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physics_en_no_proof_embedding

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1,309
A.2 The parcel volume $V(x, t)$ oscillates around the equilibrium value of $V_{0}=S \Delta x$ and has the form $$ V(x, t)=V_{0}+V_{1}(x) \cos (\omega t) . \tag{2} $$ Obtain an expression for $V_{1}(x)$ in terms of $V_{0}, a, k$ and $x$.
[ "$V_{1}(x)=a k V_{0} \\cos (k x)$" ]
[ 0.02197265625, -0.006195068359375, 0.0031585693359375, 0.00567626953125, 0.0072021484375, 0.01361083984375, 0.0240478515625, 0.0166015625, 0.00286865234375, 0.00640869140625, -0.0181884765625, -0.00860595703125, -0.0224609375, -0.025634765625, 0.02490234375, -0.0025634765625, -0.0080...
1,310
A.3 Assume that the total pressure of the gas, as a result of the sound wave, takes the approximate form $$ p(x, t)=p_{0}-p_{1}(x) \cos (\omega t) . \tag{3} $$ Considering the forces acting on the parcel of gas, compute the amplitude $p_{1}(x)$ of the pressure oscillation to leading order, in terms of the position $x...
[ "$p_{1}(x)=a \\frac{\\omega^{2}}{k} \\rho_{0} \\cos (k x)$" ]
[ -0.0022125244140625, 0.00830078125, 0.005767822265625, -0.00732421875, 0.00799560546875, 0.0184326171875, 0.00159454345703125, -0.00506591796875, 0.01165771484375, 0.00335693359375, -0.042236328125, -0.006805419921875, -0.0184326171875, -0.00830078125, 0.01165771484375, 0.0062561035156...
1,311
A.4 Use the relation above and the results of the previous tasks to obtain an expression for the speed of sound waves $c=\omega / k$ in the tube, to first order. Express your answer in terms of $p_{0}, \rho_{0}$ and the adiabatic constant $\gamma$.
[ "$c=\\sqrt{\\frac{\\gamma p_{0}}{\\rho_{0}}}$" ]
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1,312
A.5 The change in the gas temperature due to the adiabatic expansion and contraction, as a result of the sound wave, takes the form: $$ T(x, t)=T_{0}-T_{1}(x) \cos (\omega t) \tag{4} $$ Compute the amplitude $T_{1}(x)$ of the temperature oscillations in terms of $T_{0}, \gamma$, $a, k$ and $x$.
[ "$a k(\\gamma-1) T_{0} \\cos (k x)$" ]
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1,314
B.1 Consider a specific parcel of gas in the region of the stack, originally at $x_{0}=L / 4 .$ As the parcel moves within the stack, the local temperature of the nearby part of the stack changes as follows: $$ T_{\mathrm{env}}(t)=T_{0}-T_{\mathrm{st}} \cos (\omega t) \tag{5} $$ Express $T_{\mathrm{st}}$ in terms of ...
[ "$\\frac{a \\tau}{\\ell \\sqrt{2}}$" ]
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1,315
B.2 Above which critical temperature difference $\tau_{\text {cr }}$ will the gas be conveying heat from the hot reservoir to the cold one? Express $\tau_{\mathrm{cr}}$ in terms of $T_{0}, \gamma, k$ and $\ell$.
[ "$\\tau_{\\mathrm{cr}}=k \\ell(\\gamma-1) T_{0}$" ]
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1,316
B.3 Obtain the general approximate expression for the heat flow $\frac{d Q}{d t}$ into a small parcel of gas as a linear function of its volume and pressure change rates. Express your answer in terms of the rate of volume change $\frac{d V}{d t}$, the rate of pressure change $\frac{d p}{d t}$, the unperturbed equilibri...
[ "$\\frac{1}{\\gamma-1} V_{0} \\frac{d p}{d t}+\\frac{\\gamma}{\\gamma-1} p_{0} \\frac{d V}{d t}$" ]
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1,317
B.4 In order to calculate work, we will consider a change to the volume of the moving parcel as a result of the thermal contact with the stack. Let us write the pressure and the volume of the parcel under the stack's influence in the form: $$ \begin{gathered} p=p_{0}+p_{a} \sin (\omega t)-p_{b} \cos (\omega t), \\ V=V...
