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A 3-m pole is supported by a ball-and-socket joint at A and by the cables CD and CE. Knowing that the line of action of the S-kN force forms an angle $ =309 with th...

Question

A 3-m pole is supported by a ball-and-socket joint at A and by the cables CD and CE. Knowing that the line of action of the S-kN force forms an angle $ =309 with the vertical yz plane , determine (a) the tension in cables CD and CE, (b) the reaction at A_12mL2 m15 m5 kN2 m

A 3-m pole is supported by a ball-and-socket joint at A and by the cables CD and CE. Knowing that the line of action of the S-kN force forms an angle $ =309 with the vertical yz plane , determine (a) the tension in cables CD and CE, (b) the reaction at A_ 12m L2 m 15 m 5 kN 2 m



Answers

The $6-m$ pole $A B C$ is acted upon by a $455-N$ force as shown. The pole is held by a ball-and-socket joint at $A$ and by two cables $B D$ and $B E .$ For $a=3 \mathrm{m}$, determine the tension in each cable and the reaction at $A .$

For this problem on the topic off equilibrium of rigid bodies, we're told that a poll of a given length is acted upon by a force as shown where were given the magnitude of the force. The poll is held at Point a Bible. It's soccer joint and by two cables B, D and E. If we had given a value for a, we want to determine the tension in each cable as well as the reaction at point. A. So here we have a free body diagram and we have five unknowns with six equations of equilibrium. But equilibrium is maintained now. The some off moments about the poll M A C, is equal to zero. The vector R B is equal to three j as we've drawn it. Now diagram and R C is equal to six j. So these are the positions of BNC relative to the Origen. The other victors we have Firstly, CF is given in Victor notation by minus three I minus six j last two K and the victor has magnitude off 7 m. New vector B D is 1.5 hi minus P J minus three K. And this vector has magnitude E. D is equal to 4.5 m. Victor B e is 1.5 along the X direction, minus three along the wide election plus three along the K direction and this vector as magnitude off 4.5 m. So the end off e is 4.5 m. Now the force p a new labeled in diagram is equal to the magnitude of P multiplied by the vector CF over C V magnitude off C E magnitude of C f rather So the that is the direction off CF. So this is equal to the magnitude of P over seven in the direction off CF minus the I minus six j last two came the tension off BD team Edie is equal to the magnitude of the tension in the direction off BD So that's the Victor BD divided by the magnitude of meaty. So this is t BD divided by 4.5 times Director one 0.5 I minus three j minus me que and we can actually simplify this This becomes t baby over three into I my miss to j minus to king. Now the attention t v e is equal to mhm in magnitude T v e In the direction off e which is the victim e over the length off the which becomes t meaty or TVE Rather Uh huh. Cool G E over three into I minus man to J last two k. I mean, now, with all of our information, we can finally take the some off the moments about a and we know the some of the moments about point A must equal to zero. So the some of the moments about point A is our be crust would t B d and lastly vector R b cross with tension t meet you plus R C cross toe like the P and this was equal to zero. So now we'll use the method of co factors to solve this cross product. So I, j and K what here? So if we enter our victims in here at zero 30 crust with one minus two minus two RB cross T v d. And this is multiplied by T v D over three plus the cross product +030 which is R B crossed with TV which is one minus two to I used t meet you over three. That's R C Cross P, which is 060 minus three minus six and two times p over seven must give us zero. So let's look at the eye coefficients and we get from the I coefficients that minus two t baby plus two t e told me. Plus 12/7 times p must equal to zero. We'll call this creation one from the J coefficients. We get minus t be minus T e plus 18/7 times P is equal to zero. And we'll call this equation to now if we add these two equations. Equation one and the equation, too. Or actually, we take equation one and added to twice equation, too. We get minus four t, many less 48 over seven times p is equal to zero, which means we get an expression for TBD in terms of P. So T BB is equal to 12/7 times p. And if we go back into equation too, we get minus 12/7 p minus. T E plus 18/7 p is equal to zero, which means now we confined TVE in terms of P, so TV is equal to 6/7 times p. But we are given the magnitude of P. Yeah, we're told P is equal to 445 Newton's and therefore we find t b d to be 700 and 80 Newtons and t e is the from the expressions about is 390 mutants. So those are our to tension tensions as required. Now we have intentions. We want to find the reaction at a Okay, So if we take is some off all the forces we know the forces acting on this Poland equilibrium as equal to zero. That means that the tension off BD plus the tension t B in plus P plus the reaction a must equal to zero. Now let's take I coefficients for our forces. Yeah, and we get 780 over three less 390 over three minus 455 over seven times three plus the x component off. The Reaction X is equal to zero and so simplifying we'll get a X to be 195 Newtons. So here we have the x component of the reaction. Now let's take J coefficients, and we get minus 780 over three times to minus 390 over three times two minus 455 off the seven I'm six, plus the white component of the reaction A Y must be put to zero. And so similarly, we find the y component of the reaction A y to be 1117 Newton's Now. Finally, if we take take the coefficient of K, we get minus 780 over three times to plus 390 over three times two plus 455 divided by seven times two plus. The component of the reaction is that must equal to zero and begin we solved for the one unknown, we get a ZBZ component of reaction to be positive. 130 Newton's Yeah, and therefore we confined the victory action, a in unit vector notation to the minus 195 Newton's Times unit Vector I, plus 1100 and 17 Newton's times. You affected J Plus 100 and 30 Newton's Times unit Vector K, and that's our reaction vector

