Determine the magnitudes of the components of the force F=90 th acting parallel and perpendicula

Determine the magnitudes of the components of the force F=90...

Key Concepts

Resolution of Forces

This concept involves breaking down a force vector into two perpendicular components along specified directions. Typically, this is done to analyze the effect of the force along different axes in a system, such as along and perpendicular to a surface or a particular line of action. By doing so, it becomes easier to apply laws of motion or to understand the impact of the force in various directions.

Vector Decomposition

Vector decomposition is the process of expressing a vector as the sum of two or more vectors that are pointing in particular directions, often along orthogonal axes. The technique involves using geometric methods or algebraic computations, such as calculating the dot product, to determine the magnitude of each component in the specified directions.

Trigonometric Relationships in Force Analysis

Trigonometric methods are essential when decomposing forces, especially when the direction of the force is inclined at an angle relative to the coordinate axes. Using sine and cosine functions, one can relate the angles present in the problem to the respective parallel and perpendicular components of the force, facilitating precise calculations of the magnitudes.

Transcript

00:01 Okay, for this problem, we have a box or crate, and we have a diagonal from point a to b in that crate.

00:06 And if you look at the drawing, you can see point a, it's on the left corner, and point b is in the upper right corner.

00:11 And we have to determine the magnitudes of this force that's acting at point b, this force called f, if you determine the magnitude of force f acting parallel and perpendicular to this diagonal.

00:27 So first, we can determine point vectors for a and b because it gives us the dimensions of the box.

00:36 So i'll go ahead and write those down.

00:38 So a is 1 .5i plus 0j plus 0k because it's just along the x -axis.

00:55 B is 0i plus 3j plus 1.

01:00 Okay, and now that we have those, we can create a vector from a to b.

01:11 To do that, we just subtract a from b.

01:21 It will look something like this.

01:28 Okay.

01:30 So now we have to convert this to a univector, because our end goal is to eventually dot this with the force f.

01:38 So we convert this to a univector by dividing it each of the components by the magnitude, and the magnitude of ab, if you just punch it into a calculator really quickly, should come out to somewhere around 3 .5.

01:54 Okay, so now our unit vector, uab, is going to equal all these components divided by 3 .5.

02:19 Okay, so that's how univector for a, b.

02:23 Now we have to find a univector or just a vector for the force f.

02:29 We're told that the force f has a magnitude.

02:31 Of 90 pounds.

02:37 Okay.

02:38 We're also told that it forms a 60 degree angle with the x, y, plane, and a 45 degree angle from, like, the y axis, or the y plane.

02:48 So we can use those angles to find each of the components for f.

02:52 So if you start with, or if we imagine, i'm going to draw it separately down here.

03:03 Okay, so imagine this is the positive y -axis, the positive, or in the negative x -axis.

03:08 It forms a 45 degree angle.

03:11 Like if you can imagine the projection of f right along there, from the 45 degree angle there, and it forms a 60 degree angle there.

03:25 So, fx and f y in magnitude are going to be the same, but they are going to have opposite signs.

03:32 But to do that, we just have to imagine just, like imagine creating a triangle with just this projection, in the xy plane...

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