Sample ProblemsDynamicsForces between charges*Two small objects each with a net charge of Q (where Q is a positive number) exert a force of magnitude F on each other. We replace one of the objects with another whose net charge is 4Q.

(i) The original magnitude of the force on the Q charge was F; what is the magnitude of the force on the Q now?(a) 16F(b) 4F(c) F(d) F/4(e) other

(ii). What is the magnitude of the force on the 4Q charge?(a) 16F(b) 4F(c) F(d) F/4(e) other

(iii) Next, we move the Q and 4Q charges to be 3 times as far apart as they were. Now what is the magnitude of the force on the 4Q?(a) F/9(b) F/3(c) 4F/9(d) 4F/3(e) other

(iv) In the original state (2 charges Q) if the symbol Q were taken to have a negative value, how would the forces change compared to the original state?(a) stay the same(b) both would reverse(c) left one would reverse(d) right one would reverse(e) none of the above.

With a grain of saltOur knowledge of electric forces between charges gives us the opportunity to begin to understand the forces that hold matter together. As a simple example, consider a crystal of salt - Sodium Chloride (NaCl). Each molecule of salt separates into a positive sodium ion (Na+) and a negative chloride ion (Cl-). In a crystal of salt, these ions are arrayed alternately in a three-dimensional cubical array as shown in the figure at the right. We ask whether or not the electrostatic forces between the ions might be able to hold it together.

To simplify our analysis for the purpose of this problem, consider a single plane of this 3D structure. Let's look at an edge of one of these planes, shown in the figure at the left. We will analyze whether in the indicated arrangement, the net electrostatic force on a charge on the edge of the crystal is into the crystal (holding the crystal together) or out of the crystal (tending to break the crystal up).(a) First consider the force between an ion on the edge and one of its neighbors on the edge, for example, the sodium ion labeled A and the chloride ion labeled B. If the charge on the ions each have a magnitudee= 1.6 x 10-19C and the lattice spacing isd(the distance between neighboring ions along the indicated lines), write an expression for the magnitude of the force between neighboring ions,F0. Express your answer in terms of the symbolse, d, and whatever constants you require

(b) Now consider the attraction on the sodium ion at A from its three nearest chloride neighbors, labeled B, C, and D. Find the magnitude and direction of the net force on A due to these three charges. Express your answer in terms ofF0for ease of interpretation.(c) Then, consider the repulsion on the sodium ion at A from its two nearest sodium neighbors, labeled E and F. Find the magnitude and direction of the net force on A due to these two charges. Again, express your answer in terms ofF0for ease of interpretation.(d) We only considered nearest neighbors. Why do you think we did this? Do you think this is a reasonable approximation? Give a brief justification for your answer.Analyzing dipolesLots of things that are electrically neutral overall have one side that's electrically negative and one side positive (a water molecule, for example). We call such things "electric dipoles," and we can model them as pairs of particles of charge +qand -q(whereqis a positive number) separated by a distanced. Usually,dis a very small distance. (For water it would be around 10-11m, thinking of a few protons worth of charge on one end - about 6x10-19C -- and a few electrons worth at the other.) Furthermore, because of the magic of quantum mechanics, in many molecules it behaves more like a rigid rod than like a soft spring. So we can treatdas a fixed distance.*Suppose you have a dipole that's free to move in any way (including rotate - imagine it floating in space). And there's an object with chargeQa distanceraway. That distancerwould be much larger thand, the distance between the charges of the dipole, so we draw the dipole small.

a) Consider the forces between the chargeQand the dipole. If the dipole is free to rotate: Would the dipole end up attracted to the charge, repelled, or neither? Would the chargeQbe attracted, repelled, or neither to the dipole? Do your answers depend on the sign ofQ?Think of the dipole as lined up the way it's drawn below, so the -qend is a distancerfromQ, and the +qend is a distancer+d.

