Determine the magnitude and direction of the resultant force, 𝐅R measured counterclockwise from

Determine the magnitude and direction of the resultant...

Key Concepts

Vector Addition

Vector addition involves summing individual vectors by combining their corresponding components along each axis. This process results in a single vector that represents the cumulative effect of multiple forces or directions, which is fundamental in understanding how different influences combine in physics and engineering problems.

Resultant Force

The resultant force is the net force acting on an object after all individual forces have been combined through vector addition. It is a single equivalent force that has the same effect on the object as the original collection of forces, making it a key concept in dynamics and statics.

Force Components

Resolving forces into their components involves breaking a vector into perpendicular parts, typically along the x and y axes. This technique simplifies the process of vector addition and analysis, as it allows for the independent treatment of effects along each axis.

Magnitude and Direction

Determining the magnitude and direction of a vector requires calculating the length of the vector and the angle it makes with a reference axis, usually the positive x-axis. The magnitude indicates the strength of the vector, while the direction provides information about its orientation in space.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent play a crucial role in relating the components of a vector to its magnitude and direction. These functions are used to convert between the vector's component form and its polar form, aiding in the calculation of angles and lengths.

Transcript

00:01 I've already drawn the diagram, and in problem 19, no, problem 21, we have to add f1 plus f2 first.

00:12 Well, notice there's a 90 degree angle in between there.

00:17 So i can just use the pythagorean theorem to add those together.

00:25 I'm going to call that f prime.

00:28 Whoops, prime.

00:31 So f prime.

00:40 I thought i put the numbers in for these things, but apparently i didn't.

00:44 F2 is 200, f1 is 400, and f3 is 300.

01:02 All right.

01:03 I think this would actually be easier if i just resolved it into components, but i'm not going to do that because i don't think that's what we're asked to do.

01:14 So, f prime is the square root of 400 squared plus 200 squared.

01:29 And thinking about one of these angles, and i guess i'm going to call it here, tangent of theta is the opposite, which is 200, over the adjacent, which is 400.

02:05 So that's going to give me the theta value.

02:08 So let's put these into a calculator.

02:16 Square root, 400 squared plus 200 squared, 447 newton.

02:33 Theta is going to be the inverse tangent, 200 over 400, which is just a half, 26 .6 degrees.

02:52 All right.

02:58 So we know that this is 90 degrees here.

03:05 And so this angle at the bottom is 90 minus theta.

03:15 I guess i'll call it phi.

03:28 Is there an easier way to do this? maybe there is, according to alternate interior angles, this is theta here.

03:45 Okay, what am i trying to do? i'm trying to resolve this into horizontal and vertical components.

04:24 All right, if that's 150 down here, then this angle in here is 150, minus 90.

04:41 It'd be 60 degrees.

04:56 So this angle over here is going to be 30 degrees because 60 plus 90 plus another 30 makes 108.

05:14 So that's 30.

05:24 So i'm really trying to get this angle in here.

05:32 I'm going to erase what i did here because that wasn't necessary at all.

05:41 That angle there, which i'm going to call phi, is 90 minus 30 minus theta.

06:01 60 minus theta.

06:05 That's going to be phi.

06:07 So now, the resultant, no, the force, yeah, resultant in the y direction is going to be f prime, times the cosine of phi, that's in the upward direction, and that's minus the 300, f3, which is 300...

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