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physics_en_no_proof_embedding

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800
A quarantined physics student decides to perform an experiment to land a small box of mass $m=60 \mathrm{~g}$ onto the center of a target a distance $\Delta d$ away. The student puts the box on a top of a frictionless ramp with height $h_{2}=0.5 \mathrm{~m}$ that is angled $\theta=30^{\circ}$ to the horizontal on a tab...
[ "2.47" ]
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801
A wooden bus of mass $M=20,000 \mathrm{~kg}$ ( $M$ represents the mass excluding the wheels) is on a ramp with angle $30^{\circ}$. Each of the four wheels is composed of a ring of mass $\frac{M}{2}$ and radius $R=1 \mathrm{~m}$ and 6 evenly spaced spokes of mass $\frac{M}{6}$ and length $R$. All components of the truck...
[ "3.32" ]
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804
A ball is situated at the midpoint of the bottom of a rectangular ditch with width $1 \mathrm{~m}$. It is shot at a velocity $v=5 \mathrm{~m} / \mathrm{s}$ at an angle of $30^{\circ}$ relative to the horizontal. How many times does the ball collide with the walls of the ditch until it hits the bottom of the ditch again...
[ "2" ]
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805
A frictionless track contains a loop of radius $R=0.5 \mathrm{~m}$. Situated on top of the track lies a small ball of mass $m=2 \mathrm{~kg}$ at a height $h$. It is then dropped and collides with another ball (of negligible size) of mass $M=5 \mathrm{~kg}$. <image_1> Let $h$ be the minimum height that $m$ was dropped...
[ "$2.37$" ]
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809
The following diagram depicts a single wire that is bent into the shape below. The circuit is placed in a magnetic field pointing out of the page, uniformly increasing at the rate $\frac{d B}{d t}=2.34 \mathrm{~T} / \mathrm{s}$. Calculate the magnitude of induced electromotive force in the wire, in terms of the followi...
[ "$1.9$" ]
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815
For his art project, Weishaupt cut out $N=20$ wooden equilateral triangular blocks with a side length of $\ell=10 \mathrm{~cm}$ and a thickness of $t=2 \mathrm{~cm}$, each with the same mass and uniform density. He wishes to stack one on top of the other overhanging the edge of his table. In centimeters, what is the ma...
[ "21" ]
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818
The graph provided plots the $y$-component of the velocity against the $x$-component of the velocity of a kiddie roller coaster at an amusement park for a certain duration of time. The ride takes place entirely in a two dimensional plane. Some students made a remark that at one time, the acceleration was perpendicular...
[ "1" ]
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820
While exploring outer space, Darth Vader comes upon a purely reflective spherical planet with radius $R_{p}=$ $40,000 \mathrm{~m}$ and mass $M_{p}=8.128 \times 10^{24} \mathrm{~kg}$. Around the planet is a strange moon of orbital radius $R_{s}=$ $6,400,000 \mathrm{~m}\left(R_{s} \gg R_{p}\right)$ and mass $M_{s}=9.346 ...
[ "$1.33 \\cdot 10^{35}$" ]
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822
Mario is racing with Wario on Moo Moo Meadows when a goomba, ready to avenge all of his friends' deaths, came and hijacked Mario's kart. A graph representing the motion of Mario at any instant is shown below. The velocity acquired by Mario is shown on the x-axis, and the net power of his movement is shown on the $\math...
[ "40" ]
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824
An engineer has access to a tetrahedron building block with side length $\ell=10 \mathrm{~cm}$. The body is made of a thermal insulator but the edges are wrapped with a thin copper wiring with cross sectional area $S=2 \mathrm{~cm}^{2}$. The thermal conductivity of copper is $385.0 \mathrm{~W} /(\mathrm{m} \mathrm{K})$...
[ "4.62" ]
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825
Three unit circles, each with radius 1 meter, lie in the same plane such that the center of each circle is one intersection point between the two other circles, as shown below. Mass is uniformly distributed among all area enclosed by at least one circle. The mass of the region enclosed by the triangle shown above is $1...
[ "39" ]
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827
Two infinitely long current carrying wires carry constant current $i_{1}=2 \mathrm{~A}$ and $i_{2}=3 \mathrm{~A}$ as shown in the diagram. The equations of the wire curvatures are $y^{2}-8 x-6 y+25=0$ and $x=0$. Find the magnitude of force (in Newtons) acting on one of the wires due to the other. <image_1> Note: The ...