[ "$V_{b}=\\frac{1}{\\gamma} p_{b} \\cdot \\frac{V_{0}}{p_{0}}$ , $V_{a}=(-\\frac{1}{\\gamma} p_{a}-\\frac{\\gamma-1}{\\gamma} \\frac{\\beta}{\\omega} \\frac{a}{\\ell \\sqrt{2}}(\\tau-\\tau_{\\mathrm{cr}})) \\cdot \\frac{V_{0}}{p_{0}}$" ]
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1,318
B.5 Obtain an approximate expression for the acoustic work per unit volume $w$ produced by the gas parcel over one cycle. Integrate over the volume of the stack to obtain the total work $W_{\text {tot }}$ generated by the gas over one cycle. Express $W_{\text {tot }}$ in terms of $\gamma, \tau, \tau_{\text {cr }}, \bet...
[ "$\\frac{\\pi}{2 \\omega}(\\gamma-1) \\beta(\\tau-\\tau_{\\mathrm{cr}}) k a^{2} S$" ]
[ 0.013916015625, 0.039794921875, 0.0162353515625, -0.0076904296875, -0.00946044921875, 0.004180908203125, -0.00762939453125, -0.005035400390625, 0.0107421875, -0.00732421875, -0.02099609375, 0.01300048828125, -0.007537841796875, -0.00592041015625, 0.0252685546875, -0.006927490234375, ...
1,319
B.6 Obtain an approximate expression for the heat $Q_{\text {tot }}$ transported from the left side of the plane $x=x_{0}$ to the right, over a cycle. Express your answer in terms of $\tau, \tau_{\text {cr }}, \beta, \omega, a, S, \ell$
[ "$\\frac{\\pi}{2 \\omega} \\beta(\\tau-\\tau_{\\mathrm{cr}}) \\frac{a^{2} S}{\\ell}$" ]
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B.7 Find the efficiency $\eta$ of the thermoacoustic engine. The efficiency is defined as the ratio of the generated acoustic work to the heat drawn from the hot reservoir. Express your answer in terms of the temperature difference $\tau$ between the hot and the cold reservoir, the critical temperature difference $\tau...
[ "$\\frac{\\tau_{\\mathrm{cr}}}{\\tau} \\cdot \\eta_{c}$" ]
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B. The liquids A and B were poured into a vessel in which the layers shown in Fig. 1.1 were formed. The surface of the liquid B has been covered with a thin layer of a non-volatile liquid $\mathrm{C}$, which is insoluble in the liquids $\mathrm{A}$ and $\mathrm{B}$ and vice versa, thereby preventing any free evaporatio...
[ "$67$ , $100$" ]
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1.1. Determine the distance from the center of mass $G$ of the lever to the rotation axis T. It is known that GT is horizontal when the bucket is empty.
[ "0.016" ]
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1,327
1.2. Water starts flowing out of the bucket when the angle between the lever and the horizontal axis reaches $\alpha_{1}$. The bucket is completely empty when this angle is $\alpha_{2}$. Determine $\alpha_{1}$ and $\alpha_{2}$.
[ "$\\alpha_{1}=20.6^{\\circ}$ , $\\alpha_{2}=30^{\\circ}$" ]
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1.3. Let $\mu(\alpha)$ be the total torque (relative to the axis $\mathrm{T}$ ) which comes from the weight of the lever and the water in the bucket. $\mu(\alpha)$ is zero when $\alpha=\beta$. Determine $\beta$ and the mass $m_{1}$ of water in the bucket at this instant.
[ "$m=0.61 $ , $\\beta=23.6$" ]
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1.2. Express the angle $\varphi$ between this intersection and the trajectory of the particle in terms of $n$ and $\beta$.
[ "$\\varphi=\\arcsin \\frac{1}{\\beta n}$" ]
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1,330
3.1. Calculate for each of the three particle types the minimal value $P_{\min }$ of the air pressure such that they emit Cherenkov light.
[ "$16, 4.6, 0.36$" ]
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1,331
3.2. Calculate the pressure $P_{\frac{1}{2}}$ such that the ring image of kaons has a radius equal to one half of that corresponding to pions. Calculate the values of $\theta_{\kappa}$ and $\theta_{\pi}$ in this case. Is it possible to observe the ring image of protons under this pressure?
[ "$P_{\\frac{1}{2}}=6 $ , $\\theta_{\\kappa}=1.6$ , $\\theta_{\\pi}=3.2$" ]
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1,332
4.1. Calculate $\frac{\Delta \theta_{\kappa}}{\Delta p}$ and $\frac{\Delta \theta_{\pi}}{\Delta p}$, the values taken by $\frac{\Delta \theta}{\Delta p}$ in the pions and kaons cases.
