Physics Problems Q&A - Recent questions in Physics Problems
http://physics.qandaexchange.com/?qa=questions/physics-problems
Powered by Question2AnswerDiffusion with reflecting boundary
http://physics.qandaexchange.com/?qa=3254/diffusion-with-reflecting-boundary
<p><img src="https://cdn.pbrd.co/images/HNaMhdt.png" alt=""></p>
<p>What I have tried:</p>
<p><img src="https://cdn.pbrd.co/images/HNaMSPP.jpg" alt=""></p>
<p><a rel="nofollow" href="https://imgur.com/a/ESWy1k5">https://imgur.com/a/ESWy1k5</a> (sorry but I did not manage to use imgur as you instructed)</p>
<p>Note that my question is that how is it possible that I get $t = \infty$. It does not make sense for me (physically speaking). Where did I get wrong?</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3254/diffusion-with-reflecting-boundaryWed, 14 Nov 2018 20:07:35 +0000Angle made by the plane of hemisphere with inclined plane
http://physics.qandaexchange.com/?qa=3241/angle-made-by-the-plane-of-hemisphere-with-inclined-plane
<blockquote><p>A uniform thin hemispherical shell is kept static on an inclined plane of angle<br>
$\theta = 30$ as shown. If the surface of the inclined plane is sufficiently rough to prevent<br>
sliding then what is the angle $\alpha$ made by the plane of hemisphere with inclined plane .<br>
<img src="https://cdn.pbrd.co/images/HLyzG1g.jpg" alt=""></p>
</blockquote>
<p><img src="https://cdn.pbrd.co/images/HLyzXc6.jpg" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3241/angle-made-by-the-plane-of-hemisphere-with-inclined-planeSun, 04 Nov 2018 04:53:51 +0000Radial distribution of particle separation in a liquid at small distances
http://physics.qandaexchange.com/?qa=3233/radial-distribution-particle-separation-liquid-distances
<blockquote><p>Draw schematically the radial distribution function $g(r)$ for a Lennard-Jones fluid at low and high particle densities. Discuss in both cases the behavior at small $r$. At high densities $g(r)$ has some damped oscillations. Explain their origin and what would happen at long distances.</p>
</blockquote>
<p><strong>What I know:</strong></p>
<p>The radial distribution function $g(r)$ describes how density varies as a function of distance from a reference particle (<em>Wikipedia</em>).</p>
<p>Specifically in Lennard-Jones case, we have (Pasteboard still does not work for me):</p>
<p><a rel="nofollow" href="https://imgur.com/a/pvXgax4">https://imgur.com/a/pvXgax4</a></p>
<p>Here we can observe that if repulsion outweighs attraction, the curve will die out on the x axis. On the contrary, if it is the other way around, the curve will grow exponentially. For the sake of clarity, I would remark y = 0 as the Ideal Gas behaviour. </p>
<p>I have read that $g(r)$ vanishes at short distances, because the probability of finding two particles close to each other vanishes due to the repulsive part of the potential. At high densities $g(r)$ can show some damped oscillations: <a rel="nofollow" href="https://imgur.com/a/i32D39I">https://imgur.com/a/i32D39I</a>. </p>
<p>These oscillations express the preference of the particles to be found at specific distances from a reference particle at the origin. For instance in the LJ case, a first layer of particles will be localized closed to the minimum of LJ's potential, which is the origin of the first peak in $g(r)$.</p>
<p>This layer prevents other particles from getting close to it, which is what causes the first<br>
minimum in $g(r)$.</p>
<p><strong>What I do not know.</strong></p>
<p>1)The explanation: 'These oscillations express the preference of the particles to be found at specific distances from a reference particle at the origin'. <strong>How can particles have 'preference' to be at different distances? I think this is not a good physical explanation. I have been thinking about this and I would say that as there is a high density of particles at small distances, the repulsion force is exerted on these, which triggers such oscillations. Finally there is a point where they are so close to each other that the repulsion force provides them with an initial kinetic energy which will be equal to a final electrostatic potential energy of zero (selecting our zero of electrostatic potential energy at y=0) , and that is the moment when the curve dies out on the x axis. Do you agree with this explanation?</strong></p>
<p>2)Are we dealing with underdamped oscillations at high densities? Could you give an insight into these kind of oscillations (if what I have just said is not enough or incorrect)?</p>
<p>3)When we are dealing with long distances, would $g(r)$ tend to be one? What would we observe, the Ideal Gas behaviour? Why? Is it because there is no interaction between the particles?</p>
<p>As always, I am interested in explaining such phenomena from a physical point of view. </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3233/radial-distribution-particle-separation-liquid-distancesThu, 01 Nov 2018 16:00:05 +0000Liquid in a capacitor as dielectric (IE Irodov 3.144)
http://physics.qandaexchange.com/?qa=3225/liquid-in-a-capacitor-as-dielectric-ie-irodov-3-144
<blockquote><p>A parallel plate capacitor is located horizontally so that one of its plates is submerged into the liquid while the other is over its surface. The permittivity of the liquid is equal to $\epsilon$, its density is equal to $\rho$. To what height will the level of the liquid in the capacitor rise after its plates get a charge of surface charge density $\sigma$?</p>
</blockquote>
<p>This is a type of controversial problem, as it contains a different answer by the different author, two different answers provided for this are </p>
<p> $$1) \ h=\dfrac{(\epsilon-1)\sigma^2}{2\epsilon_o\epsilon\rho g}$$ $$2) \ h=\dfrac{(\epsilon^2-1)\sigma^2}{2\epsilon_o\epsilon^2\rho g}$$</p>
<p><a rel="nofollow" href="https://ibb.co/fYmek0">Solution for 1</a> and <a rel="nofollow" href="https://youtu.be/oyfo-TJEG28?t=31">solution for 2</a></p>
<p>Please help!</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3225/liquid-in-a-capacitor-as-dielectric-ie-irodov-3-144Tue, 30 Oct 2018 01:37:42 +0000Induced charge dentisity at boundry surface.