[ "$7.5398 \\cdot 10^{-6}$" ]
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829
Two electrons are in a uniform electric field $\mathbf{E}=E_{0} \hat{\mathbf{z}}$ where $E_{0}=10^{-11} \mathrm{~N} / \mathrm{C}$. One electron is at the origin, and another is $10 \mathrm{~m}$ above the first electron. The electron at the origin is moving at $u=10 \mathrm{~m} / \mathrm{s}$ at an angle of $30^{\circ}$ ...
[ "6.84" ]
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830
Consider a long uniform conducting cylinder. First, we divide the cylinder into thirds and remove the middle third. Then, we perform the same steps on the remaining two cylinders. Again, we perform the same steps on the remaining four cylinders and continuing until there are 2048 cylinders. We then connect the termina...
[ "1.017" ]
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831
A square based pyramid (that is symmetrical) is standing on top of a cube with side length $\ell=10 \mathrm{~cm}$ such that their square faces perfectly line up. The cube is initially standing still on flat ground and both objects have the same uniform density. The coefficient of friction between every surface is the s...
[ "1.07" ]
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832
During quarantine, the FBI has been monitoring a young physicists suspicious activities. After compiling weeks worth of evidence, the FBI finally has had enough and searches his room. The room's door is opened with a high angular velocity about its hinge. Over a very short period of time, its angular velocity increase...
[ "$\\frac{\\sqrt{3}}{3}$" ]
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833
A solid half-disc of mass $m=1 \mathrm{~kg}$ in the shape of a semi-circle of radius $R=1 \mathrm{~m}$ is kept at rest on a smooth horizontal table. QiLin starts applying a constant force of magnitude $F=10 \mathrm{~N}$ at point A as shown, parallel to its straight edge. What is the initial linear acceleration of point...
[ "$15.9395$" ]
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834
A regular tetrahedron of mass $m=1 \mathrm{~g}$ and unknown side length is balancing on top of a hemisphere of mass $M=100 \mathrm{~kg}$ and radius $R=100 \mathrm{~m}$. The hemisphere is placed on a flat surface such that it is at its lowest potential. For a certain value of the length of the regular tetrahedron, the o...
[ "$200\\sqrt{6}$" ]
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838
A frictionless track contains a loop of radius $R=0.5 \mathrm{~m}$. Situated on top of the track lies a small ball of mass $m=2 \mathrm{~kg}$ at a height $h$. It is then dropped and collides with another ball of mass $M=5 \mathrm{~kg}$. <image_1> The coefficient of restitution for this collision is given as $e=\frac{...
[ "$72.902$" ]
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839
Two astronauts, Alice and Bob, are standing inside their cylindrical spaceship, which is rotating at an angular velocity $\omega$ clockwise around its axis in order to simulate the gravitational acceleration $g$ on earth. The radius of the spaceship is $R$. For this problem, we will only consider motion in the plane pe...
[ "$60.2$" ]
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841
A bicycle wheel of mass $M=2.8 \mathrm{~kg}$ and radius $R=0.3 \mathrm{~m}$ is spinning with angular velocity $\omega=5 \mathrm{rad} / \mathrm{s}$ around its axis in outer space, and its center is motionless. Assume that it has all of its mass uniformly concentrated on the rim. A long, massless axle is attached to its ...
[ "0.568" ]
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843
A small toy car rolls down three ramps with the same height and horizontal length, but different shapes, starting from rest. The car stays in contact with the ramp at all times and no energy is lost. Order the ramps from the fastest to slowest time it takes for the toy car to drop the full $1 \mathrm{~m}$. For example,...
[ "213" ]
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845
In a typical derby race, cars start at the top of a ramp, accelerate downwards, and race on a flat track, and are always set-up in the configuration shown below. <image_1> A common technique is to change the location of the center of mass of the car to gain an advantage. Alice ensures the center of mass of her car is ...
[ "1.03" ]
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846
A simple crane is shown in the below diagram, consisted of light rods with length $1 \mathrm{~m}$ and $\sqrt{2} \mathrm{~m}$. The end of the crane is supporting a $5 \mathrm{kN}$ object. Point $B$ is known as a "pin." It is attached to the main body and can exert both a vertical and horizontal force. Point $A$ is known...
[ "10" ]
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848
A small block of mass $m$ and charge $Q$ is placed at rest on an inclined plane with a slope $\alpha=40^{\circ}$. The coefficient of friction between them is $\mu=0.3$. A homogenous magnetic field of magnitude $B_{0}$ is applied perpendicular to the slope. The speed of the block after a very long time is given by $v=\b...