I am everyone. This is the problem based on system Duveen Equilibrium Fuck given system, Toby In equilibrium Submission off forces must be Europe. In this problem, it is given cables. A, B and B C are tied as soon the figure they can be a maximum tension off. Yeah, it can you get We have to find the largest force p which can be applied. Right? Let us see it. This is the tension in a C. This is the tension in in B c. You can resolve that nature into its component. Resolving this force, it will be p cause off air for and peace. I know Valpo resolving this force into its component. This will be 35 and this will be 50. So here will be t A C course off 35 t A c sign up 35 t v c course of 50 and T v c. Sign off 15 Applying the condition off Equilibrium Submission off force Must be zero along XX is You can write b cause off. L for DVC cause off. 15 minus de ese cause off. 35 are must be derailed bean cause off helper T v C maximum is continued. Er cause of 15. 80 cause off 35. It's called 20 say in question. But similarly submission off course. And on y axis you will get Yeah, yeah, moment, do you bc sign off 50 D A c sign off 35 minus We sign off. Alfa Must be zero that is 80 side off, 15 less 80 Sign off 35 minus p. Sign off with video by solving equations Born in tow. Equations one and two value of Phoebe will get 1081 noted and al five. You will get 82.5. That's all. Thanks for watching.

Here we have then topped the polls to be 9 m the his location from the upper end. He's Essick was 2 m and where equals 5700. New done. If he's the attention and and is the normal forced? First we have to balance the dark so the clock way struggled out. Counter block Western balances. It says she doesn't and minus esports. I intend it was job blue Times as a way to was, I intended so. We have tension equals dubbed over Do older one minus Yes, sir. And and putting the numbers in, we get torrential equals 3700 noodle. The enormous forces be given by weight. My extension It was 5700, minus 3700 equals 2000 new done. And this equation cause I ain't Ada Kansas out. So it's valid for on

First we have to find the directions off each tension polls. The first tension force is applied along the line A B, and we have to find the unique factor off the factor R b A which is the factor from point B to point in So we can write. Meeting is a question r e minus Ali Yeah, is equivalent to e zero minus 24 minus 18. The magnitude of this vector is R B is equal to Squire it off zero square plus 24 square plus 18 square. Is it well do Kirby E. Yeah, and the unit vector is I e. Is it well too? Director R B A Upon magnitude off army is equal to zero minus port by five and pain is three by five. The force in vector bomb is F B is a call to be minus B four by five G minus B three by five games. Yeah, Similarly, the second tension force is applied along the line D c. And we have to find the unique factor off the vector RDC, which is the factor from point B to point C so we can write coats. She Lisi is a Culture RC minus already, which is equivalent to a minus 15 minus 20 zero 30 plus 15 minus joined the and zero the magnitude of the unit Victories. R Release. He is equal to a square root off 15 square plus minus joined a square plaza zero is equal to 25 a. M and the unit vector release A is equal to Victor RDC upon magnitude. RDC, which is equivalent to three by five, minus four by five and zero. And the force in vector form is if the one is equal to B three by five i minus B four by five You. The second tension force is applied along the line E and we have to find the unit off the vectors R E, which is a factor from point B to E. So let's do the same. All the calculations in the similar manner so R D is equal to e minus already is equal to E minus nine minus 20 and drug the magnitude of the factories. Really, he is equal to square root off a a into square root off nine square plus 20 square plus 12 square, which is 25 a And the unique factor is yeah, odd B is equal. Do pressure off bacteria rd two magnitude already is equivalent of minus nine upon trendy fine. Minus 20 upon 25 and 12 upon 25. The forcing factor for me is if me, too, is equal to minus B 9.25 I minus B 20 upon 25 g minus B. Well, upon 25 games, the resultant is the factors, um, off all forces acting on the body so we can write. Our is equal to Sigma i f i and the some off all forces are be six by 25. Aye. Minus B 60 upon 25 J minus B 31 25 games. Now go find the beach. We need toe. We need the moment, um, off the original forces about the origin. The moment about the origin off all forces we can write as Mm. Ooh is equal to urban cross. F B. Yes, are to cross f B one. Yes. Are three grass f need to now solving all these cross products with the help off determinant. We will get the answer be a upon 25 in bracket minus 1. 20 I Yes, When 20 King and the beach is B is equal. June odd m do off one are square which will be equivalent to minus zero point 09 Top elite eight In the third part, the excess off the wrench is defined with a position vector are. And if we put bicycle to zero Since we are looking for the Intersect with the excerpt plane, we can get our cross our physical to M Ooh minus R R So from this equation, we can write 2.4 I nice 0.24 third plus 0.12 eggs I Saudi G minus 2.4 x game is equal to minus 4.77 six ai minus zero point 237 g minus four point seven Double edge A. He combining both decides we can get 2.4. There is equal to minus 4.7768 so there will be equivalent of minus one point double nine A and minus 2.4. That is equal to minus 4.7. Double eight A. So X is equal run into to any This is eggs 2.4 eggs


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