b) Write a formula for the total force on the dipole by the chargeQ.c) If you were to double the distancerbetweenQand the dipole, what would happen to the magnitude of the force? Would it double? Cut in half? Something else?You could work on this in a couple of ways; pick whichever one you prefer. One way would be to try numbers. You could use the water molecule if you like,q=6x10-19C andd=10-11m. A reasonable amount of charge forQmight be 10-6Coulombs, andrcould be 1/2cm=5x10-3m. FindF, and then doubler. Or you could make up your own numbers - you could pick easier numbers - just be suredis much smaller thanr.The other, more advanced way is to simplify the formula you found in part b). A trick to make that easier is to ignore the termd2you'll get along the way. Remember thatdis a very small number compared tor, and sod2is a very very small number compared tor2. When you're adding numbers, and you know one is tiny compared to the others, you can often ignore it. (As in 1,000,000+2 is about equal to 1,000,000.) So - at one point you'll have (r2+2rd+d2), and you can ignore thed2- just cross it out. (Don't ignore the 2rdterm, though! Why not?)

* Water doesn't actually look like a pair of charges separated by a distance. It's more like a pair of "Mickey Mouse ears" with the oxygen atom being the head and the hydrogen atoms being the ears. The hydrogen atoms have excess positive charge and the oxygen has excess negative charge. It turns out that the difference between a "two-ball-and-stick" dipole and "Mickey Mouse ears" becomes irrelevant to the electric field or forces calculated once the distance to the molecule is more than a few molecular diameters.

Crawling AmoebasIn the picture at the right is shown a frame from a video in which a group of amoebas are spread at random on a glass slide. A small needle introduces a bit of a chemical that attracts the amoebas. As the chemical spreads through the water in which the system is immersed, the amoebas begin to move towards the source of the chemical and are soon moving at an (on the average) constant velocity. We will build a model about what forces are responsible for the amoebas horizontal motion. (Ignore gravity.)

Our simplest biological model is that the amoebas respond to a difference in the density of the chemical (more on the side near to the needle, less on the side farther away) and do something in response. What? Here are two hypotheses: H1. The amoebas crawl along the glass slide, pushing along. H2. The amoebas move off the glass and swim in the fluid in which they are imbedded.A. In order to decide which hypothesis makes sense, start by drawing a system schema that would allow either hypothesis. In addition to your usual labels, mark interactions that would only be present in hypothesis 1 with an H1 label and interactions that would only be present in hypothesis 2 with an H2.B. Now draw two free-body diagrams for a moving amoeba; one assuming hypothesis H1 and another assuming hypothesis H2. Be sure to label each force in your diagram identifying the type of force, who is causing the force, and who is feeling the force. Under each diagram indicate if any pairs of forces are equal and give a reason why you think they are equal.C. The experiment is repeated with a glass slide that is essentially frictionless. In this case the amoebas do not move towards the needle. Tell which hypothesis you think this supports (if either) and explain why you think so.

The stooping hawkHawks and gannets soar above the ground and, when they spot prey, they fold their wings and essentially drop like a stone. They have evolved a highly aerodynamic shape that lets gravity build up their speed without having to make the effort of trying to fly at a high speed. (See the figure of a diving hawk at the right. The technical term for this maneuver is stooping.) For this problem, you may approximate the strength of the gravitational field as g = 10 N/kg.A. If a hawk is slowly soaring at a height of about 150 meters and spots a vole on the ground, folds its wings and begins its dive, it will simply accelerate downward with the gravitational acceleration, 10 m/s2. With what speed will it be going when it gets to the ground? (Of course, it has to turn a bit above the ground in order not to crash. We will ignore this part of its flight path.) B. If the vole on the ground has been nervously watching the hawk, how much time does it have to get away from the instant it observes the hawk pulling in its wings? C.If the vole can run at a speed of 2 m/s and always wants to be safe, how close to a safe hole should it always stay?D. What assumptions did you need to make to solve this problem? Is it reasonable to make them?

The flying squirrel and the water fleaTypically when we describe falling objects in a physics class, we say ignore resistive forces.