http://physics.qandaexchange.com/?qa=3216/induced-charge-dentisity-at-boundry-surface
<blockquote><p>The space between the plates of a parallel plates capacitor is filled consecutively with two dielectrics layers $1$ and $2$ having the thickness $d_1$ and $d_2$ respectively and permittivities $\epsilon_1$ and $\epsilon_2$. The area of each plate is equal to $S$ find:<br>
1. The capacitance of the capacitor <br>
2. The density $\sigma'$ of the bound charges on the boundary plane if the voltage across the capacitor equals $V$ and the electric field is directed from layer $1$ to layer $2$.</p>
</blockquote>
<p>I managed to find $C_{eq}$ easily, for the second part I don't understand that word " <em>boundary plane</em>", I only know $\sigma'=\sigma\bigg(1-\dfrac{1}{\epsilon}\bigg)$ holds when there would have been just one plate, how to work on two such consecutive plates. Please help.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3216/induced-charge-dentisity-at-boundry-surfaceSun, 28 Oct 2018 05:02:22 +0000Diffusion with an even number of reflecting boundaries
http://physics.qandaexchange.com/?qa=3211/diffusion-with-an-even-number-of-reflecting-boundaries
<p><img src="https://cdn.pbrd.co/images/HKoOk3T.png" alt=""></p>
<p>a)<br>
<img src="https://cdn.pbrd.co/images/HKoQ6sP.jpg" alt=""></p>
<p>This is the behaviour I expect from both concentrations at t= 0. Based on the theory of diffusion I would say that as t goes on, the Gaussian distributions will spread out and after a large number of collisions both concentrations will end up spread out more or less evenly throughout the whole volume. Please let me know if this reasoning is wrong.</p>
<p>A doubt came to my mind as I imagine the same scenario if we had absorbing boundaries instead of reflecting ones. Therefore, does really matter the kind of boundary we are dealing with when we are trying to predict how the two concentrations (in this case) are going to spread out as time flows?</p>
<p>EDITED: </p>
<p>My initial idea about the identical scenario regarding either absorbing boundaries or reflecting ones was wrong. That is because after a long time in presence of absorbing boundaries, the concentration will end up being zero. </p>
<p>Related question: <a rel="nofollow" href="http://physics.qandaexchange.com/?qa=3113/solving-the-diffusion-equation-with-an-absorbing-boundary">http://physics.qandaexchange.com/?qa=3113/solving-the-diffusion-equation-with-an-absorbing-boundary</a></p>
<p>b)<br>
<img src="https://cdn.pbrd.co/images/HKoXG2o.jpg" alt=""><br>
Based on what I have said before, I would expect the limiting case to be that both concentrations were equal to each other.</p>
<p>EDITED: </p>
<p>Actually I was wrong as there is only one distribution of concentration, which tends to be uniform after a long time (as explained at a)).</p>
<p><strong>I have been looking for a deeper explanation and I came up with the idea that at the boundaries the current is zero:</strong></p>
<p>$$j =-D\frac{\partial c(+-a/2,t)}{\partial x} = 0$$</p>
<p><strong>Which means that c(x,t) has vanishing derivatives. How could this fact help to explain the limiting behaviour of the concentration after a long time?</strong></p>
<p>c)<br>
This household potential:<br>
<img src="https://cdn.pbrd.co/images/HKp12yu.jpg" alt=""></p>
<p>As we are dealing with the distribution of particles in thermal equilibrium, the first thing that came to mind when I read potential was the Boltzmann equation:</p>
<p>$$n = ke^{-V(x)/KT}$$</p>
<p>EDITED: </p>
<p><strong>I am coming up with the Boltzmann distribution because it can tell us the distribution of molecules. If the potential energy is known as a function of distance, then the proportion of them at different distances is given by this law. And we do know the potential, which is the square-well. So in the equilibrium concentration I should be able to check whether with the square-well potential we get:<br>
- When $|x| \le \frac{a}{2}$ we get a constant (the value of the concentration in equilibrium between the boundaries).<br>
- When $|x| > \frac{a}{2}$ we get zero (outside the barriers).<br>
But how can I check this?</strong></p>
<p>d) <br>
<strong>I came across an explanation. It is not detailed and I barely understand it. Please explain it if you see that it makes sense:</strong> (I do not know why pasteboard did not work. I had to use Imgur):</p>
<p><a rel="nofollow" href="https://imgur.com/a/8TLgeRe">https://imgur.com/a/8TLgeRe</a></p>
<p><a rel="nofollow" href="https://imgur.com/a/unkkyyQ">https://imgur.com/a/unkkyyQ</a> </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3211/diffusion-with-an-even-number-of-reflecting-boundariesSat, 27 Oct 2018 15:15:35 +0000Charges induced on conducting plates.
http://physics.qandaexchange.com/?qa=3210/charges-induced-on-conducting-plates
<blockquote><p>Two infinite conducting plates $1$ and $2$ are separated by a distance $l$. A point charge $q$ is located between the plates at a distance $x$ from the plate $1$. Find the charges induced on each plate.</p>
</blockquote>
<p>Working on author's instruction, assuming $q$ to be spread uniformly on the plane through $q$ and parallel to plates, makes it easier to calculate electric field strength. Then to bring these distance into the picture we should work on potentials (as $E$ is known). But I don't know about potentials, should I take the potential difference between conductor to be zero, please explain everything in greater details.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3210/charges-induced-on-conducting-platesSat, 27 Oct 2018 05:01:02 +0000Relation between flux through lateral surface of a cylinder and flat parts.
http://physics.qandaexchange.com/?qa=3192/relation-between-flux-through-lateral-surface-cylinder-parts
<p>On solving IE Irodov 3.21, </p>
<blockquote><p>A ball of radius $R$ is uniformly charged with the volume density $\rho$. Find the flux of the electric field strength vector across the ball's section formed by the plane located at a distance $r_{0} < R$ from the center of the ball.</p>
</blockquote>
<p>From calculus obtained result is $$\boxed{\phi_1=\dfrac{\pi\rho r_o(R^2-r_o^2)}{3\epsilon_o}}$$</p>
<p>But if consider that flat surface as a flat surface of a cylinder then radius and height of such a cylinder will be $\sqrt{R^2-r_o^2}$ and $2r_o$ respectively, then from Gaus' Law flux through <strong>whole of cylinder</strong> is given by $$\boxed{\phi_2=\dfrac{\pi\rho (2r_o)(R^2-r_o^2)}{\epsilon_o}}$$</p>
<p>This is very similar to flux $\phi_1$ obtained for flat surface. More precisely $$\phi_1=\dfrac{\phi_2}{6}$$</p>
<p>So can we prove that flux through flat part of the cylinder is one-third of the total, and that through curved part is two-thirds of the total, in general?</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3192/relation-between-flux-through-lateral-surface-cylinder-partsMon, 22 Oct 2018 12:21:17 +0000Oscillations of a rotating mass on a spring
http://physics.qandaexchange.com/?qa=3191/oscillations-of-a-rotating-mass-on-a-spring
<blockquote><p>A spring of constant $k$ is wrapped around a long rod. One end of the rod and spring are attached to a motor that rotates the rod with constant angular velocity $\Omega$. The other end of the spring is attached to a mass $m$. This mass slides along the rod as the spring expands and contracts. There is no friction or gravity in the problem. The equilibrium length of the spring is $r_{0}.$</p>
</blockquote>
<p><img src="http://i68.tinypic.com/2crm0pc.png" alt=""></p>
<blockquote><p>What is the oscillation frequency of the mass for small values of $\Omega$? What happens when $\Omega$ is large? Is the motion still simple harmonic motion?</p>
</blockquote>
<p>Since there is no gravity, the centrifugal force must equal the spring force so</p>
<p>$$m\Omega^{2}r=k(r-r_{0})$$</p>
<p>which leads to a constant $r$ since $\Omega$ is constant. So I am missing something basic here.</p>
<p>Maybe I should use energy conservation?</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3191/oscillations-of-a-rotating-mass-on-a-springSun, 21 Oct 2018 20:39:57 +0000Deducing the mass of the Sirius system - Feynman exercises 3.17
http://physics.qandaexchange.com/?qa=3185/deducing-the-mass-of-the-sirius-system-feynman-exercises-17
<p><img src="https://cdn.pbrd.co/images/HJlk9yQ.jpg" alt=""></p>
<p>I do not even know how to try to deduce the mass M of the Sirius system in terms of that of the sun, as I do not not how to interpret the given data. </p>
<p>Could you give me a hint? </p>
<p><strong>EDITED AT THIS POINT</strong></p>
<p><strong>1) Estimating the distance to the Sirius binary system.</strong></p>
<p>I calculated the hypotenuse of the triangle $S E_1 SS$ (Sun - position of the Earth located at the top of a vertical circular orbit - Sirius System):</p>
<p>$$ \theta = 0.378^{\circ} arc \times \frac{1}{3600} \times \frac{2 \pi}{360} = 1.83 \times 10^{-6}arc sec $$</p>
<p>$$D = \frac{D'}{sin(\theta)} = 8.16\times 10^{16} m$$</p>
<p>Where:</p>
<p>D = the distance from the Earth to the Sirius binary system.</p>
<p>D' = Distance from the Sirius system to the Earth ($15 \times 10^{10} m$).</p>
