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Points) large - number of electrons are removed from each of nwo small spherical balls of Styrofoam (an insulator) which are attached to identical lengths of insula...

Question

Points) large - number of electrons are removed from each of nwo small spherical balls of Styrofoam (an insulator) which are attached to identical lengths of insulating, massless string: 'The other ends of the srong are then both attached t0 the same spOt neutral wall and are balanced in cramc equilibrium as shown Assume the magnitude of the electric force is half that of the gravitational force on ball Draw an appropriately scaled and accurate free body diagram for ball number 2 Include an

points) large - number of electrons are removed from each of nwo small spherical balls of Styrofoam (an insulator) which are attached to identical lengths of insulating, massless string: 'The other ends of the srong are then both attached t0 the same spOt neutral wall and are balanced in cramc equilibrium as shown Assume the magnitude of the electric force is half that of the gravitational force on ball Draw an appropriately scaled and accurate free body diagram for ball number 2 Include and label all relevant forces



Answers

Two hard rubber spheres, each of mass $m=15.0 \mathrm{g}$ are rubbed with fur on a dry day and are then suspended with two insulating strings of length $L=5.00 \mathrm{cm}$ whose sup- port points are a distance $d=3.00 \mathrm{cm}$ from each other as shown in Figure P23.69. During the rubbing process, one sphere receives exactly twice the charge of the other. They are observed to hang at equilibrium, each at an angle of $\theta=10.0^{\circ}$ with the vertical. Find the amount of charge on each sphere.

So this question is asking us to determine uh to show that that the distance between the two, the horizontal distance between the 22 masses is approximately equal to X. This term here actually with cuba of 22 K. Q squared over N. G. Uh If we take the approximation that theater is small, so tan Nathi is approximately equal to sign of theater. And then we're also asked to find the charge given that led to 1.2 mm 0.01 kg. And actually get 0.05 kg, 0.05 m. So this is really we're gonna just analyze the forces that are all in this question to make a start on this. So let's look at the diagram. So all right, let's draw. So we have our we have one mass here. So we have weight since it's it's a massive particle. So energy going downwards. Then we also have the tension of the stream growing up in this direction. T. Yeah, and the sun angle of tita here. So then we also have that we also have the repulsive force. So if the other charges here, the repulsive force to be in this direction due to the electric force recruiting force. So F. E. In this direction. So then let's write our force equations that we get from this then. So we have in the in the X. Direction. So we're going to have the set up the axes like this why an X. To the extraction we have just so uh we have in the positive direction so we're gonna have a component of the tension in the string so that that's going to be tee times. That's the magnitude of the tension times. So we're going to have a sign of theaters and signs the X. Component. So T. Sign of theater minus F. E. It's equal to zero since we're told that it's a it's a stable the two masters aren't moving there in a stable position. Then in the why you're actually we have that Tea castle theatre. This is the component, the Y. Direction minus MG zero. So we're told that we were interested we can use the approximation to sanity because it's approximately society. So we want to get rid of the cost of the terms. So that's uh so and and we also want to get rid of cost to a charming but we also need to get rid of the tent that attitude the tension since we're not giving that either. So let's get find what tears from from this equation. So let's so T. Rearrange this we get the T. Is equal to MG. All over cost of theater MG over across the theater. And then so then now we have this expression for tea. We can sub this, substitute this back into the first equation. So the first equation is also so we can get an expression for what F. S. Or F. E. Legal to T. Sign the theater said something in for tea. We got that at M. G. Castle Theatre, my sine of theta. So then the secret to MG. Tanner theater. So then since we're told that we can approximate tanner theater with site with side of theater, we want to we need to determine mortality is equal to So we need to find this this this distance why to get the tan ratios of X over two over why? So that's good that then. So we have. Yeah. Yeah. So we know that 10 of theater is you throw uh X over wide or X over two over wide. Since it's the half of the X. Distance to speak with the X over two, although for white so we can use pythagoras here and find why? So elsewhere is equal to X over two squared plus why square? So then that means that why is equal to the square root? L squared minus X squared over four. And of course if we're assuming small X. So if we if X is small, so X is much smaller than uh X is much smaller than one, this implies that X squared Is approximately equal to zero. So then we're gonna take that approximation to them. We get that Y is approximately equal to just so it's gonna be root elsewhere now since X squared is zero. So that's just uh so now we get that using this approximation we got the talent theater is approximately equal to X over chua excellent you out. And this is approximately equal to then sign of theater. And the reason that we can do this is since since Tana theater is approximately equal society to, that implies that X is a very small number. So that then we can take this approximation. So then if we if we substitute this back into our equation for the electric force, so F E. Um actually the electric force equals which we know is equal to the same charge. So the coolant constant K bike, youth words since the same constant by the separation distance, that's X word and that is equal to MG time a theater which is equal to MG X over two out. Yeah, so then rearranging this to solve for X, we get the X cubed is equal to two K Q squared uh over MG. And this is an approximately equal society should all be approximations so, and then we get to this then this implies that X. Is approximately equal to take the cube root of this. So two K. Cute squared L. All over MG. To the power of 1/3. So that that proves part eight the property. We're asked to find the charge given some initial conditions. So crappy. So rearranging this expression to find charge. We got the ac. So so rearranging this expression actually to find the charge, we can bring everything over. So then we get the Q squared. Is he is approximately equal to ah execute X cubed MG all over to kate. Oh then if we only get the Q. Q. Of this. So you want the charts so that it's just taking the square root of both sides. We get the charity department equal to this. And we're giving some figures sub students or we're told us the signal to square. so X is equal to 0.05 m. Uh huh. M is equal it. And that's cubed is equal to 10 g. That 0.01 kg. And we're talking we know at GSGS 9.8 9.8 and the Silver two. And we know that the K is equal to 8.99 by 10-9. And it is equal to 1.2 m in this case. And then we get this is equal to uh, 2.38 x 10 to Mice, 2.38 x 10 to the -8 cool loans. So this is our answer for part B.