[ "0.6" ]
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850
A sprinkler fountain is in the shape of a semi-sphere that spews out water from all angles at a uniform speed $v$ such that without the presence of wind, the wetted region around the fountain forms a circle in the $X Y$ plane with the fountain centered on it. Now suppose there is a constant wind blowing in a direction ...
[ "2.5" ]
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852
Poncho is a very good player of the legendary carnival game known as Pico-Pico. Its setup consists of a steel ball, represented by a point mass, of negligible radius and a frictionless vertical track. The goal of Pico-Pico is to flick the ball from the beginning of the track (point $A$ ) such that it is able to travers...
[ "0.1231" ]
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854
Anyone who's had an apple may know that pieces of an apple stick together, when picking up one piece a second piece may also come with the first piece. The same idea is tried on a golden apple. Consider two uniform hemispheres with radius $r=4 \mathrm{~cm}$ made of gold of density $\rho_{g}=19300 \mathrm{~kg} \mathrm{~...
[ "0.0281" ]
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855
In the following two problems we will look at shooting a basketball. Model the basketball as an elastic hollow sphere with radius 0.1 meters. Model the net and basket as shown below, dimensions marked. Neglect friction between the backboard and basketball, and assume all collisions are perfectly elastic. <image_1> F...
[ "7.19" ]
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857
Consider a toilet paper roll with some length of it hanging off as shown. The toilet paper roll rests on a cylindrical pole of radius $r=1 \mathrm{~cm}$ and the coefficient of static friction between the role and the pole is $\mu=0.3$. <image_1> The length of the paper hanging off has length $\ell=30 \mathrm{~cm}$ an...
[ "2.061" ]
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858
In the circuit shown below, a capacitor $C=4 \mathrm{~F}$, inductor $L=5 \mathrm{H}$, and resistors $R_{1}=3 \Omega$ and $R_{2}=2 \Omega$ are placed in a diamond shape and are then fed an alternating current with peak voltage $V_{0}=1 \mathrm{~V}$ of unknown frequency. Determine the magnitude of the maximum instantaneo...
[ "0.65" ]
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859
A uniform bar of length $l$ and mass $m$ is connected to a very long thread of negligible mass suspended from a ceiling. It is then rotated such that it is vertically upside down and then released. Initially, the rod is in unstable equilibrium. As it falls down, the minimum tension acting on the thread over the rod's e...
[ "0.165" ]
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862
Colliding Conducting Slab A thin conducting square slab with side length $s=5 \mathrm{~cm}$, initial charge $q=0.1 \mu \mathrm{C}$, and mass $m=100 \mathrm{~g}$ is given a kick and sent bouncing between two infinite conducting plates separated by a distance $d=0.5 \mathrm{~cm} \ll s$ and with surface charge density $\p...
[ "25273" ]
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864
Spinning Cylinder Adithya has a solid cylinder of mass $M=10 \mathrm{~kg}$, radius $R=0.08 \mathrm{~m}$, and height $H=0.20 \mathrm{~m}$. He is running a test in a chamber on Earth over a distance of $d=200 \mathrm{~m}$ as shown below. Assume that the physical length of the chamber is much greater than $d$ (i.e. the ch...
[ "$1.25$" ]
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867
Consider a rectangular loop made of superconducting material with length $\ell=200 \mathrm{~cm}$ and width $w=2 \mathrm{~cm}$. The radius of this particular wire is $r=0.5 \mathrm{~mm}$. This superconducting rectangular loop initially has a current $I_{1}=5 \mathrm{~A}$ in the counterclockwise direction as shown in the...
[ "$1.12 \\times 10^{-3}$" ]
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874
Consider the following simple model of a bow and arrow. An ideal elastic string has a spring constant $k=10 \mathrm{~N} / \mathrm{m}$ and relaxed length $L=1 \mathrm{~m}$ which is attached to the ends of an inflexible fixed steel rod of the same length $L$ as shown below. A small ball of mass $m=2 \mathrm{~kg}$ and the...
[ "$2.23$" ]
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875
A truck (denoted by $S$ ) is driving at a speed $v=2 \mathrm{~m} / \mathrm{s}$ in the opposite direction of a car driving at a speed $u=3 \mathrm{~m} / \mathrm{s}$, which is equipped with a rear-view mirror. Both $v$ and $u$ are measured from an observer on the ground. Relative to this observer, what is the speed (in $...
[ "$8$" ]
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879
These days, there are so many stylish rectangular home-designs (see figure A). It is possible from the outline of those houses in their picture to estimate with good precision where the camera was. Consider an outline in one photograph of a rectangular house which has height $H=3$ meters (see figure B for square-grid c...