<p>Here's an illustration of what I have done:</p>
<p><img src="https://cdn.pbrd.co/images/HJtrflM.png" alt=""></p>
<p><strong>2) Measuring the semi-major axis of the ellipse and estimating the period of the orbit from the figure.</strong></p>
<p>To compute the semi-major axis I have assumed the major axis is under the scale [0'',12'']. </p>
<p>As Sammy Gerbil said :'To measure the semi-major axis from the figure, magnify the diagram to fill your screen, place the edge of a sheet of paper along the diagonal line marked on the ellipse, mark off the ends of this line on the paper, then transfer the paper edge to the scale and read off the length in arc seconds. I get 13.8" for the major axis length, so the semi-major axis is a=6.9". The calculation is then:'</p>
<p>$$a=8.16\times 10^{16}m\times \frac{6.9}{3600}\times \frac{2\pi}{360} = 2.73 \times 10^{12}m$$</p>
<p>Now let's estimate the period of the orbit.</p>
<p>Is this the right thought? </p>
<p>$$\frac{dA}{dt} = SM \times SM \times \frac{d \theta}{2dt}$$</p>
<p>It will not be accurate as 'the two sides of most of the triangles are different'.</p>
<p>$A_1$:</p>
<p><img src="https://cdn.pbrd.co/images/HK5k6ab.jpg" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3185/deducing-the-mass-of-the-sirius-system-feynman-exercises-17Sat, 20 Oct 2018 15:50:53 +0000Find the angle between the $x$-axis and a vector
http://physics.qandaexchange.com/?qa=3105/find-the-angle-between-the-%24x%24-axis-and-a-vector
<blockquote><p>The $x$ component of vector $A$ is 25.0 m and the $y$ component is 40.0 m. <br>
(a) What is the magnitude of $A$? <br>
(b) What is the angle between the direction of $A$ and the positive direction of x?</p>
</blockquote>
<p>For (b) I tried using the formula $\tan \theta = \frac{a_y}{a_x} = \frac{40}{-25} = -1.6$, thus $\arctan(-1.6)=58$ degrees which does not match the answer key: $122$ degrees.</p>
<p>Any help is appreciated.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3105/find-the-angle-between-the-%24x%24-axis-and-a-vectorSat, 20 Oct 2018 11:59:19 +0000Onset of bouncing for rolling hoop with off-centre mass (Irodov ex 1.265)
http://physics.qandaexchange.com/?qa=3173/onset-bouncing-for-rolling-hoop-with-off-centre-mass-irodov
<blockquote><p>A small body $A$ is fixed to the inside of a thin rigid hoop of radius $R$ and the mass equal to that of the body $A$. The hoop rolls without slipping over a horizontal plane, at the moments when the body $A$ gets into lower position, the center of the hoop moves with velocity $v_0$ At what values of $v_0$ will the hoop move without bouncing.<img src="https://cdn.pbrd.co/images/HJaN8cM.png" alt=""></p>
</blockquote>
<p>It's very intuitive that bouncing can happen only at the top point.</p>
<p>Let there be normal $N$ at the top between $A$ and hoop and hoop rotating with $\omega$, for centripetal force and condition that $N$ lifts up $mg$ of hoop$$N+mg=m\omega^2R\\ N=mg\\ \implies 2mg=m \omega^2R\implies \omega^2=\dfrac{2g}{R}$$</p>
<p>Now I need to conserve energy, the best option for most cases is from CM but conserving it here from CM makes it tedious, so from where and how should I conserve it to get it easily.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3173/onset-bouncing-for-rolling-hoop-with-off-centre-mass-irodovFri, 19 Oct 2018 12:56:10 +0000Normal reaction for particle at rest on a sphere
http://physics.qandaexchange.com/?qa=3171/normal-reaction-for-particle-at-rest-on-a-sphere
<p><img src="https://cdn.pbrd.co/images/HIVCKHf.png" alt=""></p>
<p>The answer is C. </p>
<p>It is not difficult to understand why force $F$ decreases as the small metal ball is raised to the top of the sphere. The normal reaction $R$ from the hemisphere bears a greater share of the weight of the metal ball. </p>
<p>But why is the normal reaction $R$ the same for all positions of the ball on the sphere? Is there an intuitive explanation?</p>
<p>See discussion in <a rel="nofollow" href="https://chat.stackexchange.com/transcript/message/47213469#47213469">Problem Solving Strategies chatroom</a> on Physics Stack Exchange.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3171/normal-reaction-for-particle-at-rest-on-a-sphereWed, 17 Oct 2018 22:19:45 +0000The length of the spring
http://physics.qandaexchange.com/?qa=3166/the-length-of-the-spring
<blockquote><p>Turns of a uniform spring of relaxed length l = 1.00 m and force constant<br>
k = 500 N/m almost touch each other. A light glue is applied evenly<br>
between every adjacent turn. Breaking strength of the glue is $F_b = 100<br>
N$. The spring is placed on a frictionless horizontal floor and pulled from<br>
one of its ends. If the pulling force is gradually increased to a value F = <br>
200 N. how much will the length of the spring become?</p>
</blockquote>
<p><img src="https://cdn.pbrd.co/images/HIzqCh6.jpg" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3166/the-length-of-the-springMon, 15 Oct 2018 13:43:26 +0000The Random Walk
http://physics.qandaexchange.com/?qa=3164/the-random-walk
<p>Consider a one dimensional random walk performing discrete jumps of length $a$ at each time step.</p>
<p>a) Calculate $P(p,N)$ the probability that the walk of $N$ steps performs $p$ steps to the right.</p>
<p><strong>I do not know if the binomial probability approach can be applied to this problem. If that is the case, why?</strong></p>
<p><img src="https://cdn.pbrd.co/images/HIksJH1.jpg" alt=""></p>
<p>The information I used:</p>
<p><img src="https://cdn.pbrd.co/images/HIktjBc.jpg" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HIktxSa.jpg" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HIktJQ3.jpg" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HIku2ST.jpg" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3164/the-random-walkSat, 13 Oct 2018 23:42:35 +0000Find angular acceleration of disc and tension in the string
http://physics.qandaexchange.com/?qa=3155/find-angular-acceleration-of-disc-and-tension-in-the-string
<blockquote><p>A uniform disc of mass m and radius R is kept on frictionless horizontal table. Two particles of mass m are connected to disc by two identical light inextensible threads as shown in figure. The particles are given velocity $v_0$ perpendicular to the length of strings. Then find angular acceleration of disc and tension in the string .<br>
<img src="https://cdn.pbrd.co/images/HHxIiMM.jpg" alt=""></p>
</blockquote>
<p><img src="https://cdn.pbrd.co/images/HHxIzPN.jpg" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3155/find-angular-acceleration-of-disc-and-tension-in-the-stringMon, 08 Oct 2018 19:35:17 +0000Maximum height after collision in a bowl
http://physics.qandaexchange.com/?qa=3146/maximum-height-after-collision-in-a-bowl
<p>Consider two masses $m, M$ that fall from the top of a hemispherical bowl (from rest) and slide down without any friction.</p>
<p>We want to find how high each of the masses will go after the collision assuming the collision is elastic.</p>
<p><img src="http://i63.tinypic.com/23vnrxv.jpg" alt=""></p>
<p><img src="http://i68.tinypic.com/5v20sn.png" alt=""></p>
<p><img src="http://i68.tinypic.com/28w1937.png" alt=""></p>
<p><img src="http://i65.tinypic.com/b49hdi.png" alt=""></p>
<p>This is how far I can go. It seems to me there are too many unknowns and I am missing some key simplification.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3146/maximum-height-after-collision-in-a-bowlMon, 08 Oct 2018 01:15:46 +0000Thevenin's theorem on Capacitor.
http://physics.qandaexchange.com/?qa=3144/thevenins-theorem-on-capacitor
<p>The time constant of the circuit shown is:<br>
<img src="https://cdn.pbrd.co/images/HHlOycn.png" alt=""><br>
I found $R_{th}=\dfrac{3R}{5}$ by following method :</p>
<p>We short-circuit the batteries to calculate Thevenin's equivalent resistance across $C$ hence $2R\parallel 2R=R$ which is parallel with $\dfrac{R}{2}+R=\dfrac{3R}{2}\implies R_{th}=R\parallel \dfrac{3R}{2}=\dfrac{3R}{5}$ <br>
So time constant $\tau=\dfrac{3RC}{5}$</p>
<p>But $\tau=\dfrac{RC}{2}$, given. Where I went wrong? </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3144/thevenins-theorem-on-capacitorSun, 07 Oct 2018 13:26:10 +0000Maximum energy stored in a spring-block system
http://physics.qandaexchange.com/?qa=3124/maximum-energy-stored-in-a-spring-block-system
<blockquote><p>A block of mass $m$ is attached with an ideal spring of spring constant $k$ is kept on a smooth horizontal surface. Now the free end the spring is pulled with a constant velocity $u$ horizontally. The maximum energy stored in the spring and block system during subsequent motion is?</p>
</blockquote>
<p>I think,<br>
Spring will keep on extending till it gain velocity $u$ to stop free end's expansion, then spring is elongated it will apply force on mass $m$ till it again gain its natural length, by now maximum work has been done, so if I calculate $v_m$ then $E=\frac{1}{2}mv^2$, so how to calculate $v_m$?</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3124/maximum-energy-stored-in-a-spring-block-systemSat, 06 Oct 2018 05:01:17 +0000Finding path for which line integral of force is zero.
