
Science
In general, the contribution to the total electric field `E` from an infinitesimal piece of charge `dE` is `dE=(k*dq)/r^2` The total field is then calculated by integrating over all the charge:...

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The Bohr formula gives `E=E_0/n^2` for the nth excited state with `E_0=13.6 eV` being the ground state energy. The relationship between energy and wavelength is `Delta E=hc/(Delta lambda)` `Delta...

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We are going to use conservation of angular momentum for this problem. We are going to need the moment of inertia for a rod rotating about its end. I will derive it here using the formula for the...

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Remember the definition of the commutation relation `[a,b]=abba` Preform this operation on `[S_x*S_y,S_y]` `[[S_x*S_y,S_y],S_z]=[[S_x*S_y*S_yS_y*S_x*S_y,S_z]` The following is the commutation...

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The speed of a point on the disk at a distance `r'` for `0lt= r' lt=r` from the axis is `v=omega r'` , so the force per unit charge is `f_(mag)=v xx B=omega r' B` radially outward. The emf is...

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Use the Lorentz force on a charge `Q` , when `Q` moves an amount `dl` . Then we also know that work is the dot product of force and distance. `dW_(mag)=F_(mag)*dl=Q(v xx B)*dl` But `dl=v dt`...

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For a continuous potential to equation is `V=int (k*dq)/r` Where `k=1/(4pi epsilon_0)` , `dq` is an infinitesimal piece of charge, and `r` is the distance away `dq` is from the point we are...

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First use conservation of energy to find `v` at the bottom of the ramp when there is no friction. `E_i=E_f` `mgh=1/2mv^2` `v=sqrt(2gh)` Now plug into a kinematic equations to find `t` after the...

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In this solution, I am going to assume that the cylinder is fixed so its center of mass can not move, even though this is not explicitly stated in the problem. This means the cylinder will not be...

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The wheel will roll up if the torque due to the horizontal force `(tau_F)` exceeds the torque due to the force of gravity `(tau_g)` . `tau_g=Mgsin(alpha)=Mg[sqrt(R^2(Rh)^2)/R]`...

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When the warmer objects (pieces of iron and copper) are placed in contact with the colder object (water), there will be a transfer of heat so that the iron and copper will cool off and the water...

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This is a tricky problem because we do not know what the mixture of steam and ice look like after they come to the thermal equilibrium. Since there is such a large amount of ice (8 kg, compared to...

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In this situation, we ignore the heat loss between the calorimeter and the environment. Then the heat that an unknown solid lost is the same as the heat that copper and water gain. Denote the heat...

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The force on block 1 from the first spring is `F_1=k_1x` Where `x` is the position to the right of the equilibrium position. Let y be the distance to the right of block 2's equilibrium position....

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The equations of motion is to find the EulerLagrange equations. These equations have the form: `d/dt (dL)/(d (dq)/(dt))=(dL)/(dq)` First, `(dL)/(d (dq)/(dt))=(d)/(d (dq)/(dt)) [ 1/2 A...

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ATP is short for adenosine triphosphate. It is the universal unit of energy used by biological cells, and it is produced in the mitochondria. When a cell needs energy, it gets it from ATP. The...

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The quick answer is that you need conditions where (1) females mate with multiple males, (2) different variants of the trait must result in different degrees of reproductive success, and (3) these...

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There isn't a definitive number of years that it would take for the environment to return to "normal" if all usage of fossil fuels stopped immediately. There are too many variables at work to give...

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Antibacterial soap has risen in popularity since the 1980's. It comes in many forms, but most commonly comes in liquid soap with the added ingredient of triclosan, which is an antibacterial and...

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As noted by Olivier Walusinski: "Yawning is a stereotyped, and often repetitive, motor action characterised by gaping of the mouth accompanied by a long inspiration of breath, a brief acme, and...

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It is important to make sure that when conducting a controlled experiment to test an hypothesis, all the variables are the same in two test groups except for the independent variable. In a...

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This might seem like something from a science fiction story. But there is actually strong evidence that plants respond to sound. In one intriguing study, researchers recorded the sound of a...

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There seems to be little consensus among the scientific community regarding the subject of birth order. Some experts believe that birth order merely decides personalities while others believe that...

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The term "sleep cycle" refers to the sequence of stages we pass through as we sleep. Typically, a sleeper repeats the cycle several times each night. Sleep cycles have been observed in many species...

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The magnitude of the Poynting vector `S` can be found using the following relation. `S=1/mu_0(E xx B)` `E` ands `B` are perpendicular here so `sin(theta)` in the cross product is one. Therefore,...