Hi. In the given problem Land of the threads, which are joined at the same point from the upper edge of the trails and their lower ends are joined the two boats. No, the length of these threads has been given US 0.25 m. Suppose they are joined at 0.8 here. This ball this is B and here this is C. The distance between them is B C, which is for the timing not known to us if you draw vertical lines, which separates the two wars at the midpoint between them. At this point, we d the angle between the two threads when a charge Q is given to both the balls. So suppose a charge given to the balls is skewed identical charge due to which using columns law, the walls will repel each other to the force. Electrostatic force of repulsion F e mass of the ball has given us 8.0 into 10 days par minus four kilogram. So here this is the weight of the ball mg. This is the tension p in that threat due to due to this identical charges for both the boss, the boss will repel each other. Then they will make an angle between them, which has given us 36 degrees. Yeah. So the angle made by one of the threads with the vertical means Tita is half of this 36 degree means this is 18 degree Hence, if this angle is set up, this angle made by the vertical component of the tension with that thread will also be theater. So this vertical component will be D'Costa and horizontal component will be See signed, Peter. No. Here in this triangle a B D, which is the right angle triangle in bright and gold triangle A B B If you want to find this BD yeah, this b d by a bee means perpendicular by hyper tennis is actually scientist to So this B d comes out to be a b into scientist to or we can say this is l scientific data. Yeah, this l is 0.25 and this is sign 18 degree. So the distance between the two walls becomes twice of BD means this is two times of 0.25 into sign 18 degree meters. Or we can say this is 0.5 and to sign 18 degree meter? No, as this small is an equilibrium. So equating the horizontal and vertical components because we know in equilibrium the net force along horizontal means along X axis and net force along via access. Both of them should be zero. So we can say this P scientist to should be equal to electrostatic force f and because theta should be equal to the weight of the ball. If you divide these three questions now we get tan theta is equal to f e electrostatic force by MG. So, using the expression for electrostatic force, Stan Theater comes out to be okay into Q Square by our square. Using Ghoulam's law divided by M G. Hence expression produce square consult to be M G into our square and Peter invited by. Okay, so finally, the expression for Q comes out to be are into square Ruto mg 10 theater divided by Kate So finally plugging in well known values here for our this is 0.5 and to sign 18 degree square root of mass, which is eight into 10 days. Par minus 4 kg into G, which is an appointed into 10 theta means 10 tangent of 18 degrees, divided by K, which is nine in 2. 10.9. So yeah, it becomes 0.16 And in the square root, this is 0.2 83 into end par minus 12. So it comes out to be 0.16 into 0.532 into 10. Dish par minus six. So finally the charge over the ball comes out to be 8.2 into 10 for minus eight column and here it becomes the answer for the first part of the problem. Yeah, this is the first part of the problem. Now, in the second part of the problem, we have to find the tension in that thread for which we will just use the expression because that I was too mg. The net vertical force is zero. So he comes out to be mg by casita for him again. This is it into tended bar minus 4 kg, 9.8 m per second squared for G and this is co sign off 18 degrees. So it becomes 78.4 into Danish, one minus four, divided by 6.95 So finally This is 82.5 in to tenders. Par minus for Newton or 8.2 into 10. Dish par minus three. No. Which becomes the answer for the second part of the problem. Thank you. Uh huh.