http://physics.qandaexchange.com/?qa=3123/finding-path-for-which-line-integral-of-force-is-zero
<blockquote><p>A particle is constrained to move from initial point $O$ to final point $C$ along three different smooth horizontal tracks namely $OBC, OPC, OAC$. If the particle moves under the influence of an external force $F$ such that the initial and final speeds are same then:<br>
1. There necessarily exists a path along which line integral of force $F$ is zero.<br>
2. $F$ is conservative<br>
3. $F$ cannot be conservative<br>
4. There is no closed path along which line integral of force $F$ is zero.</p>
</blockquote>
<p>Only correct answer is $(1)$.</p>
<p>What does this mean?</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3123/finding-path-for-which-line-integral-of-force-is-zeroSat, 06 Oct 2018 03:17:35 +0000Average of the internal energy of a system
http://physics.qandaexchange.com/?qa=3122/average-of-the-internal-energy-of-a-system
<p><img src="https://cdn.pbrd.co/images/HH4nYLV.png" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HH4odz4.png" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HH4oOvr.jpg" alt=""></p>
<p>I am struggling to get the right value for $Z_1$, which is $\beta^{\frac{-5}{2}}$ multiplied by constants. Once you have the value for the one particle partition function it is just about applying the definition of average energy. </p>
<p>$$\langle E \rangle = -\frac{ \partial \ln(Z_1)}{\partial \beta}$$</p>
<p>I got :</p>
<p>$$ \langle E \rangle = \frac{9}{4\beta}$$</p>
<p>The right answer is though:</p>
<p>$$\langle E \rangle = \frac{5}{2\beta}$$</p>
<p>NOTE: I know this is more a mathematical issue and I have already asked for this question in MSE, but I could not get the right value for < E > yet. If you want to have a look:</p>
<p><a rel="nofollow" href="https://math.stackexchange.com/questions/2941009/solving-a-partition-function-in-polar-coordinates">https://math.stackexchange.com/questions/2941009/solving-a-partition-function-in-polar-coordinates</a></p>
<p>More information about the partition function and more definitions if needed be. Please let me know if you need information that is not stated:</p>
<p>$$\beta = \frac{1}{k_B T}$$</p>
<p><img src="https://cdn.pbrd.co/images/HH4rh4c.png" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HH4rrF1L.png" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HH4rGeA.png" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3122/average-of-the-internal-energy-of-a-systemFri, 05 Oct 2018 17:17:37 +0000The ratio of temperatures of two sides of a disk near a concave mirror
http://physics.qandaexchange.com/?qa=3117/the-ratio-of-temperatures-two-sides-disk-near-concave-mirror
<blockquote><p>A wide homogeneous beam of light falls on a concave spherical mirror of<br>
radius R parallel to the optical axis. A small opaque disc of radius r (r<br>
<< R) made of a perfectly heat insulating material is placed at a distance<br>
R/4 from the pole of the mirror perpendicular to the optical axis. In<br>
steady state, both the surfaces of the disc acquire different temperatures<br>
slightly higher than the surroundings. Find the ratio $\delta T_1 /\delta T_2$, where<br>
$\delta T_1$. and $\delta T_2$; are the temperature differences of the left and the right<br>
surfaces of the disc and the surroundings.<br>
<img src="https://cdn.pbrd.co/images/HGqS17f.jpg" alt=""></p>
</blockquote>
<p><img src="https://cdn.pbrd.co/images/HGqSh7X.jpg" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3117/the-ratio-of-temperatures-two-sides-disk-near-concave-mirrorMon, 01 Oct 2018 12:24:00 +0000Is centripetal force not always true for Planets?
http://physics.qandaexchange.com/?qa=3114/is-centripetal-force-not-always-true-for-planets
<p><img src="https://cdn.pbrd.co/images/HGi2DtH.png" alt=""></p>
<p><em>Source :</em> IE Irodov: 1.207.</p>
<p>I used, force by gravitational pull of Sun should provide necessary centripetal force at one extreme $r_1$ where velocity is $v_1$ ( I took that moment as circle as $r_1\perp v_1$) ,$$\implies\dfrac{GM_sm}{r_1^2}=\dfrac{mv_1^2}{r_1}\implies v_1=\sqrt{\dfrac{GM_s}{r_1}}$$</p>
<p>But when I use energy conservation then, $$v_1=\sqrt{\dfrac{2GM_sr_2}{r_1(r_1+r_2)}}$$</p>
<p>Second expression gives me the correct answer, also if I put $r_1=r_2$ second expression reduces to first one, which says to me that we apply centripetal force method only when we have complete circle, not that instantaneous circles, why?</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3114/is-centripetal-force-not-always-true-for-planetsSun, 30 Sep 2018 11:04:59 +0000Solving the diffusion equation with an absorbing boundary
http://physics.qandaexchange.com/?qa=3113/solving-the-diffusion-equation-with-an-absorbing-boundary
<p><img src="https://cdn.pbrd.co/images/HGatckL.png" alt=""></p>
<p>There is a one-dimensional diffusion process in which particles start running at $t = 0$ and from $x_o > 0$.</p>
<p><strong>When particles reach x = 0 they are removed from the system, thus the total concentration is not conserved anymore.</strong></p>
<p>I have to solve the diffusion equation, which is the following partial differential equation:</p>
<p>$$\frac{\partial P (R, t)}{\partial t} = D\triangledown^2P(R,t) $$</p>
<p>Where $P(R, t)$ is the probability that the particles arrive at R at time t.</p>
<p>We have the initial conditions:</p>
<p> $$c(x,0) = N\delta(x - x_o)$$</p>
<p> $$c(0, t) = 0$$</p>
<p>I have been doing some research in how to do so and I came across with a method which is based on a particular Gaussian function:</p>
<p>$$G (R, t) = (\frac{1}{4\pi Dt})^{\frac{d}{2}} e^{\frac{(R-R_0)^2}{4Dt}}$$</p>
<p>Where d is the dimensionality of the system.</p>
<p>But the issue here is that we are working with an 'absorbing boundary' that makes the condition $c(0, t) = 0$ useless because we work from $x_o > 0$. </p>
<p>Then how could I solve the probability (i.e this differential equation)? We have been suggested the method of images, but not sure how it works with differential equations.</p>
<p>NOTE: I know this is more a mathematical issue (solving a differential EQ.) but thought that as it is a mechanics problem it could be useful asking here. </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3113/solving-the-diffusion-equation-with-an-absorbing-boundarySat, 29 Sep 2018 18:43:19 +0000Angle of pendulum in accelerated incline
http://physics.qandaexchange.com/?qa=3107/angle-of-pendulum-in-accelerated-incline
<p>Consider the following system consisting of a box sliding down a plane. The coefficient of friction between the plane and the box is $\mu$. A pendulum is attached to the top of the box as shown.</p>
<p><img src="http://i65.tinypic.com/1z37azr.png" alt=""></p>
<p>The acceleration of the box+pendulum is $a=g(\sin\theta-\mu\cos\theta)$ I believe.<br>
In a non-inertial frame attached to the box, the free-body diagram for the pendulum is</p>
<p><img src="http://i63.tinypic.com/2eyyx3s.png" alt=""></p>
<p>My goal is to find the angle $\phi$ from the equilibrium of these 3 forces. I have to pick x and y axes to decompose these forces. If I pick the x axis along the fictitious force and the y perpendicular to it I get (I think) </p>
<p>$$m_P a+T\sin\phi=m_P g\sin\theta$$ </p>
<p>and </p>
<p>$$T\cos\phi=m_{P}g\cos\theta$$</p>
<p> which I can then solve for $\phi$ :</p>
<p>$$\tan\phi=\frac{g\sin\theta-a}{g\cos\theta}=\mu$$</p>
<p>Is this right?</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3107/angle-of-pendulum-in-accelerated-inclineFri, 28 Sep 2018 04:00:23 +0000Minimum distance from the mirror should the boy be to see his full image
http://physics.qandaexchange.com/?qa=3099/minimum-distance-from-the-mirror-should-the-boy-see-full-image
<blockquote><p>A plane mirror with its bottom edge on the floor is tilted at an angle $\theta$<br>
