1. The problem statement, all variables and given/known data The threading die is screwed onto the end of the fixed pipe, which is bent through an angle of 20º. Replace the two forces by an equivalent force at O and a couple M. Find M and calculate the magnitude M' of the moment which tends to screw the pipe into the fixed block about its angled axis through O. 2. Relevant equations M=r x f M=fd 3. The attempt at a solution I think that F= 30-40 j= -10j, I don´t know if this is right because the book doesn´t give the answer. I calculated correctly M=136.46i -679.56k lb*inch, according to the book, but I have some doubts: 1.- lb inches are not in the SI system, I don´t understand why the force is given in mass unities. I´m used to use Newtons, so I´d need to transform lb to kg, then multiply by 9.8 to get Newtons, then inches to meters and finally I´d get N*m. But Why is this book using mass units to talk about forces? 2.- The second part says: "and calculate the magnitude M' of the moment which tends to screw the pipe into the fixed block about its angled axis through O" I did that using vector n=(sin(20), 0, cos(20)) and I got -591.90 lb inch which according to the book was wrong. The correct result is 685 lb inch and I deduce from that the fact that they are using this vector n=(sin(20), 0, -cos(20)). When you need to project the moment M onto the angled axis you can choose two vectors on the same line but oppossite sense, WHy is correct to use a minus sign on the z component of the vector n, and not the oppossite? How do you know that? 3.- If I use the general formula for M=Sum r x f, I get the right result. But I tried to do a much more simple calculation and I failed, I´d like to understand why. I have read that the vector M is a free vector, and I think I don´t understand that well, What does free vector means?. I thought that I could calculate the moment M around a point, like the origin G of the axis x,y,z and that it should be the same around point O, so I thought M=40*10 - 30*10 k= 100k. Obviously that is not the case, that´s false because the distances between G and the forces are different to the distances from O to the forces. So ok, I understand that but in this problem: I calculated the total moment around point O, calculating the moment created by the 1200 force from point G, and adding the 240 Nm moment vector to that. In that case I applied the 240 Nm moment to O without any change, as if O were the point G because I thought that it was a free vector and I could use it in any point and the result would be the same. So now it´s a bit confusing to see that I can´t calculate the moment around G and use it around the point O, What´s the difference in both cases? As you can see it´s not clear to me what a free vector is, hope your help. THanks!