[ "$0.9$" ]
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880
Consider a thin rigid wire-frame MNPP'N'M' in which MNN'M' and NPP'N' are two squares of side $L$ with resistance per unit-length $\lambda$ and their planes are perpendicular. The frame is rotated with a constant angular velocity $\omega$ around an axis passing through $\mathrm{NN}$ ' and put in a region with constant ...
[ "$6.58$" ]
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882
A scale of uniform mass $M=3 \mathrm{~kg}$ of length $L=4 \mathrm{~m}$ is kept on a rough table (infinite friction) with $l=1 \mathrm{~m}$ hanging out of the table as shown in the figure below. A small ball of mass $m=1 \mathrm{~kg}$ is released from rest from a height of $h=5 \mathrm{~m}$ above the end of the scale. F...
[ "$18.21$" ]
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886
The following information applies for the next two problems. A circuit has a power source of $\mathcal{E}=5.82 \mathrm{~V}$ connected to three elements in series: an inductor with $L=12.5 \mathrm{mH}$, a capacitor with $C=48.5 \mu \mathrm{F}$, and a diode with threshold voltage $V_{0}=0.65 \mathrm{~V}$. (Of course, the...
[ "$2.446 \\times 10^{-3}$" ]
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887
At Hanoi-Amsterdam High School in Vietnam, every subject has its own flag (see Figure A, taken by Tung X. Tran). While the flags differ in color, they share the same central figure. Consider a planar conducting frame of that figure rotating at a constant angular velocity in a uniform magnetic field (see Figure B). The ...
[ "$0.864$" ]
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889
A tesseract is a 4 dimensional example of cube. It can be drawn in 3 dimensions by drawing two cubes and connecting their vertices together as shown in the picture below: <image_1> Now for the 3D equivalent. The lines connecting the vertices are replaced with ideal springs of constant $k=10 \mathrm{~N} / \mathrm{m}$ ...
[ "2.35" ]
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892
An open electrical circuit contains a wire loop in the shape of a semi-circle, that contains a resistor of resistance $R=0.2 \Omega$. The circuit is completed by a conducting pendulum in the form of a uniform rod with length $\ell=0.1 \mathrm{~m}$ and mass $m=0.05 \mathrm{~kg}$, has no resistance, and stays in contact ...
[ "$145$" ]
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895
Consider an optical system made of many identical ideal (negligible-thickness) halflenses with focal length $f>0$, organized so that they share the same center and are angular-separated equally at density $n$ (number of lenses per unit-radian). Define the length-scale $\lambda=f / n$. A light-ray arrives perpendicular ...
[ "$1.05$" ]
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896
For black body radiation, Wien's Displacement Law states that its spectral radiance will peak at $$ \lambda_{\text {peak }}=\frac{b}{T} $$ where $b=2.89777 \times 10^{-3} \mathrm{mK}$, and $T$ is the temperature of the object. When QiLin tried to reproduce this in a lab, by working with a tungsten-filament lightbulb a...
[ "0.08" ]
[ 0.018310546875, -0.0069580078125, 0.007476806640625, 0.007049560546875, 0.02197265625, 0.02734375, -0.01318359375, -0.00909423828125, 0.006622314453125, 0.00927734375, -0.031494140625, 0.03955078125, 0.015625, 0.007476806640625, 0.00162506103515625, 0.0091552734375, -0.03076171875, ...
904
A coin of uniform mass density with a radius of $r=1 \mathrm{~cm}$ is initially at rest and is released from a slight tilt of $\theta=8^{\circ}$ onto a horizontal surface with an infinite coefficient of static friction. The coin has a thicker rim, allowing it to drop and rotate on one point. With every collision, the c...
[ "0.716" ]
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906
Here is a Physoly round button badge, in which the logo is printed on the flat and rigid surface of this badge. Toss it in the air and track the motions of three points (indicated by cyan circles in the figure) separated a straight-line distance of $L=5 \mathrm{~mm}$ apart. At a particular moment, we find that these al...
[ "$4.1411$" ]
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913
In general, we can describe the quadratic drag on an object by the following force law: $$ F_{D}=\frac{1}{2} C_{D} \rho A v^{2} $$ where $A$ is the cross-sectional area of the object exposed to the airflow, $v$ is the speed of the object in a fluid, and $C_{D}$ is the drag coefficient, a dimensionless quantity that va...