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We will use conservation of energy to solve this problem. We need to consider the rotational energy of the cylinder and the translational energy of the center of mass. `E_i=E_f`...

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The magnetic flux through the loop is the magnetic field times the component of the area vector that is parallel to field. `Phi_B=B_0*A cos(theta)=B_0*L^2 cos(omega t)` This generates an...

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Hello! The mass conservation law will help us in this problem. Denote `rho_1` the known density of 95% sulfuric acid, `rho_2` the known density of 40% sulfuric acid (the given numbers are in...

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When a cell replicates, it duplicates its genetic material called the DNA to yield two daughter cells containing exactly the same genetic information. The DNA is a doublestranded molecule. When a...

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Significant figures can be annoying, but the fundamental principle underlying significant figures is a very important one: We don't want to lie about the precision of our results.Significant figure...

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The colossal impact of global warming is particularly evident when we consider that 2016 marked the boreal ice cap reaching its second lowest measured amount of ice ever seen without any direct or...

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Probably because there's more "dark" matter than there is "regular" matter! We call it "dark" because we can't see it, nor can we clearly pinpoint its location by any other means; but our best...

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First, we do not know precisely how many galaxies there are in the universe. There are several reasons why a precise count cannot exist with current technology and funding. Our way to determine the...

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A writer named Rowan Jacobsen recently wrote and posted an obituary for the Great Barrier Reef on a website known as Outside Online. Most experts agree that the Great Barrier Reef is not dead, but...

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Contrary to popular belief, diamonds do not actually from coal, at least not usually. Coal is a relatively recent phenomenon in geological terms, as it required land plants to live, die, and then...

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Chromatophores are a type of cell (or group of cells) which are capable of reflecting light and which contain pigments. The functioning of these cells vary from one organism to another. For...

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Use the relativistic doppler shift for light. `f_r=sqrt(1+beta)/sqrt(1beta)*f_0` where, `f_r` is the received frequency,` f_0` is the original frequency, and `beta=v/c=(0.6 c)/c=0.6`...

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This is a magnetic dipole. A magnetic dipole in a vacuum radiates an average power by the following relation. `ltPgt =(mu_0 m_0^2 omega^4)/(12 pi c^3)` `mu_0` is the magnetic permeability of a...

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The mechanical work done on this system is going to be equal to the change in energy of the system. The energy stored in the electric field of the capacitor is: `E=1/2 CV^2` A dielectric also...

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The capacitance per unit length is `C/L=Q/(VL)` . So lets find the voltage of this configuration. In order to do that use Gausses' law to find `E` from the charge distribution. Make a Gaussian...

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After the switch is closed, the charge on the capacitor decreases as `Q(t)=Q_0e^(t/(RC))` The initial energy of the capacitor is `U=Q_0^2/(2C)` So we want to solve for a time such that...

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This is essentially the problem of the simple pendulum applied to a current loop. The dipole moment p will want to align itself to an external magnetic field. The torque is: `tau=m xx B=iaB...

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The magnetic field produced by a current carrying wire is: `B=(mu_0 I)/(2pi r)=(mu_0 I)/(2pi d)` Then use the lorentz force law per unit length is `F/L=q/L(v xx B)=lambda (v xx B)=(lambda*v xx...

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Here we need the conservation of momentum law. I suppose that before the explosion the block was in rest thus its momentum was zero. The law states that after the explosion the total momentum of...

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There are 4 forces acting on the block:gravitational force `mg` downwards,the force `F_0` downwards,the reaction force `N` upwards,the friction force `F_f` horizontally. The net force `F_(n et),`...

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There are three forces acting on the lead cube:gravitational force with the magnitude `Mg` downwards,buoyant (Archimedes') force `mg` upwards,lifting force of unknown magnitude `F` upwards. Here...

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I will solve this problem in a general way that would work for any one body parabolic, elliptic or circular orbits. The total energy of an arbitrary classical orbit is derived from its Lagrangian....

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We must apply conservation of relativistic 4momentum, `P^(mu)=[E_(t o t)/c,p]` . Where` p=gamma*mv` is the relativistic spatial momentum. `P_i^(mu)=P_f^(mu)` `P_(i,1)^(mu)+P_(i,2)^(mu)=P_f^(mu)`...

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Use conservation of linear momentum. `P_i=P_f` `mv_(m,i)=m v_(m,f)+M v_(M,f)` (1) `m(v_(m,i)v_(m,f))=M v_(M,f)` Use conservation of kinetic energy since this collision is fully elastic....

Science
Every substance is comprised of molecules, atoms, and/or other subatomic particles of that substance. These particles are in constant motion, colliding with each other and perpetually moving. These...