=\\rho(r)$ of the sphere.", "answer_latex": " $\\frac{1}{2 \\pi}$", "answer_number": "0.1591549431", "unit": " $\\frac{C}{G r}$", "source": "class", "problemid": " Problem 5.2", "comment": " "}, {"problem_text": "An Earth satellite has a perigee of $300 \\mathrm{~km}$ and an apogee of $3,500 \\mathrm{~km}$ above Earth's surface. How far is the satellite above Earth when it has rotated $90^{\\circ}$ around Earth from perigee?", "answer_latex": "1590", "answer_number": "1590", "unit": "$\\mathrm{~km}$ ", "source": "class", "problemid": " Problem 8.24", "comment": " "}, {"problem_text": "Two masses $m_1=100 \\mathrm{~g}$ and $m_2=200 \\mathrm{~g}$ slide freely in a horizontal frictionless track and are connected by a spring whose force constant is $k=0.5 \\mathrm{~N} / \\mathrm{m}$. Find the frequency of oscillatory motion for this system.", "answer_latex": " 2.74", "answer_number": "2.74", "unit": "$\\mathrm{rad} \\cdot \\mathrm{s}^{-1}$ ", "source": "class", "problemid": " Problem 3.6", "comment": " "}, {"problem_text": "A particle moves with $v=$ const. along the curve $r=k(1+\\cos \\theta)$ (a cardioid). Find $\\ddot{\\mathbf{r}} \\cdot \\mathbf{e}_r=\\mathbf{a} \\cdot \\mathbf{e}_r$.", "answer_latex": "$-\\frac{3}{4}$", "answer_number": "-0.75", "unit": " $\\frac{v^2}{k}$", "source": "class", "problemid": " Problem 1.26", "comment": " "}, {"problem_text": "Calculate the minimum $\\Delta v$ 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). Consider only the gravitational attraction of the Sun. ", "answer_latex": " 5275", "answer_number": "5275", "unit": "$\\mathrm{~m} / \\mathrm{s}$ ", "source": "class", "problemid": " Problem 8.38", "comment": " "}, {"problem_text": "Find the ratio of the radius $R$ to the height $H$ of a right-circular cylinder of fixed volume $V$ that minimizes the surface area $A$.", "answer_latex": " $\\frac{1}{2}$", "answer_number": "0.5", "unit": " $H$", "source": "class", "problemid": " Problem 6.10", "comment": " "}, {"problem_text": "Find the dimension of the parallelepiped of maximum volume circumscribed by a sphere of radius $R$.", "answer_latex": " $\\frac{2}{\\sqrt{3}}$", "answer_number": "1.154700538", "unit": "$R$", "source": "class", "problemid": " Problem 6.8", "comment": " "}, {"problem_text": "A potato of mass $0.5 \\mathrm{~kg}$ moves under Earth's gravity with an air resistive force of $-k m v$. (a) Find the terminal velocity if the potato is released from rest and $k=$ $0.01 \\mathrm{~s}^{-1}$. (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 $120 \\mathrm{~m} / \\mathrm{s}$.", "answer_latex": " 1000", "answer_number": "1000", "unit": "$\\mathrm{~m} / \\mathrm{s}$  ", "source": "class", "problemid": " Problem 2.54", "comment": " "}, {"problem_text": "A particle of mass $m$ and velocity $u_1$ makes a head-on collision with another particle of mass $2 m$ at rest. If the coefficient of restitution is such to make the loss of total kinetic energy a maximum, what are the velocities $v_1$ after the collision?", "answer_latex": "  $\\frac{1}{3}$", "answer_number": "0.33333333", "unit": "$u_1$", "source": "class", "problemid": " Problem 9.32", "comment": " "}, {"problem_text": "The height of a hill in meters is given by $z=2 x y-3 x^2-4 y^2-18 x+28 y+12$, where $x$ is the distance east and $y$ is the distance north of the origin. What is the $x$ distance of the top of the hill?", "answer_latex": " -2", "answer_number": "-2", "unit": "m ", "source": "class", "problemid": "Problem 1.40 ", "comment": " "}, {"problem_text": "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 to cool the lead bullets. Many such shot towers were built in New York State. Assume a shot tower was constructed at latitude $42^{\\circ} \\mathrm{N}$, and the lead fell a distance of $27 \\mathrm{~m}$. In what direction and how far did the lead bullets land from the direct vertical?", "answer_latex": "2.26", "answer_number": "2.26", "unit": " $\\mathrm{~mm}$", "source": "class", "problemid": "Problem 10.22", "comment": " "}, {"problem_text": "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 force field. Express the result in terms of the force constant of the field and the semimajor axis of the ellipse. ", "answer_latex": "-1", "answer_number": "-1", "unit": " $\\frac{k}{a}$", "source": "class", "problemid": " Problem 8.4", "comment": " "}, {"problem_text": "A simple harmonic oscillator consists of a 100-g mass attached to a spring whose force constant is $10^4 \\mathrm{dyne} / \\mathrm{cm}$. The mass is displaced $3 \\mathrm{~cm}$ and released from rest. Calculate the natural frequency $\\nu_0$.", "answer_latex": " 6.9", "answer_number": "6.9", "unit": " $10^{-2} \\mathrm{~s}^{-1}$", "source": "class", "problemid": " Problem 3.2", "comment": " "}, {"problem_text": " Find the center of mass of a uniformly solid cone of base diameter $2a$ and height  $h$", "answer_latex": " $\\frac{3}{4}$", "answer_number": "0.75", "unit": "$h$", "source": "class", "problemid": " Problem 9.2", "comment": " "}, {"problem_text": "A particle is projected with an initial velocity $v_0$ up a slope that makes an angle $\\alpha$ with the horizontal. Assume frictionless motion and find the time required for the particle to return to its starting position. Find the time for $v_0=2.4 \\mathrm{~m} / \\mathrm{s}$ and $\\alpha=26^{\\circ}$.", "answer_latex": "2", "answer_number": "2", "unit": "$\\frac{v_0}{g \\sin \\alpha}$", "source": "class", "problemid": " Problem 2.16", "comment": " "}, {"problem_text": " Use the function described in Example 4.3, $x_{n+1}=\\alpha x_n\\left(1-x_n^2\\right)$ where $\\alpha=2.5$. Consider two starting values of $x_1$ that are similar, 0.9000000 and 0.9000001 . Make a plot of $x_n$ versus $n$ for the two starting values and determine the lowest value of $n$ for which the two values diverge by more than $30 \\%$.", "answer_latex": " 30", "answer_number": "30", "unit": " ", "source": "class", "problemid": " Problem 4.14", "comment": " "}, {"problem_text": "A gun fires a projectile of mass $10 \\mathrm{~kg}$ of the type to which the curves of Figure 2-3 apply. The muzzle velocity is $140 \\mathrm{~m} / \\mathrm{s}$. Through what angle must the barrel be elevated to hit a target on the same horizontal plane as the gun and $1000 \\mathrm{~m}$ away? Compare the results with those for the case of no retardation.", "answer_latex": " 17.4", "answer_number": "17.4", "unit": "$^{\\circ}$ ", "source": "class", "problemid": "Problem 2.20 ", "comment": " "}, {"problem_text": "A spacecraft is placed in orbit $200 \\mathrm{~km}$ above Earth in a circular orbit. Calculate the minimum escape speed from Earth. ", "answer_latex": " 3.23", "answer_number": "3.23", "unit": " $ \\mathrm{~km} / \\mathrm{s}$", "source": "class", "problemid": " Problem 8.30", "comment": " "}, {"problem_text": "Find the value of the integral $\\int_S(\\nabla \\times \\mathbf{A}) \\cdot d \\mathbf{a}$ if the vector $\\mathbf{A}=y \\mathbf{i}+z \\mathbf{j}+x \\mathbf{k}$ and $S$ is the surface defined by the paraboloid $z=1-x^2-y^2$, where $z \\geq 0$.", "answer_latex": "$-\\pi$", "answer_number": "-3.141592", "unit": "", "source": "class", "problemid": "Problem 1.38", "comment": " "}, {"problem_text": "A skier weighing $90 \\mathrm{~kg}$ starts from rest down a hill inclined at $17^{\\circ}$. He skis $100 \\mathrm{~m}$ down the hill and then coasts for $70 \\mathrm{~m}$ along level snow until he