[ "$\\frac{\\Delta \\theta_{\\kappa}}{\\Delta p}=0.51 $ , $\\frac{\\Delta \\theta_{\\pi}}{\\Delta p}=0.02$" ]
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4.2. When the separation between the two ring images, $\theta_{\pi}-\theta_{\kappa}$, is greater than 10 times the half-width sum $\Delta \theta=\Delta \theta_{\kappa}+\Delta \theta_{\pi}$, that is $\theta_{\pi}-\theta_{\kappa}>10 \Delta \theta$, it is possible to distinguish well the two ring images. Calculate the max...
[ "$0.3 $" ]
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1,334
5.1. Find out the minimal kinetic energy $T_{\min }$ of a particle with a rest mass $M$ moving in water, such that it emits Cherenkov light. The index of refraction of water is $n=1.33$.
[ "$0.517 M c^{2}$" ]
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1,335
5.2. The radioactive source used by Cherenkov emits either $\alpha$ particles (i.e. helium nuclei) having a rest mass $M_{\alpha}=3.8 \mathrm{GeV} / c^{2}$ or $\beta$ particles (i.e. electrons) having a rest mass $M_{\mathrm{e}}=0.51 \mathrm{MeV} / c^{2}$. Calculate the numerical values of $T_{\min }$ for $\alpha$ part...
[ "1.96, 0.264" ]
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6.1. Consider a beam of pions with definite momentum of $10.0 \mathrm{GeV} / \mathrm{c}$ moving in air at pressure $6 \mathrm{~atm}$. Find out the angular difference $\delta \theta$ associated with the two ends of the visible range.
[ "$0.033$" ]
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1,337
6.2.1. Calculate the broadening due to dispersion (varying refraction index) and that due to achromaticity of the beam (varying momentum).
[ "$0.017^{\\circ}$" ]
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1,351
5.1. Derive the differential equation determining the CO pollutant concentration $C(t)$ as a function of time.
[ "$\\frac{d C}{d t}+\\frac{u}{W} C(t)=\\frac{M}{L W H}$" ]
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5.2. Write down the solution of that equation for $C(t)$.
[ "$C(t)=\\frac{M}{L H u}\\left[1-\\exp \\left(-\\frac{u t}{W}\\right)\\right]$" ]
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5.3. Calculate the numerical value of the concentration $C(t)$ at 8:00 a.m. Given $L=15 \mathrm{~km}, W=8 \mathrm{~km}, u=1 \mathrm{~m} / \mathrm{s}$.
[ "$2.3 $" ]
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B.5 From the figure, estimate $f_{\mathrm{GW}}(t)$ at $$ t_{\overline{\mathrm{AB}}}=\frac{t_{\mathrm{B}}+t_{\mathrm{A}}}{2} \quad \text { and } \quad t_{\overline{\mathrm{CD}}}=\frac{t_{\mathrm{D}}+t_{\mathrm{C}}}{2} \tag{13} $$ Assuming that (12) is valid all the way until the collision (which strictly speaking is n...
[ "$30$ ,$69$" ]
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1,369
B.1 Find an equation which relates the square of the $W^{+}$boson mass, $m_{\mathrm{W}}^{2}$, with the neutrino and anti-muon momentum components presented in the table above. Express your answer in terms of the neutrino and anti-muon transverse momentum, $$ \vec{p}_{\mathrm{T}}{ }^{(\nu)}=p_{x}{ }^{(\nu)} \hat{\imath...
[ "$m_{W}^{2}=\\frac{1}{c^{2}}(2 p^{(\\mu)} \\sqrt{(p_{\\mathrm{T}}^{(\\nu)})^{2}+(p_{z}^{(\\nu)})^{2}}-2 \\vec{p}_{\\mathrm{T}}^{(\\mu)} \\cdot \\vec{p}_{\\mathrm{T}}^{(\\nu)}-2 p_{z}^{(\\mu)} p_{z}^{(\\nu)})$" ]
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1,370
B.2 Assuming a $W^{+}$boson mass of $m_{\mathrm{W}}=80.4 \mathrm{GeV} / c^{2}$ calculate the two possible solutions for the neutrino momentum along the $z$-axis, $p_{z}{ }^{(\nu)}$. Express your answer in $\mathrm{GeV} / \mathrm{c}$.
[ "$p_{z}^{(\\nu)}=74.0 $ , $ p_{z}^{(\\nu)}=-188.3$" ]
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1,371
B.3 Calculate the top quark mass for each one of the two previous solutions. Express your answer in $\mathrm{GeV} / c^{2}$. IIf you did not obtain the two solutions in B.2, use $$ \left.p_{z}^{(\nu)}=70 \mathrm{GeV} / c \text { and } p_{z}{ }^{(\nu)}=-180 \mathrm{GeV} / c .\right] $$
[ "$m_{\\mathrm{t}}=169.3 $ and $m_{\\mathrm{t}}=311.2$" ]
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1,372
B.4 According to the top quark mass distribution, which one of the two previous solutions is more likely to be the right one? Estimate the probability for the most likely solution.