to the vertical (see figure). A boy whose eyes are at a height h above the<br>
floor is standing in front of the mirror. At what minimum distance from<br>
the mirror should the boy be to see his full image in the mirror?<br>
<img src="https://cdn.pbrd.co/images/HFvJzFb.jpg" alt=""></p>
</blockquote>
<p><img src="https://cdn.pbrd.co/images/HFvJOfm.jpg" alt=""><br>
To get the minimum d the rays should go parallel to floor and then get reflected to the boy eyes<br>
Answer given = $(h\sec\theta)/2$</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3099/minimum-distance-from-the-mirror-should-the-boy-see-full-imageTue, 25 Sep 2018 10:59:43 +0000Lengths of vertical cables and maximum tension - Feynman exercises 2.35
http://physics.qandaexchange.com/?qa=3088/lengths-vertical-cables-maximum-tension-feynman-exercises
<p><img src="https://cdn.pbrd.co/images/HFc4uGb.png" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HFc0vgU.jpg" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HFojRUM.jpg" alt=""></p>
<p><strong>a) Find the proper lengths for the remaining vertical cables A and B.</strong></p>
<p>I focused on the left hand side of the diagram and drew the right triangle BEC. Once here I drew the bisector of the right angle located at C, originating the right triangle CED. Now to get the angle $\theta$ we just need to know that the three angles of a triangle sum 180 degrees. As you can see I applied that doing:</p>
<p>angle E + angle CDE + angle DCE = 180 (where angle E = $\theta$ )</p>
<p>So now it is just about solving for $\theta$, which I got is 45 degrees. This is not correct so I am wondering what I did wrong here. Note after I applied tan $\theta$ in order to get the length BC which, as expected, did not make sense.</p>
<p>I am sure my mistake has to be on calculating $\theta$ but I do not see where is the flaw exactly.<br>
It might be on calculating the angle DCE but it has to be 45 degrees because is the bisection of a right angle.</p>
<p>To find the distance GC (the second largest cable, the one related to letter A in the original exercise) I could not find the proper method.</p>
<p>Given answers A = 5 m; B = 11 m.</p>
<p><strong>b) Find maximum tension in the two longitudinal cables</strong></p>
<p>$$\sum F_y = mg $$ </p>
<p>$$T sin 45 - mg = 0$$</p>
<p>$$T = 6.66 x 10^5 N$$</p>
<p>But the given answer is T = 34 x 10^3 kg-wt</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3088/lengths-vertical-cables-maximum-tension-feynman-exercisesSun, 23 Sep 2018 09:19:27 +0000Statics - Minimum angle before sliding
http://physics.qandaexchange.com/?qa=3084/statics-minimum-angle-before-sliding
<p><img src="https://imgur.com/a/mixJS6f" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3084/statics-minimum-angle-before-slidingThu, 20 Sep 2018 08:01:27 +0000Spinning 4 connected springs system - Feynman exercises 14.20
http://physics.qandaexchange.com/?qa=3059/spinning-connected-springs-system-feynman-exercises-14-20
<p><img src="https://cdn.pbrd.co/images/HDS2fGT.jpg" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HDS2IAS.jpg" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HE1vRUX.jpg" alt=""></p>
<p>b) Based on the Potential energy stability test the condition to have stable equilibrium is: second derivative < 0.</p>
<p>I quote from Wikipedia: ‘The potential energy is at a local minimum. This is a stable equilibrium. The response to a small perturbation is forces that tend to restore the equilibrium. If more than one stable equilibrium state is possible for a system, any equilibria whose potential energy is higher than the absolute minimum represent metastable states‘.</p>
<p>So the equation should be</p>
<p>$2K - m\omega^2 < 0$ </p>
<p>But the answer is based on the second derivative > 0, which is unstable equilibria.</p>
<p>What am I misunderstanding here?</p>
<p>Solutions:</p>
<p>a) $\frac{ML\omega^2}{2K - m\omega^2}$</p>
<p>b) Only if $\frac{M\omega^2}{k}$ < 2</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3059/spinning-connected-springs-system-feynman-exercises-14-20Fri, 14 Sep 2018 16:18:27 +0000Find the range of a projectile on an inclined plane
http://physics.qandaexchange.com/?qa=3055/find-the-range-of-a-projectile-on-an-inclined-plane
<p>Consider the following problem:<br>
<img src="http://i64.tinypic.com/vya1rl.png" alt="dd"><br>
A hill makes an angle $\alpha$ with the horizontal.<br>
An arrow is shot from a point on the hill with initial velocity $v_{0}$ and at an angle $\beta>\alpha$ with the horizontal.</p>
<p>(a) Find the time needed for the arrow to land in terms of $\alpha,\beta,v_{0}$ and the gravitational acceleration.</p>
<p>(b) Show that the distance between the origin and the place of landing is given by <img src="http://i67.tinypic.com/kbwor7.png" alt=""></p>
<p>I am unable to get the expected answer for part (b). Since part (b) depends on parts (a) I am imagining my reasoning in (a) is faulty somewhere, but where?</p>
<p>Here it is my answer for part (a) <img src="http://i64.tinypic.com/2eap15l.png" alt=""></p>
<p>and for part (b) I get <img src="http://i68.tinypic.com/14tqb1t.png" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3055/find-the-range-of-a-projectile-on-an-inclined-planeThu, 13 Sep 2018 03:00:41 +0000Find velocity of approach of the centres
http://physics.qandaexchange.com/?qa=3053/find-velocity-of-approach-of-the-centres
<blockquote><p>A thread is wound on two identical bobbins placed on a horizontal floor<br>
with their axes parallel. Radius of the outer flanges of the bobbins is $n$<br>
times of that of the inner spools. The midpoint of the thread is pulled<br>
vertically upwards with a constant velocity u. If the bobbins roll on the<br>
floor without slipping, find velocity of approach of their centres when<br>
angle between thread segments becomes $2\theta$.<br>
<img src="https://cdn.pbrd.co/images/HDxtZo2.png" alt=""></p>
</blockquote>
<p><img src="https://cdn.pbrd.co/images/HDxuoZW.png" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3053/find-velocity-of-approach-of-the-centresWed, 12 Sep 2018 11:53:36 +0000A 5.0kg object slides 2.0m down a smooth plane. Find its potential energy and component of weight parallel to the plane
http://physics.qandaexchange.com/?qa=3036/object-slides-potential-energy-component-weight-parallel
<p>A 5.0kg block of ice is at rest at the top of a smooth inclined plane. The block is released and slides 2.0m down the plane. Assuming there is no friction between the block and the surface, calculate:<br>
1. The gravitational potential energy at the top of the plane,<br>
2. The component of the weight parallel to the plane,<br>
3. The acceleration of the block,<br>
4. The velocity of the block at the bottom of the plane,<br>
5. The kinetic energy at the bottom of the plane.</p>
<p>Since there's no friction, the change in potential energy will be equal to the kinetic energy at the end. Am I right in thinking that the work done on the block is also equal to the kinetic energy gained by the block?</p>
<p>If so, that gives: <br>
$$W = E_k = \Delta E_p$$<br>
$$F \cdot s \cdot \cos\theta = \frac{m \cdot v^2}{2} = m \cdot g \cdot \Delta h$$ <br>
Substituting in values gives: <br>
$$98.1 \cos \theta = 2.5v^2 = 49.05 \Delta h$$<br>
Without knowing the angle the slope is on, I'm not sure where to go from here.<br><br>
I've been assuming that the block travels at an angle, 2 metres down the slope. Perhaps it goes down 2 metres vertically? <br>
$$49.05 \cdot s \cdot \cos \theta = 2.5v^2 = 98.1$$<br>
This would help with parts 1, 4, and 5; but I wouldn't know how to answer parts 2 and 3.</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3036/object-slides-potential-energy-component-weight-parallelFri, 07 Sep 2018 12:24:58 +0000Find maximum charge on ball and maximum current through inductor .