[ "$4.98$" ]
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914
The following information applies for the next two problems. Pictured is a wheel from a 4wheeled car of weight $1200 \mathrm{~kg}$. The absolute pressure inside the tire is $3.0 \times 10^{5} \mathrm{~Pa}$. Atmospheric pressure is $1.0 \times 10^{5} \mathrm{~Pa}$. Assume the rubber has negligible "stiffness" (i.e. a ne...
[ "$1.00 \\times 10^{9}$" ]
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916
A hollow sphere of mass $M$ and radius $R$ is placed under a plank of mass $3 M$ and length $2 R$. The plank is hinged to the floor, and it initially makes an angle $\theta=\frac{\pi}{3}$ rad to the horizontal. Under the weight of the plank, the sphere starts rolling without slipping across the floor. What is the spher...
[ "$2.76$" ]
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920
On a flat playground, choose a Cartesian Oxy coordinate system (in unit of meters). A child running at a constant velocity $V=1 \mathrm{~m} / \mathrm{s}$ around a heart-shaped path satisfies the following order- 6 algebraic equation: $$ \left(x^{2}+y^{2}-L^{2}\right)^{3}-L x^{2} y^{3}=0, L=10 \text {. } $$ When the chi...
[ "$0.066591$" ]
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921
A boy is riding a tricycle across along a sidewalk that is parallel to the $x$-axis. This tricycle contains three identical wheels with radius $0.5 \mathrm{~m}$. The front wheel is free to rotate while the last two wheels are parallel to each other and to the main body of the tricycle. See the diagram. <image_1> The f...
[ "$0.16912$" ]
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924
For any circuit network made of batteries and resistors, if we know the voltages of all the batteries and the resistance values of all the resistors, we can calculate all the electrical currents. However, if we know the voltages of all the batteries and all the currents, it is still not enough to uniquely determine the...
[ "$4.0741 \\times 10^{10}$" ]
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925
Two carts, each with a mass of $300 \mathrm{~g}$, are fixed to move on a horizontal track. As shown in the figure, the first cart has a strong, tiny permanent magnet of dipole moment $0.5 \mathrm{~A} \cdot \mathrm{m}^{2}$ attached to it, which is aligned along the axis of the track pointing toward the other cart. On th...
[ "25.2" ]
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927
The logo of OPhO describes two objects travelling around their center-of-mass, following the same oval-shape trajectory. For simplicity, we assume these objects are point-like, have identical mass, and interacts via an interacting potential $U(d)$ depends on the distance $d$ between them. Choose the polar coordinates $...
[ "-0.6864" ]
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930
Consider a uniform isosceles triangle prism $\mathrm{ABC}$, with the apex angle $\theta=110^{\circ}$ at vertex $\mathrm{A}$. One of the sides, $\mathrm{AC}$, is coated with silver, allowing it to function as a mirror. When a monochrome light-ray of wavelength $\lambda$ approaches side $\mathrm{AB}$ at an angle of incid...
[ "1.5436" ]
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931
Field-drive is a locomotion mechanism that is analogous to general relativistic warp-drive. In this mechanism, an active particle continuously climbs up the field-gradient generated by its own influence on the environment so that the particle can bootstrap itself into a constant non-zero velocity motion. Consider a fie...
[ "$4 \\times 10^{-2}$" ]
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932
1. Restore equilibrium A wooden plank of length $1 \mathrm{~m}$ and uniform cross-section is hinged at one end to the bottom of a tank as shown in figure. The tank is filled with water up to a height $0.5 \mathrm{~m}$. The specific gravity of the plank (or ratio of plank density to water density) is 0.5 . Find the an...
[ "$\\theta=45$" ]
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933
(a) After a very short moment, the shock front caused by an explosion sweeps up the material around it, increasing the mass which is blasted outwards. At early times, the energy $E \sim M v^{2} / 2$ can be thought of as roughly conserved. Let $R$ be the radius of the explosion at some time $t$. Estimate the speed of t...
[ "$\\left(\\frac{E}{\\rho}\\right)^{1 / 5} t^{2 / 5}$" ]
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934
(b) Air has a density $\rho \simeq 1 \mathrm{~kg} \mathrm{~m}^{-3}$. Using the provided image, roughly estimate the yield of the Trinity explosion in kilotons (of TNT). Note that 1 kiloton is equal to approximately $4.2 \times 10^{12} \mathrm{~J}$.
[ "$11$" ]
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935
(c) In 1054, Chinese astronomers observed and documented a supernova which was bright enough to be visible during the day for around a month. The rubble left behind is a rapidly expanding supernova remnant called the Crab Nebula, which is intensely studied today. <image_1> Figure 2: Photographs of the Crab Nebula ta...