stops. Find the coefficient of kinetic friction between the skis and the snow. ", "answer_latex": "0.18", "answer_number": "0.18", "unit": " ", "source": "class", "problemid": " Problem 2.24", "comment": " "}, {"problem_text": "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 $\\operatorname{Sun}$ is $\\beta r_E$, where $r_E$ is the radius of Earth's (assumed) circular orbit and where $\\beta<1$, show that the time the comet spends within the orbit of Earth is given by\r\n$$\r\n\\sqrt{2(1-\\beta)} \\cdot(1+2 \\beta) / 3 \\pi \\times 1 \\text { year }\r\n$$\r\nIf the comet approaches the Sun to the distance of the perihelion of Mercury, how many days is it within Earth's orbit?", "answer_latex": " 76", "answer_number": "76", "unit": "$ \\text { days }$ ", "source": "class", "problemid": " Problem 8.12", "comment": " "}, {"problem_text": "An automobile drag racer drives a car with acceleration $a$ and instantaneous velocity $v$. The tires (of radius $r_0$ ) are not slipping. For what initial velocity in the rotating system will the hockey puck appear to be subsequently motionless in the fixed system? ", "answer_latex": " 0.5", "answer_number": "0.5", "unit": " $\\omega R$", "source": "class", "problemid": " Problem 10.4", "comment": " "}, {"problem_text": " A British warship fires a projectile due south near the Falkland Islands during World War I at latitude $50^{\\circ} \\mathrm{S}$. If the shells are fired at $37^{\\circ}$ elevation with a speed of $800 \\mathrm{~m} / \\mathrm{s}$, by how much do the shells miss their target and in what direction? Ignore air resistance.", "answer_latex": " 260", "answer_number": "260", "unit": " $\\mathrm{~m}$", "source": "class", "problemid": " Problem 10.18", "comment": " "}, {"problem_text": "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?\r\n", "answer_latex": " 9", "answer_number": "9", "unit": "$10^7 \\mathrm{yr}$ ", "source": "class", "problemid": " Problem 8.46", "comment": " "}, {"problem_text": "To perform a rescue, a lunar landing craft needs to hover just above the surface of the moon, which has a gravitational acceleration of $g / 6$. The exhaust velocity is $2000 \\mathrm{~m} / \\mathrm{s}$, but fuel amounting to only 20 percent of the total mass may be used. How long can the landing craft hover?", "answer_latex": "273", "answer_number": "273", "unit": " $\\mathrm{~s}$", "source": "class", "problemid": " Problem 9.62", "comment": " "}, {"problem_text": "In an elastic collision of two particles with masses $m_1$ and $m_2$, the initial velocities are $\\mathbf{u}_1$ and $\\mathbf{u}_2=\\alpha \\mathbf{u}_1$. If the initial kinetic energies of the two particles are equal, find the conditions on $u_1 / u_2$ such that $m_1$ is at rest after the collision and $\\alpha$ is positive. ", "answer_latex": " $3 \\pm 2 \\sqrt{2}$", "answer_number": "5.828427125", "unit": " ", "source": "class", "problemid": " Problem 9.36", "comment": " "}, {"problem_text": "Astronaut Stumblebum wanders too far away from the space shuttle orbiter while repairing a broken communications satellite. Stumblebum realizes that the orbiter is moving away from him at $3 \\mathrm{~m} / \\mathrm{s}$. Stumblebum and his maneuvering unit have a mass of $100 \\mathrm{~kg}$, including a pressurized tank of mass $10 \\mathrm{~kg}$. The tank includes only $2 \\mathrm{~kg}$ of gas that is used to propel him in space. The gas escapes with a constant velocity of $100 \\mathrm{~m} / \\mathrm{s}$. With what velocity will Stumblebum have to throw the empty tank away to reach the orbiter?", "answer_latex": "11", "answer_number": "11", "unit": "$ \\mathrm{~m} / \\mathrm{s}$ ", "source": "class", "problemid": " Problem 9.12", "comment": " "}]