http://physics.qandaexchange.com/?qa=3046/find-maximum-charge-ball-maximum-current-through-inductor
<blockquote><p>An uncharged conducting ball of radius r is earthed through an inductor<br>
L as shown in the figure. A horizontal electron beam swoops on the ball<br>
from a great distance. The density of electrons in the incident beam is n<br>
and all these electrons are moving with velocity u that is much smaller<br>
than the speed of light so the relativistic effects can be neglected . <br>
Find maximum charge on ball and maximum current through inductor .<br>
<img src="https://cdn.pbrd.co/images/HCk2Wul.jpg" alt=""> </p>
</blockquote>
<p><img src="https://cdn.pbrd.co/images/HCk3d2R.jpg" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3046/find-maximum-charge-ball-maximum-current-through-inductorTue, 04 Sep 2018 11:55:24 +0000Minimum width of belt and minimum speed of the disc
http://physics.qandaexchange.com/?qa=3040/minimum-width-of-belt-and-minimum-speed-of-the-disc
<blockquote><p>A horizontal conveyor belt is running at a constant speed $v_b$ = 3.0 m/s.<br>
A small disc enters the belt moving horizontally with a velocity $v_0$ = 4.0<br>
m/s that is perpendicular to the velocity of the belt. Coefficient of friction<br>
between the disc and the belt is 0.50.<br>
<img src="https://cdn.pbrd.co/images/HBIfrHI.jpg" alt=""><br>
<img src="https://cdn.pbrd.co/images/HBIfHEy.jpg" alt=""></p>
</blockquote>
<p>My try :<br>
<img src="https://cdn.pbrd.co/images/HBIfWtQ.jpg" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3040/minimum-width-of-belt-and-minimum-speed-of-the-discFri, 31 Aug 2018 11:44:55 +0000$\Psi$ in function of 'position'
http://physics.qandaexchange.com/?qa=3028/%24-psi%24-in-function-of-position
<p>The general exercise:<br>
<img src="https://cdn.pbrd.co/images/HApljRF.jpg" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HAQZH0G.jpg" alt=""></p>
<p>Let's focus on c). </p>
<p>The value of $\sigma$:</p>
<p>$$\sigma =\sqrt{ < x^2 > -< x >^2 }= \frac{1}{\lambda \sqrt{2}}$$</p>
<p>We are asked to plug in $\sigma$:</p>
<p>$$| \Psi (\sigma ) | = \lambda e^{\frac{-2}{\sqrt{2}}}$$</p>
<p>Where $\lambda$ is a positive constant.</p>
<p>I calculated < x > and its value is 0. Check this link to obtain further details:</p>
<p><a rel="nofollow" href="https://math.stackexchange.com/questions/2893056/misleading-expected-value">https://math.stackexchange.com/questions/2893056/misleading-expected-value</a></p>
<p><img src="https://cdn.pbrd.co/images/HBKGTII.png" alt=""></p>
<p><strong>About the probability</strong></p>
<p>I have difficulties solving this probability as this time I got:</p>
<p>$P_i = \int_{-\sigma}^{\sigma} \lambda e^{-2 \lambda |x|} dx =$ </p>
<p>$= \lambda \int_{-\sigma}^{0} e^{-2 \lambda |x|} dx$ </p>
<p>$+ \lambda \int_{0}^{+\sigma} e^{-2 \lambda |x|} dx = 0$</p>
<p>Where did I get wrong? I have been trying to get this probability but none outcome seems to make sense.</p>
<p>$$P_o = 1 - P_i $$</p>
<p>Where $P_i$ and $P_o$ mean inside and outside respectively</p>
<p>Thank you</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3028/%24-psi%24-in-function-of-positionSat, 25 Aug 2018 21:27:13 +0000The amplitude of the resulting oscillation is $\frac{xqQ}{8\pi ^2 \epsilon _0 a^3 k}$ .Find the value of 'x'?
http://physics.qandaexchange.com/?qa=3022/amplitude-resulting-oscillation-frac-xqq-epsilon-find-value
<blockquote><p>In the given figure ABC is a non-conducting semi-circular wire of<br>
radius 'a' carrying a total charge Q uniformly distributed on it and a<br>
point charge q is at its centre. Ends of the wire are attached to two<br>
separate springs each having spring constant k as shown in the<br>
figure. In the given position system is in equilibrium. Now the point<br>
charge is suddenly removed. The amplitude of the resulting oscillation<br>
is $\frac{xqQ}{8\pi ^2 \epsilon _0 a^3 k}$ . Ignoring any force other than spring force and<br>
electrostatic force find the value of 'x'?<br>
<img src="https://cdn.pbrd.co/images/HACbFAe.jpg" alt=""></p>
</blockquote>
<p>My try :<br>
<img src="https://cdn.pbrd.co/images/HACbVoL.jpg" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3022/amplitude-resulting-oscillation-frac-xqq-epsilon-find-valueFri, 24 Aug 2018 06:32:17 +0000Normalization of a wave function
http://physics.qandaexchange.com/?qa=3018/normalization-of-a-wave-function
<p><img src="https://cdn.pbrd.co/images/HApljRF.jpg" alt=""></p>
<p>In order to normalize the wave function I am aware of the following condition:</p>
<p>$$\int_{-\infty}^{\infty} | \psi (x,t) |^2 dx = 1$$</p>
<p>If I am not mistaken what I have to do first is obtaining A so as to ensure that this value meets the stated condition. But what I got does not make sense:</p>
<p><img src="https://cdn.pbrd.co/images/HApq2UW.jpg" alt=""></p>
<p>There has to be a mistake here but I do not see where.</p>
<p>My book has the following information about normalization:</p>
<p><img src="https://cdn.pbrd.co/images/HApsJqz.jpg" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HApt7c0.jpg" alt=""></p>
<p>Thank you</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3018/normalization-of-a-wave-functionWed, 22 Aug 2018 22:10:05 +0000Oblique collision of three disks
http://physics.qandaexchange.com/?qa=3005/oblique-collision-of-three-disks
<p><img src="https://cdn.pbrd.co/images/HzQjIyY.png" alt=""></p>
<p>Can you tell me how to solve this question. I have solved somewhat till the final velocity of the blocks and my final diagram looks like this : </p>
<p><img src="https://cdn.pbrd.co/images/HzQkXMY.png" alt="http://imgur.com/a/z6etzT5"></p>
<p>But I have trouble calculating coeffecient of restitution. For starters do we calculate coefficient of restitution along the line of collision only, because impulse is imparted along this line so that means all velocities should be resolved along this line? </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3005/oblique-collision-of-three-disksSun, 19 Aug 2018 04:35:53 +0000Obtaining standard deviation using different equations
http://physics.qandaexchange.com/?qa=3002/obtaining-standard-deviation-using-different-equations
<p><img src="https://cdn.pbrd.co/images/HzMkIAP.jpg" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HzMl0jU.jpg" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HzMlIhZ.jpg" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HzMlZTM.jpg" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/HzMmdJf.jpg" alt=""></p>
<p>As you can see I used two equations in order to obtain $\sigma$ (standard deviation) but did not obtain the same result. </p>