[ "$1.7 \\times 10^{43}$" ]
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937
(b) If the concave part is filled with water of refractive index $4 / 3$, find the distance through which the pin should be moved, so that the image of the pin again coincides with the pin. <image_1>
[ "1.16" ]
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951
(a) What is the total energy stored in the capacitor before any fluid rises as a function of its height $l$ ?
[ "$U=\\frac{\\pi \\epsilon_{0} l}{\\log (b / a)} V^{2}$" ]
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952
(b) How high $h$ does the dielectric fluid rise against the force of gravity given by acceleration $g$ ? (Note: it is very easy to get the correct answer while describing the problem incorrectly. Take care for full credit.)
[ "$h=\\frac{\\epsilon_{0}(\\kappa-1) V^{2}}{\\log (b / a) \\rho g\\left(b^{2}-a^{2}\\right)}$" ]
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953
(c) Calculate the pressure difference $P$ above atmospheric pressure needed to suck the fluid to the top of the cylinder, assuming this is possible. Assume $l \gg h$ and neglect fluid dynamical and thermodynamic effects. You will find that $P$ consists of a term depending on $l$ and a term depending on $\kappa-1$. Prov...
[ "$P=\\frac{\\rho g l}{2}-\\frac{\\epsilon_{0} V^{2}(\\kappa-1)}{\\left(b^{2}-a^{2}\\right) \\log (b / a)}$" ]
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958
(a) Find an expression for the $y_{e}$, the position where the block experiences net zero vertical force.
[ "$y_{e}=y_{0}-\\frac{g M}{\\beta Q}$" ]
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959
(b) What is the force acting on the block when it is in contact with the incline plane?
[ "$F(y)=\\left(M g-Q \\beta\\left(y_{0}-y\\right)\\right) \\sin (\\alpha)$" ]
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962
(e) Once the block has lost contact with the plane it begins to oscillate in the field. What frequency does it oscillate at?
[ "$f=\\frac{1}{2 \\pi} \\sqrt{\\frac{\\beta Q}{m}}$" ]
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990
(a) Consider a small time interval $\Delta t$ in the rocket's frame. $\Delta t$ is small enough that the rocket's frame can be considered an inertial frame (i.e., the frame has no acceleration). The amount of fuel ejected in this time is $\Delta m=|\mu| \Delta t$. (See Figure 2.) <image_1> Figure 2: Rocket gaining spe...
[ "$\\mathbf{u} \\Delta m+m \\Delta \\mathbf{v}=0$ , $\\mathbf{a}=-\\frac{\\mu}{m} \\mathbf{u}$" ]
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991
(b) Suppose that the rocket is stationary at time $t=0$. The empty rocket has mass $m_{0}$ and the rocket full of fuel has mass $9 m_{0}$. The engine is turned on at time $t=0$, and fuel is ejected at the rate $\mu$ and relative velocity $\mathbf{u}$ described previously. What will be the speed of the rocket when it ru...
[ "$u \\ln 10$" ]
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993
(a) Suppose that the product of combustion is just one species of molecule $\left(\mathrm{H}_{2} \mathrm{O}\right.$, for example) with mass $m_{p}$ and average kinetic energy $E$. What is the upper limit of exhaust speed $u$ when these molecules are ejected from the rocket?
[ "$\\sqrt{\\frac{2 E}{m_{p}}}$" ]
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995
(c) The enthalpy of combustion of the reaction $2 \mathrm{H}_{2}+\mathrm{O}_{2} \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}$ is approximately $15.76 \times 10^{6} \mathrm{~J} / \mathrm{kg}$. That is, for every kilogram of water $\left(\mathrm{H}_{2} \mathrm{O}\right)$ produced, the energy released by the reaction is $1...
[ "$\\sqrt{2} \\times 3970$" ]
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996
(d) Now, consider the effects of gravity. Given a rocket of mass $m_{0}$ and mass ejection rate $\mu$, what ejection speed $u$ would be required to launch satellites in the Earth's gravitational field, of strength $g$ ? (An estimate, ignoring numerical factors, is acceptable.) What about launching space probes to other...
[ "$\\sqrt{g R}$ , $\\sqrt{2 g R}$" ]
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1,000
d - Calculate the distance between two neighbouring $\mathrm{K}$ ions in the crystal.
[ "0.44" ]
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1,001
$a_{1}$ - Deduce for which value(s) of $\alpha$ the configuration of the spaceship and satellite remain unchanged (with respect to the earth)? In other words, for which value(s) of $\alpha$ is $\alpha$ constant?
