
Science
We can use the definition of entropy change to find the change in entropy of the liquid water as it freezes. The change of entropy of the water is given by: `DeltaS=Q_(rem)/T` Where the heat...

Science
The efficiency of a heat engine is the ratio of work done per cycle `W` to the heat absorbed from the high temperature reservoir `Q_h` . The efficiency epsilon is therefore: `epsilon=W/Q_h` The...

Science
We can express the rms speed of Argon atoms by combining the ideal gas law with the equation for the root mean square velocity of an atom. The ideal gas law is: `PV=nRT` The rms velocity of a...

Science
There are two main forms of nuclear energy, fission, which involves splitting atoms apart, and fusion, which involves releasing energy when 2 atomic nuclei are forced close together and fuse into a...

Science
The de Broglie relation is `eq. (1) >` `P=h/lambda` Now get the momentum `P` in terms of the kinetic energy `K_E` . `K_E=1/2 mv^2` `K_E=P^2/(2m)` `eq. (2) >` `P=sqrt(2mK_E)` Therefore the...

Science
The change in the wavelength, `Delta lambda` , is given by the famous Compton formula. `Delta lambda = h/(m_e c)(1cos(theta))` Now plug in the values for the electrons mass `m_e` , the speed of...

Science
The work function of a metal `phi` , the wavelength of the incident photon lambda required to eject an electron with the maximum kinetic energy `K_(max)` is expressed by Einsteins photoelectric...

Science
This can be solved by applying Poiseuille's law. `Delta P=(8 eta L)/(pi r^4)*I_v` where: `eta` = viscosity `L` = length of the pipe `r` = radius of the pipe `I_v` = volume flow rate Therefore...

Science
The pressure applied to an enclosed liquid is transmitted undiminished to every point in the fluid and to the walls of the container. Hence we can equate the pressure produced by the force applied...

Science
The pressure due to a column of height `h` of a liquid of density `rho` is given by its weight per unit area. The pressure as a function of height is then: `P(h)=W/A=(mg)/A=((rhoV)g)/A=rho g h `...

Science
We can use the continuity equation to compare the flow rates at the two locations. `A_A*v_A=A_B*v_B` This equation relates the area and velocity of pipe `A` to that of `B` . Solve for `v_A` ....

Science
The water level in the pond will fall slightly. When the anchor is in the boat, the boat displaces enough water so that the buoyant force on it equals the sum of the weight of the boat, your...

Science
The attached diagram shows the displacement of the pendulum bob, through an angle theta, as a consequence of the complete absorption of the incident radiation. Now use conservation of mechanical...

Science
We can use the definition of pressure to express the radiation force on the balloon. We'll assume that the gravitational force on the balloon is approximately its weight at the surface of Earth,...

Science
The static force opposes the motion of the object, and the maximum value of the staticfriction force is proportional to the normal force `F_n` . The normal force is equal to the vertical...

Science
First lets set up our coordinate system. Let the xaxis follow the string and have it be +x up the ramp for block 1 and +x for block 2 be downward. We will then have the yaxes for block 1 be...

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:...

Science
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...

Science
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...

Science
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...

Science
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...

Science
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`...

Science
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...

Science
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...

Science
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...

Science
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]`...

Science
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...

Science
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...

Science
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...

Science
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....

Science
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...

Science
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...

Science
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...

Science
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...

Science
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...

Science
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...

Science
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...

Science
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...

Science
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...

Science
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...

Science
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,...

Science
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`...

Science
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...

Science
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...

Science
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...

Science
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...

Science
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...

Science
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...

Science
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...

Science
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...