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6-54. Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive 'axis:Fz 26 kNFi = 30 kN...

Question

6-54. Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive 'axis:Fz 26 kNFi = 30 kN

6-54. Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive 'axis: Fz 26 kN Fi = 30 kN



Answers

If $F=5 \mathrm{kN}$ and $\theta=30^{\circ},$ determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive $x$ axis.

All right. I'm gonna try to use what I already have here, and it tells me that F is five killer Nunes and Fada is 30 degrees. I already knew that data was 30 but I did not know that F waas uh five. Ah, what is the magnitude of the resultant and its direction? Well, I believe that I can still use everything that I've got. So let's go ahead and go to a calculator. Gonna put in the ah resultant square root of parentheses six plus f, which is five sign 30 minus four Signed 15 square that for us five co sign 30 was four who signed 15 squared. That gives me these were my answers to the old question The result in force of 11.1 killer news. Let me just verify that That's correct. 59. Yes, it is. And so, um, think I wrote data twice here. That's annoying. I apologize for that. This should be fi. So let's go ahead and put that into a graphing calculator. The inverse tangent of and I already have this stuff in here. So the why or why over a far X, I gives me 47 0.7 degrees. So what it's saying is that the result now is like this

Okay, so the given four sectors can be expressed in terms of the horizontal physical vertical components as a 20 co sign off 45 um, plus 20. So this is I plus 20 times the sign off 45 j. This is equal to 10 times the square it oak Thio I plus 10 times the square root of two J because Senate 45 this word to two as the site for five. So therefore we can rewrite this 16 co signed 30 i minus 16 sign third J is equal to eight Bruce three I minus eight j The result in force at the some of these two vectors. So F is equal to 10 routes to plus eight Route three I plus 10 route to minus eight j, which is equal to 28 I plus 6.11 for J. So therefore we have the magnitude of F is equal to the square root of 28 square plus 6.14 square, which is approximately equal to 28.7 pounds. If we let Fada is equal to the angle that f meets with positive X axis, then we can write that data is equal to the arc. Tangent off 10 routes to minus 8 10 Route minus eight, divided by 10 routes to UM plus eight point plus eight Route three, and this is approximately equal to 12.4 degrees.

S so far, This section seems to be exactly the same as the previous section. I don't know what is different. Seriously, um, so or showing that this is 30 degrees were also shown that this down here is 45 degrees. Um, but I would rather mark this is 45 degrees and it would be 45 degrees because it all it's a right angle was the magnitude and direction of the resultant force. Well, um, force resultant in the X direction is going to be 400 co sign 30 plus 800 co signed 45 force resulting in the Y direction is going to be 400. Sign 30 minus 800. Sign 45. The resultant forces going to be the square root of f R X squared plus f are why squared and the the the angle is going to be the inverse tangent of f R. Why, over f r X Put all of these into a calculator is important that you do the 1st 2 so that you know exactly where that data is, whether it's ah, downward or upward from the horizontal. And so, um, going to a calculator, you 400 co sign. 30 plus 800. Could sign 45 912 Nunes. Okay, next one 400 Sign 30 minus 800. Signed. 45 negative. 366 News. Ah, square E of the 1st 1 Weird. Plus 2nd 1 Where is Ah, 983 news. And they'd, uh, is the inverse tangent of the 2nd 1 or the Y direction over the X direction. Negative. 20 to 21.8 degrees. So we're supposed to measure clockwise from the positive X axis, so that would be correct. So those two are my answers. Let's check in the book. 1983 21.8. Yeah, it would be positive. 20.8 as measured from in the clockwise direction.

All right. I noticed that the answer is not in the back of the book for this one. Um, so we're gonna have to be perhaps more careful. Especially the three mistakes I made on the last one. Um, but I don't intend to make any mistakes on this one. Not I intended to make mistakes in the previous one. £40. Okay, there. 34 five, triangle 5 12 13 Triangle down here. I definitely didn't draw it like that. Although I'm not sure the book did either. Okay, so we just need the magnitude in the direction of the resultant force. So the resultant force in the X direction is going to be 30 obviously plus 3/5 of the 40 plus 5/13 just using proportions of the 91. And that's gonna be in pounds. The result in force in the Y direction is going to be of the 40 is going upwards. So that's gonna be 4/5 of 40 minus 12 13th of 91. So let's go ahead and calculate both of those. I was gonna put that in a calculator. 30 plus 3/5 of 40 plus 5/13 of 91 89. 4/5 of 40 minus 12 13th of 91 negative 52. So the result in just using the Pythagorean Theriault is going to be the square root of 89 squared plus negative 52 squared is going to give me £103 and the angle Inverse tangent. I can visualize this one in my brain, but anyway, why is negative? 52 x is 89 going to be 30.3 degrees measured clockwise from the horizontal, which is what we wanted to dio. So those are the answers.


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