[ "$0,\\pi,\\frac{\\pi}{2},\\frac{3\\pi}{2}$" ]
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1,005
$\mathrm{c}_{2}$ - How long must this current be maintained to change the altitude of the orbit by 10 m. Assume that $\alpha$ remains zero. Ignore all contributions from currents in the magnetosphere.
[ "$t=\\frac{1}{2} \\cdot \\frac{m \\cdot \\Omega \\cdot \\Delta R}{BIL}$" ]
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1,007
$\mathrm{b}$ - Calculate the average radius of the liquid interior, using the data of Fig. 1. Make the approximation that the densities of the crust and the interior are the same. (Ignore the change in shape of the interior).
[ "$0.95$" ]
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1,008
1.1 Two gravitating masses $M$ and $m$ are moving in circular orbits of radii $R$ and $r$, respectively, about their common centre of mass. Find the angular velocity $\omega_{0}$ of the line joining $M$ and $m$ in terms of $R, r, M, m$ and the universal gravitational constant $G$.
[ "$\\omega_{0}=\\sqrt{\\frac{G m}{R(R+r)^{2}}}$" ]
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1,009
1.2 A third body of infinitesimal mass $\mu$ is placed in a coplanar circular orbit about the same centre of mass so that $\mu$ remains stationary relative to both $M$ and $m$ as shown in Figure 1. Assume that the infinitesimal mass is not collinear with $M$ and $m$. Find the values of the following parameters in terms...
[ "$R+r$, $R+r$, $\\sqrt{r^{2}+r R+R^{2}}$" ]
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1,010
1.3 Consider the case $M=m$. If $\mu$ is now given a small radial perturbation (along $\mathrm{O} \mu$ ), what is the angular frequency of oscillation of $\mu$ about the unperturbed position in terms of $\omega_{0}$ ? Assume that the angular momentum of $\mu$ is conserved.
[ "$\\frac{\\sqrt{7}}{2} \\omega_{0}$" ]
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1,011
1.4 In the plane containing the three spacecrafts, what is the relative speed of one spacecraft with respect to another?
[ "$996 $" ]
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1,020
3.1 Calculate the electric field intensity $\vec{E}_{p}$ at a distance $r$ from an ideal electric dipole $\vec{p}$ at the origin $\mathrm{O}$ along the direction of $\vec{p}$ in Figure 2. $p=2 a q, \quad r \gg a$ <image_1> FIGURE 2
[ "$E_{p}=\\frac{2 p}{4 \\pi \\varepsilon_{0} r^{3}}$" ]
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1,021
3.2 Find the expression for the force $\vec{f}$ acting on the ion due to the polarised atom. Show that this force is attractive regardless of the sign of the charge of the ion.
[ "$\\vec{f}=-\\frac{\\alpha Q^{2}}{8 \\pi^{2} \\varepsilon_{0}^{2} r^{5}} \\hat{r}$" ]
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1,022
3.3 What is the electric potential energy of the ion-atom interaction in terms of $\alpha, Q$ and $r$ ?
[ "$-\\frac{\\alpha Q^{2}}{32 \\pi^{2} \\varepsilon_{0}^{2} r^{4}}$" ]
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1,023
3.4 Find the expression for $r_{\min }$, the distance of the closest approach, as shown in Figure 1.
[ "$r_{\\min }=\\frac{b}{\\sqrt{2}}\\left[1+\\sqrt{1-\\frac{\\alpha Q^{2}}{4 \\pi^{2} \\varepsilon_{0}^{2} m v_{0}^{2} b^{4}}}\\right]^{\\frac{1}{2}}$" ]
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1,024
3.5 If the impact parameter $b$ is less than a critical value $b_{0}$, the ion will descend along a spiral to the atom. In such a case, the ion will be neutralized, and the atom is, in turn, charged. This process is known as the "charge exchange" interaction. What is the cross sectional area $A=\pi b_{0}^{2}$ of this "...
[ "$\\pi\\left(\\frac{\\alpha Q^{2}}{4 \\pi^{2} \\varepsilon_{0}^{2} m v_{0}^{2}}\\right)^{\\frac{1}{2}}$" ]
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1,027
c) Compute the temperature of the air close to the ground in b) assuming that your eyes are located $1.60 \mathrm{~m}$ above the ground and that the distance to the "water" is $250 \mathrm{~m}$. The refractive index of the air at $15{ }^{\circ} \mathrm{C}$ and at normal air pressure $(101.3 \mathrm{kPa})$ is 1.000276 ....