[ "$m_{\\mathrm{t}}=169.3$" ]
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1,373
B.5 Calculate the distance traveled by the top quark before decaying, using the most likely solution. Assume the top quark has a mean lifetime at rest of $5 \times 10^{-25}$ s.
[ "$d=2 \\times 10^{-16}$" ]
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1,374
A.1 Obtain an expression for the volumetric flow rate, $Q_{i}$, in a vessel at any level $i$, as a function of the total number of levels $N$, of the viscosity $\eta$, of the radius $r_{0}$ and length $\ell_{0}$ of the first vessel, and of the difference $\Delta P=P_{0}-P_{\text {cap }}$ between the pressure at the art...
[ "$Q_{i}=\\Delta P \\frac{\\pi r_{0}^{4}}{2^{i+3} N \\ell_{0} \\eta}$" ]
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1,375
A.2 Calculate the numerical value of the volumetric flow rate $Q_{0}$ of the arteriole at level 0 , if its radius is $6.0 \times 10^{-5} \mathrm{~m}$ and its length is $2.0 \times 10^{-3} \mathrm{~m}$. Consider that the pressure at the arteriole inlet is $55 \mathrm{mmHg}$ and the vessel network has $N=6$ levels linkin...
[ "$1.5$" ]
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1,377
A.4 For the vessel network in A.2 estimate the maximum arteriole wall thickness $h$ so that the condition established in A.3 is satisfied (consider that $h$ is level independent).
[ "$h=8 \\times 10^{-5}$" ]
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1,378
B.1 The mass of normal cells is not altered while the tumour is growing. Obtain the ratio between the tumour volume and the total tissue volume, $v=V_{\mathrm{T}} / V$, as a function of the ratio between the tumour mass $\left(M_{\mathrm{T}}\right)$ and the normal tissue mass $\left(M_{\mathrm{N}}\right), \mu=M_{\mathr...
[ "$v=\\frac{1+\\mu-\\sqrt{(1+\\mu)^{2}-4 \\mu(1-\\kappa)}}{2(1-\\kappa)}$" ]
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1,379
B.2 Obtain, for the stationary state, the temperature at the centre of the tumour as a function of $\mathcal{P}, k$, the human body temperature and the tumour radius, $R_{\mathrm{T}}$.
[ "$310.15+\\frac{\\mathcal{P} R_{\\uparrow}^{2}}{2 k}$" ]
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1,380
B.3 Obtain the minimum power per unit volume, $\mathcal{P}_{\min }$, needed to heat up all tumour cells in a tumour with $5.0 \mathrm{~cm}$ radius to a temperature larger than $43.0^{\circ} \mathrm{C}$. Take the thermal conductivity of the tissue to be equal to $k=0.60 \mathrm{~W} \mathrm{~K}^{-1} \mathrm{~m}^{-1}$.
[ "$\\mathcal{P}_{\\min }=4.3$" ]
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1,381
B.4 In the linear regime (i.e. consider that $p-P_{\text {cap }}$ is very small), express the relative drop in the flow rate, $\frac{\delta Q_{N-1}}{Q_{N-1}}$, in these thinnest vessels, as a function of the tumour volume ratio $v=V_{\mathrm{T}} / V$ and $K_{N}, N, p_{\mathrm{c}}, \delta r_{\mathrm{c}}, r_{N-1}, P_{\te...
[ "$-\\frac{2}{N} \\frac{K_{\\mathrm{N}} v-(1-v) P_{\\text {cap }}}{(1-v)\\left(p_{\\mathrm{c}}-P_{\\mathrm{cap}}\\right)} \\frac{\\delta r_{\\mathrm{c}}}{r_{N-1}}$" ]
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i. What is the horizontal velocity of the muons in Bob's reference frame?
[ "$v=\\beta c$" ]
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1,388
ii. What is the vertical velocity of the muons in Bob's reference frame?
[ "$c \\sqrt{1-\\beta^{2}}$" ]
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1,389
iii. How long does it take the muons to reach the ground in Bob's reference frame?
[ "$\\Delta t=\\frac{h}{c \\sqrt{1-\\beta^{2}}}$" ]
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1,390
b. In Bob's reference frame, how much time is there between when he sees Alice first fire the beam, and when he sees the beam first hit the ground? (Hint: remember to account for the travel time of light to Bob's eyes.)