<p>I think my mistake has to be at b), calculating the average value of $(\Delta j)^2$.</p>
<p>First method to compute $\sigma$ :</p>
<p>Standard deviation is defined: </p>
<p>$$\sigma^2 = <(\Delta j)^2>$$</p>
<p>And we also know:</p>
<p>$$\Delta j = j - < j > $$</p>
<p>So first of all I calculated < j >, the average value of a discrete function:</p>
<p>$$< j > = \sum_{j = 0}^{\infty} j P(j) = \frac{1}{N} \sum_{j = 0}^{\infty} j N(j) -------> EQ1$$ </p>
<p>Which equals to 21.</p>
<p>Then I computed each $\Delta j$ for each sample (age 14,15...)</p>
<p>At this point I applied EQ1 in order to obtain $<(\Delta j)^2>$ and subsequently $\sigma^2$</p>
<p>As you can see I got $\sigma$ = 3.117 using method 1</p>
<p>Second method to compute $\sigma$ :</p>
<p>Another equation involving $\sigma$:</p>
<p>$$\sigma^2 = < j^2 > -< j >^2$$</p>
<p>I computed $< j^2 >$ using EQ1 and squared < j >. Afterwards I applied the formula and got:</p>
<p>$\sigma$ = 4.309</p>
<p>There has to be a mistake in method one (I think) but I do not see it</p>
<p>What do you think?</p>
<p>Thank you</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3002/obtaining-standard-deviation-using-different-equationsSat, 18 Aug 2018 18:43:55 +0000Rising balloons
http://physics.qandaexchange.com/?qa=3000/rising-balloons
<p>There are N balloons full of helium separated by equal intervals attached to a weightless rope of length L. Each balloon has a rising force F.</p>
<p><img src="https://cdn.pbrd.co/images/HzL3FVT.jpg" alt=""> </p>
<p><img src="https://cdn.pbrd.co/images/HzLbjM0.jpg" alt=""></p>
<p>a) The second figure shows force diagram of $i$ balloon. Prove horizontal component of $T_i$ (named $T_H$) is equal in all segments.</p>
<p>b)Deduce $F$ equation.</p>
<p>c) Prove $tg\theta_o = -tg\theta_{N+1} = \frac{NF}{2T_H}$</p>
<p>d) Based on $tg\theta_o = -tg\theta_{N+1} = \frac{NF}{2T_H}$ and both diagrams prove the following expressions:</p>
<p>$$tg\theta_i = \frac{(N-2i)F}{2T_H}$$</p>
<p>$$y_i = \frac{L}{N + 1}\sum_{j=0}^{i-1} cos\theta_j$$</p>
<p>$$y_i = \frac{L}{N + 1}\sum_{j=0}^{i-1} sin\theta_j$$</p>
<p>What I tried:</p>
<p>a) We know the balloons are in equilibrium:</p>
<p>$\sum F_y = 0$ and $\sum F_x = 0$</p>
<p>The exercise tells us that:</p>
<p>$$T_{i-1}cos\theta_{i-1} = T_o cos\theta_o = T_H$$</p>
<p>But I am not able to find a general trigonometric equation so as to prove it. Any clue?</p>
<p>b) Based on second diagram ($i$ balloon):</p>
<p>$$\sum F_y = 0$$ </p>
<p>$$F = T_{i-1}sin\theta_{i-1} - T_i sin\theta_i$$</p>
<p>c) By extending segment 0 (red line pointing down with $\theta_o$ angle) and applying trigonometry:</p>
<p>$$tg\theta_o = \frac{NF}{2T_H}$$</p>
<p>But what I do not get is why:</p>
<p>$$tg\theta_o = -tg\theta_{N+1} = \frac{NF}{2T_H}$$</p>
<p>I know this issue is related to symmetry but I do not see it.</p>
<p>d) In order to proof the following expression: </p>
<p>$$tg\theta_i = \frac{(N-2i)F}{2T_H}$$</p>
<p>We have to assume:</p>
<p>$$tg\theta_{i-1} = tg\theta_o$$</p>
<p>Once we do it and apply $\frac{F}{T_H}$ we get it.</p>
<p>However my issue here is: We know horizontal component is the same in both cases. What I do not know is why the vertical component is $\frac{NF}{2}$ in balloon $i$ case. How could we proof $tg\theta_{i-1} = tg\theta_o$?</p>
<p>In order to prove the last two we have to start with defining the length between two balloons which I saw is:</p>
<p>$$l = \frac{L}{N +1}$$</p>
<p>The reason for N+1 is that whatever balloon we pick there is another to account for in order to have the segment? I am not sure why it is divided by N + 1.</p>
<p>Once here I do not how to carry on in order to get: </p>
<p>$$y_i = \frac{L}{N + 1}\sum_{j=0}^{i-1} cos\theta_j$$</p>
<p>$$y_i = \frac{L}{N + 1}\sum_{j=0}^{i-1} sin\theta_j$$</p>
<p>Thank you</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=3000/rising-balloonsSat, 18 Aug 2018 17:12:15 +0000What is electromotive force induced between its ends
http://physics.qandaexchange.com/?qa=2998/what-is-electromotive-force-induced-between-its-ends
<blockquote><p>Consider a quarter circular conducting ring of large radius r with its<br>
centre at the origin, where a magnetic dipole of moment m is placed as<br>
shown in the figure. If the ring rotates at a constant angular velocity $\omega$<br>
about the y-axis, then what is electromotive force induced between its ends?<br>
<img src="https://cdn.pbrd.co/images/HzHpARo.png" alt=""></p>
</blockquote>
<p>My try :</p>
<p><img src="https://cdn.pbrd.co/images/HzHpTJX.png" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=2998/what-is-electromotive-force-induced-between-its-endsSat, 18 Aug 2018 06:02:41 +0000Finding the matricial form of Brinkmann's metric
http://physics.qandaexchange.com/?qa=2994/finding-the-matricial-form-of-brinkmanns-metric
<p>I have the following problem: given Brinkmann's metric expressed as<br>
$$ds^2 = du dv - \delta_{i j} dx^i dx^j - K_{i j}(u) x^i x^j du^2$$<br>
and $i,j=1,2, $ I have to find it's matricial form; my question is the following: how I'm supposed to do that?</p>
<p>I know that the general expression is $$ds^2 = g_{i j} dx^i dx^j$$<br>
but I do not see how to write $g_{ij}$.</p>
<p><em>Subsidiary question: why is this metric useful to study gravitational waves in the void?</em></p>
<p>Thanks you!</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=2994/finding-the-matricial-form-of-brinkmanns-metricMon, 13 Aug 2018 22:15:44 +0000Mutual indutance of two long parallel wires
http://physics.qandaexchange.com/?qa=2990/mutual-indutance-of-two-long-parallel-wires
<blockquote><p>Two long parallel wires whose centres are a distance <strong>d</strong> apart carry equal currents in opposite directions. If the flux within the wires is neglected, the inductance of such arrangement of wire of length <strong>l</strong> and radius <strong>a</strong> will be?</p>
</blockquote>
<p>I am not able to relate this question to the definition of mutual inductance. In mutual inductance, we basically change the current in one component so as to induce emf in the other component but here both of the components are carrying currents in opposite directions. </p>
<p>I know that: </p>
<p>$\phi = MI$ </p>
<p>Please help me with the physical interpretation of the question and provide a hint on how to solve it. </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=2990/mutual-indutance-of-two-long-parallel-wiresFri, 10 Aug 2018 16:27:27 +0000Average velocity Of the particle in a large time interval.