Here's an example of using Newton's second law with the electric force. It is a very common example using equilibrium in particular that the some of the forces on two suspended charges Has to Equal zero in Equilibrium. And of course the force on the two masses that are usually interacting is due to columns law. So the electric force in magnitude is K Q one Q 2 with an absolute sign around them. Over the separation squared The direction of that force is a long line joining the two charges. So here we have two charges, but they have different amounts on them. Uh but equal masses. Um So they should hang at an equal angle to the vertical as they interact if they are suspended by a string. And that is because of Newton's 3rd law which says that the force of electrical ah Interaction on one is equal and opposite the interaction on the other. But in addition because the masses are the same uh they need to be balance the same way by the tension in the string. Okay, so here I am going to draw a free body diagram. So the technique to use is to first come with a free body or forced diagram And then you can apply the summer force is equal to zero in a methodical way. So we will do a force diagram on the second mass with a slightly more charge on it. Um And it's got mt downwards. The electrical force is acting along the two points the way they are joined. And so we can put in the magnitude of that as well as the direction. And we'll work a little bit on the separation which will have a small uh change to it from the distance between the support points. And of course there's the attention from the string lifting up. That's probably a little bit longer that it needs to be because that tension um We'll see has to balance the other two. Uh huh. And we have a faded degree. The Y component is tension times co sign data along the Y. And tension sign of data along the X. And that is because the angle is typically given with respect to the vertical. Okay, now we're going to go through and solve for quantities in terms of symbols using our ballots equation and usually it helps make a little table and to write down X. And Y. Components of each of the forces in the diagram. So the MG is an easy one. Uh It's all why in the negative direction using the convention plus X to the right plus why to the positive up direction? Um The electric force is all X. Mhm. And I'll do a little aside and work on the R. For this particular situation. Um Sometimes the to support points are joined but here they are separated by a large distance D. But also include the basis, the fact that the strings are pushing a little bit further apart. So that is an added amount to the distance D. in the amount two times the length of the string time sign of data. Let me make sure I've shown the length of the string. Yeah in the diagram. Okay so yeah the R. Is a little bit bigger than D. And there was no why component. And finally the tension has two components. Um E uh huh. A negative X. Component and a positive why component. And now we can see how things are adding up to zero. Um Each of those components has a negative part at a positive part. And so there will be a balance. Um for the X. Equation the balance is Teesside data equals two Q squared. Let's put the key 1st. K. is the electrical constant in columns law sometimes just called 1/4 pi epsilon not. And the white balance is t co sign data equals mg mm. Okay now it may not be clear what the knowns are and the unknowns. So um we'll take a look at a case in trying to solve the system. So we'll take a case where we know the masses. The equal masses two masses Are each 15 g or zero 15 kg G. of course we do know the Theta is 10°. Um the big the big distance between them is three m. That's a large distance And they like that. The string is five cm or .5 m and are unknown. Is the cube. Oh so we can see working through this um we want to take the Y. Equation first and solve it for tension and then we can sub tension back into the X. Equation and saw for Q. It's kind of a process in algebra that we want to use. So let's go ahead and take the Y equation. Attention equals MG over cosign data. And we have the mass. Of course we have G. And we have, The data is 10° And not a big amount of force but 0.149 newtons. Okay, let's take the X. Equation and I'll play around with it. Um The R. Is the Plus two sides data and we want to square it and multiply by T science data. And if I buy two K. Kind of doing a cross multiplication as well as working out what are is and that should be cute squared. And to make things a little bit clearer, I think what I'll do is solve the are separate and then we can plug it in as a full are. So are is the separation between the two spheres And it's a little over three m. I keep forgetting the length of the string. Uh In fact I'll put that in green. So I know it is the ill from the diagram. Mhm. So that's a small amount to add And that separation is almost nearly three m. So plugging this in, I'll show an intermediate step just so that we're all on the same page here. K is the electrical constant, 1/4 pi Absolute zero is sometimes the way that is written, but it is a well known constant in S. I. Units. Um and that should give us the square up the charge and working that out, we get um a small number good and that should be in Coolum squared, so that's an intermediate number, taking a square root. We get of the order of micro columns, which is a reasonable amount. That's a lot of charge, but feasible to rub a small sphere and get that amount on.