[ "$\\frac{h}{c} \\frac{1+\\beta \\cos \\theta}{\\sqrt{1-\\beta^{2}}}$" ]
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1,391
c. In this part, suppose that $\beta=1 / 2$. Does there exist a value of $\theta$ so that the time it takes the muons to hit the ground in Alice's frame is equal to the time taken according to Bob's eyes, in Bob's frame? If so, find the value of $\theta$ in degrees. If not, briefly explain why not.
[ "$105.5$" ]
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d. Suppose Alice is carrying a radio transmitter set to frequency $f$. To what frequency would Bob have to set his radio receiver in order to receive Alice's transmission?
[ "$f^{\\prime}=\\frac{\\sqrt{1-\\beta^{2}}}{1+\\beta \\cos \\theta} f$" ]
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a. The ant walks an angular displacement $\theta$ along the edge of the disk. Then it walks radially inward by a distance $h \ll R$, tangentially through an angular displacement $-\theta$, then back to its starting point on the disk. Assume the ant walks with constant speed $v$. <image_1> Through what net angle d...
[ "$\\frac{2 m \\theta}{M}\\left(\\left(1-\\frac{h}{R}\\right)^{2}-1\\right)$", "$-\\frac{4 m}{M} \\frac{h \\theta}{R}$" ]
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1,399
b. Now suppose the ant walks with speed $v$ along a circle of radius $r$, tangent to its starting point. <image_1> Through what net angle does the disk rotate?
[ "$\\frac{4 m}{M} \\frac{\\pi r^{2}}{R^{2}}$" ]
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1,405
ii. What is the total radial force (i.e., normal to the surface of the hemisphere) on the mass at $t=0$ ? Express your answer in terms of $m, v, R, g$, and $\theta_{0}$.
[ "$F_{r}=m g \\cos \\theta_{0}-\\frac{m v^{2}}{R}$" ]
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c. A cylinder of radius $r \ll R \theta_{0}$ is placed on top of the sphere. Suppose the mass is launched at an angle $\alpha$ away from the direction of the spring's displacement with kinetic energy $K$, as shown. What is the maximum angle $\alpha_{\max }$ at which the mass can be launched such that it can still hi...
[ "$\\alpha=\\frac{r}{R \\sin \\theta_{0}} \\sqrt{1-\\frac{m g R\\left(1-\\cos \\theta_{0}\\right)-(1 / 2) m g R \\theta_{0} \\sin \\theta_{0}}{K}}$" ]
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a. Find an expression of the final angular velocity $\omega$ of the cylinder in terms of the symbols given and other constants.
[ "$\\omega(T)=\\frac{\\frac{\\lambda a^{2}}{2 I} B_{0}}{1+\\mu_{0} \\frac{\\lambda^{2} a}{8 \\pi I}}$" ]
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1,408
b. You may be surprised that the expression you find above is not zero! However, the electric and magnetic fields can have angular momentum. Analogous to the "regular" angular momentum definition, the EM field angular momentum per unit volume at a displacement $\mathbf{r}$ from the axis of rotation is: $$ \mathcal{L...
[ "$\\alpha=\\epsilon_{0}$" ]
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1,409
a. For John, what are the approximate equations describing the motion of the pendulum in the $x-y$ plane? You may assume that the amplitude of the oscillations is small. We define the coordinates of the pendulum at rest as $(0,0)$.
[ "$a_{x}+\\omega^{2} x=0 , a_{y}+\\omega^{2} y=0$" ]
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1,410
b. What will the coordinates $x, y$ in Jonh's frame be at a later time $t$ ?
[ "$x(t)=A \\cos (\\omega t), y(t)=\\frac{V}{\\omega} \\sin (\\omega t)$" ]
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1,411
c. Ella, an observer resides at the North Pole, is also looking at the pendulum. What are the coordinates, $\tilde{x}(t)$ and $\tilde{y}(t)$, as observed by Ella? Assume that at time $t=0$, the coordinate systems of John's and Ella's overlap.
[ "$\\tilde{x}=A \\cos (\\omega t) \\cos (\\Omega t)+\\frac{V}{\\omega} \\sin (\\omega t) \\sin (\\Omega t)$,$\\tilde{y}=-A \\cos (\\omega t) \\sin (\\Omega t)+\\frac{V}{\\omega} \\sin (\\omega t) \\cos (\\Omega t)$" ]
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1,412
d. What is the speed of the pendulum bob observed by Ella at $t=0$ ?
[ "$\\tilde{v}_{x}=0$ , $\\tilde{v}_{y}=V-\\Omega A$" ]
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1,414
e. Find the initial conditions for $A, V$, such that as measured in Ella's frame: ii. it has a "spike" at the points of maximal amplitude (see figure below) instead of a "rounded" trajectory.