[ "328" ]
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1,028
<image_1> In certain lakes there is a strange phenomenon called "seiching" which is an oscillation of the water. Lakes in which you can see this phenomenon are normally long compared with the depth and also narrow. It is natural to see waves in a lake but not something like the seiching, where the entire water volume ...
[ "$\\frac{\\pi L}{\\sqrt{3 h}}$" ]
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1,029
A.1 What is the $x$-component of the net force $F_{x}$ acting on the right plate (magnitude and direction)?
[ "$F_{x}=\\left(\\frac{\\rho_{0}}{\\rho_{\\mathrm{oil}}}-1\\right) \\frac{\\rho_{0} g h^{2} w}{2}$" ]
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1,030
A.2 Assuming that the crust is isotropic, find how its density $\rho$ depends on its temperature $T$. Assuming that $\left|k_{l}\right| \ll 1$, write your answer in the approximate form $$ \rho(T) \approx \rho_{1}\left[1+k \frac{T_{1}-T}{T_{1}-T_{0}}\right] $$ where terms of order $k_{l}^{2}$ and higher are neglected...
[ "$\\rho_{1}\\left(1+3 k_{l} \\frac{T_{1}-T}{T_{1}-T_{0}}\\right)$ , $k=3 k_{l}$" ]
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1,031
A.3 By assuming that mantle and water each behave like an incompressible fluid at hydrostatic equilibrium, express the far-distance crust thickness $D$ in terms of $h, \rho_{0}, \rho_{1}$, and $k$. Any motion of the material can be neglected.
[ "$D=\\frac{2}{k}(1-\\frac{\\rho_{0}}{\\rho_{1}}) h$" ]
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1,032
A.4 Find, to the leading order in $k$, the net horizontal force $F$ acting on the right half $(x>0)$ of the crust in terms of $\rho_{0}, \rho_{1}, h, L, k$ and $g$.
[ "$\\frac{2 g L h^{2}}{3 k} \\frac{\\left(\\rho_{1}-\\rho_{0}\\right)^{2}}{\\rho_{1}}$" ]
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1,033
A.5 By using dimensional analysis or order-of-magnitude analysis, estimate the characteristic time $\tau$ in which the difference between the upper and lower surface temperatures of the crust far away from the ridge axis is going to approach zero. You can assume that $\tau$ does not depend on the two initial surface te...
[ "$\\frac{c \\rho_{1} D^{2}}{\\kappa}$" ]
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1,034
B.1 Consider a single ray emitted by the earthquake that makes an initial angle $0<\theta_{0}<\pi / 2$ with the $z$-axis and travels in the $x-z$ plane. What is the horizontal coordinate $x_{1}\left(\theta_{0}\right) \neq 0$ at which this ray can be detected at the surface of the planet? It is known that the ray path i...
[ "$x_{1}(\\theta_{0})=2 z_{0} \\cot \\theta_{0}$" ]
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1,035
B.2 Find how the energy density per unit area $\varepsilon(x)$ absorbed by the surface depends on the distance along the surface $x$.
[ "$\\varepsilon(x)=\\frac{2 E z_{0}}{\\pi(4 z_{0}^{2}+x^{2})}$" ]
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1,036
B.3 At what distance $x_{\max }$ along the surface from the source is the furthest point that the signal does not reach? Write your answer in terms of $\theta_{0}, \delta \theta_{0}$ and other constants given above.
[ "$x_{\\max }=\\frac{2 z_{0} \\cos ^{2} \\theta_{0}}{\\delta \\theta_{0}}$" ]
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1,037
A.1 Calculate the electrostatic potential $\Phi(z)$ along the axis of the ring at a $z$ distance from its center (point A in Figure 1).
[ "$\\Phi(z)=\\frac{q}{4 \\pi \\varepsilon_{0}} \\frac{1}{\\sqrt{R^{2}+z^{2}}}$." ]
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1,038
A.2 Calculate the electrostatic potential $\Phi(z)$ to the lowest non-zero power of $z$, assuming $z \ll R$.
[ "$\\Phi(z) \\approx \\frac{q}{4 \\pi \\varepsilon_{0} R}(1-\\frac{z^{2}}{2 R^{2}})$" ]
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1,039
A.3 An electron (mass $m$ and charge $-e$ ) is placed at point A (Figure 1, $z \ll R$ ). What is the force acting on the electron?
[ "$F(z)=-\\frac{q e}{4 \\pi \\varepsilon_{0} R^{3}} z$" ]
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