In this problem, well estimate which resistive force dominates for different objects. If we have a sphere moving in a fluid of density,dfluid, and the object has a radius,R, then it will experience two resistive forces, drag and viscosity as given by the equations:

For air and water, here are the values of the parameters that occur in these equations.Density (d)Drag coefficient (C)Viscosity ()

Air1.2 kg/m30.202.0 x 10-5N-s/m2

Water1.0 x103kg/m30.250.8 x 10-3N-s/m2

Since we are only interested in the approximate scale of the forces, we will model our objects by spheres,even though they are not really spherical. (Corrections can be expected to be less than a factor of 10.)A. Consider the larva of a daphnia, a small crustacean, shown at the right. It lives in water and at certain times in its lifecycle, it goes dormant and sinks passively to the bottom. Model it as a sphere of radius=0.3mm, density=1.3g/cm3, and mass=0.3 mg. Typical observed passive sinking velocities are on the order of 2mm/s. Draw a free body diagram showing all the forces acting on the daphnia larva as it sinks, being careful to label each of the forces. Calculate the ratio of the drag to the viscous force,

R=Fdrag/Fviscousand tell which force is more important for the passive sinking of the larva.

B. As our second example, lets consider a northern flying squirrel, shown in the figure at the right. For this problem, we will model it as a sphere (!) of radius=15cm and mass=200g. Typical observed velocities are on the order of 50 cm/s. In the space below calculate the ratio of the drag to the viscous force,

R=Fdrag/Fviscousand mark which force is more important for the falling of the squirrel. Show your work.

Comfort with forcesA biology student who has never taken college physics recently said to me, "One of my first impressions of forces was that they were completely made up. I recognized that everything in science is made up. But with forces I felt a real lack of correspondence to 'real stuff' or 'real processes' in the world. Something about forces seemed suspicious - it still does." Do you agree? Disagree? If you agree, discuss how your sense of "forces in a physics class" is different from your everyday experiences with forces. If you disagree, explain why and how.Estimating chargeA. When a DNA molecule is in water, each base pair grabs a pair of electrons from surrounding water molecules, releasing a pair of hydrogen ions. As a result, the DNA molecule has a net charge. On the web, find a reliable site (and say why you think it is reliable) that tells you the number of base pairs in a typical human chromosome. (Not the Y chromosome!) To get a sense of the charge involved, imagine that you had two coiled up chromosomes, each with a charge of 2 extra electrons per base pair. Suppose you held them fixed in a vacuum one micrometer apart. For simplicity, model the chromosomes as point charges. Estimate the electric force between the two chromosomes in this example. Explain why this kind of electrostatic repulsion is not a problem when DNA is in its natural environment.B.We know that within the limits of measurement, the magnitudes of the negative charge on the electron and the positive charge on the proton are equal. Suppose, however, that the magnitude of the charge on the proton was bigger than that on the electron. On the web, find a reliable site (and say why you think it is reliable) that tells you the number of electrons and protons in a copper atom and the mass of a copper atom. From this data, estimate the total number of protons and electrons in a penny, and estimate the net charge there would be on a penny ifthe charge on the proton was bigger than the charge on an electron by 1 part in a million.With what force would two copper pennies then repel each otherif they were held one meter apart?Since the size of a penny is small compared to one meter, we can treat the pennies as point charges.Since pennies do NOT repel each other with any detectable force, what can you conclude about the charges on a proton and on an electron?

C. You should have found that the force between the DNA molecules was very small and the force between the pennies was very large. This might seem surprising at first, since the number of base pairs (and therefore the number of electrons) was so large, while the discrepancy between the charges assumed in part B were so small. From the calculations you have carried out, figure out WHY the result in part A is so much smaller than the result in part B and explain why the result actually makes sense.You may take the Coulomb constant to bekC ~ 9 x 109N-m2/C2.

Water coat forcesAn important part of the functioning of a biological membrane is its ability to selectively pass either sodium ions (Na+) or potassium ions (K+) through channels in the membrane. Because these ions have the same charge (+e), the electric force exerted on them by the membrane will be similar.In addition, they are about the same size (RNa+~ 0.12 nm, RK+~ 0.15 nm).So how can channels in the membrane distinguish them?One mechanism that has been proposed to account for the fact that the membrane channels treat these two ions differently is the suggestion that the ions attract water molecules providing a sphere of water that magnifies the small size difference between the ions. This coating of water is illustrated in the figure at the right.