[ "$V=\\Omega A$" ]
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1,415
f. How long would it take for the plane of oscillation of Foucault's pendulum to return to its initial value in Paris, which has a latitude of about $49^{\circ}$.
[ "16" ]
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c. The water-air interface has some surface tension, $\sigma$. The effect of surface tension is to change the pressure in the stream according to the Young-Laplace equation, $$ \Delta P=\sigma\left(\frac{1}{r}+\frac{1}{R}\right) $$ where $\Delta P$ is the difference in pressure between the stream and the atmosphere ...
[ "$\\frac{1}{2} \\rho v_{0}^{2} \\frac{r_{0}^{4}}{r^{4}}+\\rho g y=\\sigma\\left(\\frac{1}{r_{0}}-\\frac{1}{r}\\right)$" ]
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a. On the right wall, an interference pattern emerges. What is the distance $y$ between the bottom corner and the closest bright fringe above it? Hint: you may assume $\lambda \ll y \ll L$ as well.
[ "$y=\\frac{\\lambda L}{8 d}$" ]
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c. Now suppose we place a transparent hemispherical shell of thickness $s$ and index of refraction $n$ over the source such that all light from the source that directly strikes the right wall passes through the shell, and all light from the source that strikes the mirror first does not pass through the shell. hemisphe...
[ "$y=\\frac{L}{4 d}\\left((n-1) s-\\lambda\\left\\lfloor\\frac{(n-1) s}{\\lambda}-\\frac{1}{2}\\right\\rfloor-\\frac{\\lambda}{2}\\right)$" ]
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1,426
d. Now, suppose the hemispherical shell is removed, and we instead observe the interference pattern on the top wall. To the nearest integer, what is the total number of fringes that appear on the top wall? You may assume that $d \ll L$.
[ "$2 \\cdot \\frac{2 d}{\\lambda}\\left(1-\\frac{2}{\\sqrt{5}}\\right)$" ]
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1,433
Determine the range of values for $F$ so that the the top wedge does not slip on the bottom wedge. Express your answer(s) in terms of any or all of $m, g, \theta$, and $\mu$.
[ "$[3 m g \\frac{\\left(1+\\mu^{2}\\right) \\tan \\theta}{1+\\mu \\tan \\theta}, 3 m g \\frac{2 \\mu+\\left(1-\\mu^{2}\\right) \\tan \\theta}{1-\\mu \\tan \\theta}]$" ]
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1,439
c. Derive an expression for the magnetic field $\overrightarrow{\mathbf{B}}_{i}$ from the iron a distance $d \gg a$ from the center of the ball. Note that there will be a component directed radially away from the ball and a component directed tangent to a circle of radius $d$ around the ball, so using polar coordinates...
[ "$B_{r}=2 B_{e} \\frac{m K_{i}}{d^{3}} \\cos \\phi$" ]
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d. If placed directly to the right and left of the ship compass, the iron balls can be located at a distance $d$ to cancel out the error in the magnetic heading for any angle(s) where $\delta \theta$ is largest. Assuming that this is done, find the resulting expression for the combined deviation $\delta \theta$ due to ...
[ "0" ]
[ 0.0174560546875, 0.006103515625, 0.00130462646484375, 0.01416015625, -0.0283203125, 0.0294189453125, 0.0084228515625, 0.004669189453125, -0.037841796875, 0.0216064453125, -0.049072265625, 0.00927734375, -0.002960205078125, 0.005340576171875, 0.01226806640625, 0.01300048828125, -0.016...
1,444
a. Suppose we give the domino a sharp, horizontal impulsive push with total momentum $p$. i. At what height $H$ above the table is the impulse $p$ required to topple the domino smallest? ii. What is the minimum value of $p$ to topple the domino?
[ "$H=h$ , $p_{\\min }=\\frac{1}{\\sqrt{3}} \\frac{m}{h} \\sqrt{g\\left(\\sqrt{t^{2}+h^{2}}-h\\right)\\left(h^{2}+t^{2}\\right)}$" ]
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1,445
b. Next, imagine a long row of dominoes with equal spacing $l$ between the nearest sides of any pair of adjacent dominos, as shown above. When a domino topples, it collides with the next domino in the row. Imagine this collision to be completely inelastic. What fraction of the total kinetic energy is lost in the collis...
[ "50" ]
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1,446
c. After the collision, the dominoes rotate in such a way so that they always remain in contact. Assume that there is no friction between the dominoes and the first domino was given the smallest possible push such that it toppled. What is the minimum $l$ such that the second domino will topple? You may work to lowes...
