Determine the magnitude and coordinate direction angles of the resultant force, and sketch this

Determine the magnitude and coordinate direction angles of...

Key Concepts

Vector Diagram in the Coordinate System

A vector diagram is a visual representation of vectors on a coordinate system. It involves plotting the vector components along the appropriate axes and drawing the resultant vector accurately to scale. This diagrammatic approach aids in visualizing the magnitude and direction of forces, making the analysis of vector problems more intuitive.

Direction Angles

Direction angles specify the orientation of a vector relative to the coordinate axes. They are determined by using trigonometric relationships, such as the inverse cosine or inverse tangent functions, to find the angles between the vector and each axis. These angles help in fully characterizing the vector’s direction in space.

Vector Addition and Resultant Force

This concept involves combining multiple vectors to obtain a single vector that represents the overall effect, known as the resultant force. In many physics and engineering problems, forces are added by summing their respective components along each coordinate axis. The resultant force is a crucial idea in understanding net effects in a system when multiple forces are applied.

Magnitude of a Vector

The magnitude of a vector quantifies its length or size, regardless of its direction. It is calculated using the Pythagorean theorem by taking the square root of the sum of the squares of its components. This measure is essential in determining the strength of the resultant force in vector problems.

Transcript

00:01 For me, it almost seems like the most difficult part of this is understanding the drawing.

00:14 So i'm going to try to draw it here.

00:18 Looks like we've got an f1 going this direction, and we've got an f2 going that direction.

00:26 Whoa, that direction.

00:41 So let me try drawing these directions in blue.

00:45 So the angle with y is 6 .5.

00:48 So i understand that one.

00:51 The angle with the negative z axis is 60, and i think it's telling me that the angle with the x -axis is 45.

01:17 Well, this one, which is the angle with the negative x -axis, i'm going to change it to be this angle, which is just going to be 180 minus 60, which would be 120.

01:37 So then i'm going to ignore this one.

01:44 So now those would be alpha, beta, and gamma.

01:49 So let's see what i can do with the other ones.

01:54 The other ones, it's showing that on the xy plane, it is 20 degrees.

02:11 But it also forms a 3 ,4 ,5 triangle.

02:21 Okay, i believe i understand what's going on here.

02:25 And so, let's go ahead and do f1x, f1y, and f1z.

02:38 This is more straightforward than two.

02:43 They're all just going to be 400, which is f1, and times the cosine of alpha.

02:55 It's 45 degrees with the x -axis, cosine of 60 degrees with the y -axis, and cosine of 120 degrees with the z -axis.

03:11 So let me put that into a calculator.

03:18 I probably should think about cosine of 60 degrees and whatever, but i have a calculator.

03:25 Makes it easier.

03:26 I don't have to think.

03:29 Okay, so this one is going to be 282 .8 newton.

03:41 Cosine 60 is going to be 200.

03:47 And yeah, i should have known that.

03:51 And this one, 120, negative 200.

03:58 That makes sense because it's headed downward.

04:01 Okay, let's go with f2 now.

04:04 So the first thing i want to do is f2 on the x, y plane.

04:15 So that's going to be the horizontal part, which is four -fifths of f -2, which is 125.

04:30 But then f -2 -x is the four -fifths of 125 times in the x direction that's going to be the cosine of 20.

04:52 F2y is going to be four -fifths of 125 times the sign of 20, except it's negative.

05:09 It's going in the negative y direction.

05:12 And then in z direction, go back to our 3 ,4 -5 triangle, and it's going in the positive z direction, 3 -5ths of 125.

05:26 So three -fifths of 125 is 75 newtons.

05:35 The other ones i'm going to use a calculator.

05:37 So four -fifths of 125 is 100, by the way, times cosine of 20.

05:52 93 .9, no, 94 .0.

06:05 And sign 20, negative 34 .2.

06:13 So, all right.

06:18 So now i've got all the component vectors, and it does appear that i'm going to check my signs.

06:34 F1 looks like it is going positive x, positive y, and negative z.

06:40 F2 looks like it's going positive x, negative y, positive z.

06:45 So that makes sense.

06:47 Okay...

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