http://physics.qandaexchange.com/?qa=2986/average-velocity-of-the-particle-in-a-large-time-interval
<blockquote><p>In a region of space, a uniform static magnetic field of induction $B_1$ is established above the x-z plane and another uniform static magnetic field of induction $B_2 \gt B_1$ is established below the x-z plane. Both fields are in the positive z-direction. <br>
A particle of mass m and charge q is projected from the origin with velocity v making angle $\theta$ with x-axis as shown in the figure. <br>
Find the average velocity of the particle in a large time interval. </p>
</blockquote>
<p><img src="https://cdn.pbrd.co/images/Hyp4I45.png" alt=""></p>
<p><img src="https://cdn.pbrd.co/images/Hyp5dHjw.png" alt=""></p>
<p>But the answer is given as :<br>
<img src="https://cdn.pbrd.co/images/Hyp5vqs.png" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=2986/average-velocity-of-the-particle-in-a-large-time-intervalThu, 09 Aug 2018 17:37:14 +0000Both molar specific heat of an ideal monatomic gas at constant volume and pressure
http://physics.qandaexchange.com/?qa=2984/both-molar-specific-ideal-monatomic-constant-volume-pressure
<blockquote><p>2 mols of oxygen are heated up from a temperature of 20 C and a pressure 1 atm to a temperature of 100 C. Compute:<br>
a) Having constant volume, how much heat must be provided to the gas throughout the process?<br>
b) Having constant pressure, how much work is done throughout the process?<br>
c) Compute in a) and b) cases the variation in the internal energy.</p>
</blockquote>
<p>My try:</p>
<p>The molar specific heat of an ideal gas at constant volume is:</p>
<p>$$C_v = \frac{3}{2}R$$</p>
<p>The molar specific heat of an ideal gas at constant pressure is:</p>
<p>$$C_p = \frac{5}{2}R$$</p>
<p>Then at a) it is just about using $Q = nC\Delta T$. My issue here is that the given outcome at a) uses n = 1:</p>
<p>$$Q = nC_v\Delta T = 997.68J$$ </p>
<p>I obtained 1995.36J (I used n=2).</p>
<p>To compute Q at constant pressure:</p>
<p>$$Q = nC_p\Delta T = 1662.80J$$ </p>
<p>I obtained 3325.6J (I used n=2).</p>
<p>I do not understand why it uses n=1 as it is specified that the gas is formed by 2 mols of oxygen. </p>
<p>b) When heat is supplied at constant pressure the gas disseminates and exerts a positive work (such as on a piston).</p>
<p>However the answer is given as a negative number, which means work is done by the system and not over it. Thus the given answer:</p>
<p>$$W = -nR\Delta T = -1330.24J$$ </p>
<p>Here I got the same but positive.</p>
<p>c) Using the first principle of thermodynamics:</p>
<p>$$ \Delta E_i = Q + W$$</p>
<p>When volume is constant:</p>
<p>$$ \Delta E_i = Q_v + W = -332.56J$$</p>
<p>When pressure is constant:</p>
<p>$$ \Delta E_i = Q_p + W = 332.56J$$</p>
<p>What I got before seeing the answers:</p>
<p>When volume is constant:</p>
<p>$$ \Delta E_i = Q_v + W = 3325.60J$$</p>
<p>When pressure is constant:</p>
<p>$$ \Delta E_i = Q_p + W = 4655.84J$$</p>
<p>As you can see all comes down to the definition: $Q = nC_v\Delta T$ I have made some research and in Tipler and $C_v$, $C_p$ and Q are defined: </p>
<p>$$C_v = n\frac{3}{2}R$$</p>
<p>$$C_p = n\frac{5}{2}R$$</p>
<p>$$Q = C\Delta T$$</p>
<p>Eventually it is the same, since I multiplied by 2 (number of mols) in the definition: $Q = nC\Delta T$ and I did not do it my first $C_v$ and $C_p$ definitions. Therefore I think my outcomes are right but what do you think?</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=2984/both-molar-specific-ideal-monatomic-constant-volume-pressureThu, 09 Aug 2018 08:59:16 +0000Mean free path of ideal gas
http://physics.qandaexchange.com/?qa=2974/mean-free-path-of-ideal-gas
<p>There's a box with a wall in mid dividing it in two sections, each filled with same ideal gas in both sections at 150 K in one section and at 300 K in another. How am I supposed to calculate ratio of mean free paths (MFP)in 2 sections?</p>
<p>My attempt: MFP ~ Volume / Number of particles</p>
<pre><code> => MFP ~ Temperature / Pressure
</code></pre>
<p>Now, Assuming pressure to be same on both sections, ratio must be half. But that is incorrect. Why?</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=2974/mean-free-path-of-ideal-gasTue, 07 Aug 2018 13:38:41 +0000Angular acceleration of conducting ring in magnetic field
http://physics.qandaexchange.com/?qa=2960/angular-acceleration-of-conducting-ring-in-magnetic-field
<p>><img src="https://cdn.pbrd.co/images/HxvPkDB.png" alt=""></p>
<p>Attempt: </p>
<p>$\tau = \mu \times B$ <br>
$\tau = I\alpha = MR^2 \alpha$</p>
<p>$\mu = iA= i\pi R^2$</p>
<p>$\implies \alpha = \dfrac{\pi i B}{M} = 20 \pi\text{rad/s^2}$ </p>
<p>But answer given is $a) 40 \pi$</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=2960/angular-acceleration-of-conducting-ring-in-magnetic-fieldFri, 03 Aug 2018 21:02:49 +0000Particle will definitely return to the origin
http://physics.qandaexchange.com/?qa=2956/particle-will-definitely-return-to-the-origin
<blockquote><p><img src="https://cdn.pbrd.co/images/Hxv7xcR.png" alt=""></p>
</blockquote>
<p>Attempt: </p>
<p>I know that the path will be helical and the particle will move initially in negative y direction then stop and then return. I don't think that the method to solve it is that the pitch should be an integer because I couldn't see any problem if pitch is not an integer. </p>
Physics Problemshttp://physics.qandaexchange.com/?qa=2956/particle-will-definitely-return-to-the-originFri, 03 Aug 2018 19:15:01 +0000Dipole moment and torque on a rectangular wire
http://physics.qandaexchange.com/?qa=2950/dipole-moment-and-torque-on-a-rectangular-wire
<blockquote><p>The rectangular wire of the image is placed on yz plane and is immersed in a magnetic field which magnitude and direction is :<br>
$$\vec B = \frac{0.05}{\sqrt{2}}(\hat i + \hat j)$$<br>
a) Compute the magnetic dipole moment and torque over the wire when the current going through is 5A<br>
b) Compute the magnetic dipole moment and torque over the wire when it turns counterclockwise 45 degrees respect to z axis and the current is also 5A</p>
</blockquote>
<p><img src="https://cdn.pbrd.co/images/HxetzHh.png" alt=""></p>
<p>My try:</p>
<p><img src="https://cdn.pbrd.co/images/Hxet54C.jpg" alt=""></p>
<p>a) Pretty confident with it.</p>
<p>b) I have been thinking about 'counterclockwise 45 degrees with respect to z axis means that you look along the +z direction from the origin. The plane of the loop then lies along the $-(\hat{i}+\hat{j})$ direction so μ and B are perpendicular. Torque is now a maximum' and I do not have a clear idea yet. I think we are in a situation liken to the one depicted here:</p>
<p><img src="https://cdn.pbrd.co/images/HxDaFrd.jpg" alt=""></p>
<p>But here as you can see μ (normal vector) and B are not perpendicular. Could you explain this with more detail?</p>
<p>Thank you</p>
Physics Problemshttp://physics.qandaexchange.com/?qa=2950/dipole-moment-and-torque-on-a-rectangular-wireThu, 02 Aug 2018 01:11:41 +0000Finding current through resistor as a function of time in the presence of 2 inductors
http://physics.qandaexchange.com/?qa=2940/finding-current-through-resistor-function-presence-inductors
<blockquote><p><img src="https://cdn.pbrd.co/images/HwwoDme.png" alt=""><br>
After long time switch is shifted from position 1 to position 2. Find $i$ through the resistor as a function of time. </p>
</blockquote>
<p>Attempt: </p>
<p>I know that the total inductance would become $L+ 2L= 3L$. But after that I am unable to understand the mechanism that is happening. So basically, since "long" time has passed, the current is constant in circuit and equal to $\dfrac{\varepsilon} R$, also $\dfrac{d\phi}{dt}$ through L is 0. </p>
<p>Now, when switch is shifted, there's flux change through both the inductors. Both will develop polarity with positive part on the same side as the positive side of the battery </p>
<p>From KVL,<br>
At any instant when current is $i$<br>
$\varepsilon = 3L\dfrac{di}{dt}+ iR$</p>
<p>I am just confused about the limits of current during integration. To make the function we will take the limt=its from $i_o$ to $i$ but how to find $i_o$. That's my question. </p>
<p><img src="https://cdn.pbrd.co/images/Hwyry5O.png" alt=""></p>
Physics Problemshttp://physics.qandaexchange.com/?qa=2940/finding-current-through-resistor-function-presence-inductorsSat, 28 Jul 2018 05:23:43 +0000