[ "$l=\\frac{3}{2} t$" ]
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1,448
a. Assuming the dome is charged to $500 \mathrm{kV}$, determine the strength of the electric field at the surface of the dome.
[ "$10^{6}$" ]
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1,449
b. Assuming the proton beam is off, determine the time constant for the accelerating dome (the time it takes for the charge on the dome to decrease to $1 / e \approx 1 / 3$ of the initial value.
[ "0.556" ]
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1,450
c. Assuming the $25 \mu \mathrm{A}$ proton beam is on, determine the surface charge density that must be sprayed onto the charging belt in order to maintain a steady charge of $500 \mathrm{kV}$ on the dome.
[ "$37.5$" ]
[ 0.0260009765625, -0.0186767578125, -0.003997802734375, 0.005035400390625, -0.006866455078125, 0.0128173828125, 0.00689697265625, -0.0118408203125, -0.0194091796875, 0.008544921875, -0.033935546875, -0.005859375, -0.0291748046875, -0.034912109375, -0.00421142578125, 0.00080108642578125,...
1,451
d. The proton beam enters the electromagnet and is deflected by an angle $\theta=10^{\circ}$. Determine the magnetic field strength. <image_1>
[ "0.0894" ]
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1,452
e. The electromagnet is composed of layers of spiral wound copper pipe; the pipe has inner diameter $d_{i}=0.40 \mathrm{~cm}$ and outer diameter $d_{o}=0.50 \mathrm{~cm}$. The copper pipe is wound into this flat spiral that has an inner diameter $D_{i}=20 \mathrm{~cm}$ and outer diameter $D_{o}=50 \mathrm{~cm}$. Assumi...
[ "33" ]
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1,453
f. Hollow pipe is used instead of solid conductors in order to allow for cooling of the magnet. If the resistivity of copper is $\rho=1.7 \times 10^{-8} \Omega \cdot \mathrm{m}$, determine the electrical resistance of one spiral.
[ "0.079" ]
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1,454
g. There are $N=24$ coils stacked on top of each other. Tap water with an initial temperature of $T_{c}=18^{\circ} \mathrm{C}$ enters the spiral through the copper pipe to keep it from over heating; the water exits at a temperature of $T_{h}=31^{\circ} \mathrm{C}$. The copper pipe carries a direct $45 \mathrm{Amp}$ cur...
[ "0.07" ]
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1,455
h. The protons are fired at a target consisting of Fluorine atoms $(Z=9)$. What is the distance of closest approach to the center of the Fluorine nuclei for the protons? You can assume that the Fluorine does not move.
[ "$2.59 \\times 10^{-14}$" ]
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1,456
b. Derive an expression for the total distance block $A$ has moved from its original position right after its $n^{\text {th }}$ collision, in terms of $\ell$ and $n$.
[ "$d_{A}(n)=\\ell \\left(2 n^{2}-2 n+1\\right)$" ]
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1,457
At time $t=1 \mathrm{~s}$, how far has block $A$ moved from its original position?
[ "$25$" ]
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1,467
a. Suppose that, after some amount of work is done by the ion pumps, the charges on the outer and inner surfaces are $Q$ and $-Q$, respectively. What is the thickness $d$ of the membrane?
[ "$d=d_{0}-\\frac{Q^{2}}{2 \\epsilon A k}$" ]
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1,468
b. Derive an expression for the voltage difference $V$ between the outer and inner surfaces of the membrane in terms of $Q$ and the other parameters given.
[ "$V=\\frac{Q}{\\epsilon A}\\left(d_{0}-\\frac{Q^{2}}{2 \\epsilon A k}\\right)$" ]
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1,469
c. Suppose that the ion pumps are first turned on in the uncharged state, and the membrane is charged very slowly (quasistatically). The pumps will only turn off when the voltage difference across the membrane becomes larger than a particular value $V_{\text {th }}$. How large must the spring constant $k$ be so that th...
[ "$\\left(\\frac{3}{2}\\right)^{3} \\frac{V_{\\mathrm{th}}^{2} \\epsilon A}{d_{0}^{3}}$" ]
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1,470
i. $k$ is infinitesimally larger than the value derived in part (c).
[ "$W=\\frac{5}{18} k d_{0}^{2}$" ]
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1,471
ii. $k$ is infinitesimally smaller than the value derived in part (c).
[ "$W=\\frac{1}{2} k d_{0}^{2}$" ]
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1,472
a. Derive an expression for the force in the rod when it is horizontal, as shown at left above.
[ "$T=M g$" ]
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1,473
b. Derive an expression for the force in the rod when the ball is directly below the bead, as shown at right above.