And this problem. We have been given two identical balls, which have been given a charge. Q. For each of them turned, they're hugging from a single point through a threat, and the threads make an angle to resist. Agree to each other so that the angle that the balls, or rather, the threats that make with the horizontal axis 72. No, we have to find out the amount of charges in each of the balls. Do that. Let's focus on one bowl and let's look at the forces acting on this ball first, because but friend, there is a tension force that goes, man go out port, then series, of course, gravitational force and G tourist the bottom. And finally, there's repulsion force. Let's write that as a cue given by the other ball. Now all of this forces must balance each other out, which means the vertical component off the tension must back balance with the crab additional attraction, whereas the horizontal component must balance out the forced you to the church. No, because of the Duncan, right. If Q is because to tee times call sign 72 degrees and you can write MG because to tee times sign 72 Dick Prints Now taking this equation on this aggression and dividing them by one another. We see that we have. If Q over MG goes to co sign 72 degrees, defied it. But sign 72 degrees, which means we can write one. I took excellent times Square Bye. The distance are square. Where are is the distance between the two balls? Times want my mg close to one by 10 72 degrees. Now the distance are. If you try Aw, political from the center, you will see that half broke. This distance can be given by Cyril 0.25 which is the slender times. No sign of the sandwiches you're open to five times will sign 72 degrees, which means our is twice that length on this given with Europe. On five times we'll sign 72 tickets. So if we rearing its execution, we're left to it. Q. Because two square root for I said, I'm not times our score kinds mg diverted by tan 72 degrees. So when you can call it that place along with the others, we have one by nine times 10 to the core nine or one Upsell enough and then seal coin finds no sign. 72 degrees. We're square for our square. We have eight times 10 to the minus pool kg for the man's thank 9.8. Gravitational acceleration functioned, then divide it. Turn seven two degrees. If you conclude this, this turns out to be about 8.22 times turned to the bone minus eight columns. So this answers that first part off our problem. Second, put up the problem asks us to find this tension. Now we know that T signed 22. Different is given by M. G. So we'll just use that question too. Right? That team, It's M g, defended by Sign off 20 70 to decrease land that if you substitute that alums, this expression turns out to be 8.24 times 10 to the power minus three futons. And that's the answer to the second part of the problem.


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