To actually calculate the size of the ions water coat is difficult. (for example, we would have to use quantum mechanics to include a repulsion force that keeps the atoms from getting too close to each other.) But we can get a first idea of what is happening by exploring the electric force between the ion and one water molecule as shown in the figure at the right.We label the ion A, the oxygen in the water B, and the hydrogen atoms in the water C and D.

(a) To simplify the calculation, well make a simple physics model take the ion and the two hydrogens as each having a charge +e, while the oxygen has a charge -2e. Treat each of these as point charges. On the diagram below, draw arrows indicating the direction and relative magnitude of the force the ion exerts on each of the three atoms in water, B, C, and D.

(b) Assuming that atoms of the water are held together in the arrangement shown without collapsing onto a point (quantum mechanics prevents them from getting too close to each other!), is the net force that A exerts on the molecule (B+C+D) attractive or repulsive? Why do you say so?(c) If the grid spacing in the figure above is 1 nm, calculate the magnitude of the force that the ion (labeled A) exerts on the water molecule in picoNewtons (10-12N) by adding together the force it exerts on all three atoms of the water. (Remember that forces add asvectors!) How does the force that the water molecule exerts on the ion compare to this?

(d) We would like to find not just the force on the water molecule for one particular position, but as a function of the distance between the two. Suppose the parameters of the size of the water molecule areaandbas shown in the figure at the right. We want to find the force between the water molecule and the ion as a function ofx, depending on parametersaandb. First findr, the distance between the ion and the hydrogens as a function ofx, a,andb. Then find the magnitude and components of the force the ion exerts on each of the water atoms. Then, taking components, find the net electric force that the ion, A exerts on the whole water molecule (B+C+D) as a function ofx, a,andb. You might find it useful in taking components to define an angle and use sines and cosines to get components. But when you have put everything together, express your sines and cosines in terms ofx, a,andb.

The farmer and the donkeyAn old Yiddish joke is told about a farmer in Chelm, a town famous for the lack of wisdom of its inhabitants. One day the farmer was going to the mill to have a bag of wheat ground into flour. He was riding to the mill on his donkey, with the sack of wheat thrown over the donkey's back behind him. On his way, he met a friend. His friend chastised him. "Look at you! You must weigh 200 pounds and that sack of flour must weigh 100. That's a very small donkey! Together, you're too much weight for him to carry!" On his way to the mill the farmer thought about what his friend had said. On his way home, he passed his friend again, confident that this time the friend would be satisfied. The farmer still rode the donkey, but this time he carried the 100 pound bag of flour on his own shoulder!

Our common sense and intuitions seem to suggest that it doesn't matter how you arrange things, they'll weigh the same. Let's be certain that the Newtonian framework we are developing yields our intuitive result. Analyze the problem by considering the following simplified picture: two blocks resting on a scale. One block weighs 10 N, the other 25 N. In case 1 the blocks are arranged on the scale as shown in the figure on the left In case 2 the blocks are arranged as shown on the right. Each system has come to rest. Analyze the forces on the blocks and on the scale in the two cases by isolating the objects -- each block and the scale -- and using Newton's laws, show that according to the principles of Newton's laws, the total force exerted on the scale by both blocks together must be the same in both cases. (Note: It's not enough to say: "They have to be the same." That's just restating your intuition. We need to see thatreasoning using only the principles of our Newtonian frameworkleads to the same conclusion.)Forces swimming on a parameciumA paramecium swimming through a fluid is moving at approximately a constant velocity as a result of wiggling its cilia. What can you say about the various forces that are being exerted while it is doing this? Complete each of the sentences below by filling in the blank with the one of the following symbols that is closest to being true: >> (significantly greater than), > (a little greater than), = (equal to), < (a little less than), ), is equal to (=), is less than (