[ "$T=9 Mg$" ]
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1,474
c. Let $\theta$ be the angle the rod makes with the vertical, so that the $\operatorname{rod} \operatorname{begins}$ at $\theta=0$. Find the angular velocity $\omega=d \theta / d t$ as a function of $\theta$.
[ "$\\omega^{2}=\\frac{4 g}{R} \\frac{1-\\cos \\theta}{1+\\sin ^{2} \\theta}$" ]
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1,479
a. At $t=0$ the pressure inside the chamber is $P_{a}$. Find an equation for the pressure at a later time $t$.
[ "$P(t)=P_{a} e^{-R t / V}$" ]
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1,480
b. Find an expression for the temperature of the gas as it is emitted from the pump cylinder into the atmosphere. Your answer may depend on time.
[ "$T_{a} e^{2 R t / 5 V}$" ]
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1,481
c. Find an expression for the minimum possible pressure in the chamber, $P_{\min }$.
[ "$P_{a}\\left(1+\\frac{\\Delta V}{V_{0}}\\right)^{-\\gamma}$" ]
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1,495
a. Find the volume of the lower chamber $V_{0}$ when the gas starts to leak between the chambers.
[ "$V(1+x)^{-3 / 5}$" ]
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1,496
b. Find the temperature $T_{1}$ in the upper chamber when the piston touches the diaphragm.
[ "$\\frac{T}{2}\\left(\\left(1+(1+x)^{3 / 5}\\right)^{5 / 3}-x\\right)$" ]
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1,497
c. Find the temperature $T_{2}$ in the lower chamber immediately before the piston touches the diaphragm.
[ "$T\\left(1+(1+x)^{3 / 5}\\right)^{2 / 3}$" ]
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1,498
T2: Thread around a cylinder One end of a thread is tied into a loop of length $L>2 \pi R$, and a cylinder of radius $R$ is put through the loop. The coefficient of friction between the thread and the cylinder is $\mu$. The free end of the thread is being pulled parallel to the axis of the cylinder (as shown by arrow ...
[ "$2 \\pi R$" ]
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1,499
a. Find the force $F_{1}$ acting on the solenoid when its head $O_{1}$ is positioned in the loop centre $O$. What is the force $F_{2}$ acting on the solenoid when its tail $O_{2}$ is located in the centre of the loop?
[ "$F_{1}=\\frac{\\mu_{0} N I A \\mathcal{E}}{2 \\ell R r}$, $F_2=-F_1$" ]
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1,500
## T2: Mechanical accelerator A massless thread makes $N$ turns around statically fixed cylinder, as shown in the figure. Initially, the free (unwound) ends of the thread are parallel to the axis $X$. Then, a heavy point-like object $P$ is attached to one end of the thread while the other end is pulled with a constant...
[ "$v_{\\max }=u(2 \\pi N+1)$" ]
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1,501
## T3: Cat eyes You may have noticed that in darkness, when a cat is within the light beam of a headlamp, its eyes appear very bright, see the photo below (left). This phenomenon can be modelled by a lens setup, see the photo on right, and the diagram beneath the photos. <image_1> The photo on right was taken by a di...
[ "$80$" ]
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1,503
(a) Express $V_{1}$ and $V_{2}$ in terms of the other parameters defined above.
[ "$V_{1}=\\sqrt{R_{1} \\alpha\\left(T_{c}-T_{0}\\right)}$ , $V_{2}=\\sqrt{R_{2} \\alpha\\left(T_{c}-T_{0}\\right)}$" ]
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1,504
(b) Assuming that $V_{1}<V<V_{2}$, sketch qualitatively how the temperature of the resistor $T$ depends on time $t$, and find the ratio $\left(T_{\max }-T_{0}\right) /\left(T_{\min }-T_{0}\right)$, where $T_{\max }$ and $T_{\min }$ denote the maximal and minimal values of $T$, respectively.
[ "$\\frac{R_{2}^{2}}{R_{1}^{2}}$" ]
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1,505
(c) Find the period of oscillations if $V=\sqrt{V_{1} V_{2}}$ and $R_{2}=16 R_{1}$.
[ "$\\frac{L}{R_{1}}(\\ln (\\frac{7}{4})+\\frac{1}{16} \\ln (7))$", "$0.68 \\frac{L}{R_{1}}$" ]
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1,508
(c) Given $\vec{v}_{0}=0$, find the minimal $\omega_{0}=\omega_{\text {min }}$ necessary for the dipole to reverse its orientation during the motion.
[ "$2 \\frac{q B}{m}$" ]
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