Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is 60.0°. If Rover exerts a force of 270 N and Fido exerts a force of 300 N, find the magnitude of the resultant force and the angle it makes with Rover's rope.

Video transcript

Hey everyone. So today we're dealing with some vector problems and doing some vector decomposition. But let's first take a look at our problem itself. So we're being told that a gardener has two bulls, and the gardener is using the two bulls to uproot a stump that is in his garden. Now each bull is pulling in a different direction with separate forces on the stump using two different ropes that has an angle of 40° between them. With this, we're being asked to determine the magnitude and direction of the resultant force measured from white's rope. So let's uh let's write this out a little bit, right? Let's conceptualize this a little further. So let's say we have, let's just put this on a grid. So we have oops, let's just say we have an xy grid, right? So this is why this is X. This is positive Y. This is positive X. So let's say this point, the origin is our stump, right? And let us say that the white ball, right? Let us say the white pool is pulling this way. This is the force by the white ball. And let us say that the black bull has a force going this way and between them, there's an angle, there's an angle right here. So just make this look a little pretty. But this angle theta is equal to 40° F. Y. is equal to 7:15 Newtons and effort. The black or Oops do that backwards. The White Bull is 860 mutants, my bad, 860 Newtons. And the Black Bull has a force of 715 mutants. There we go. So with all this in mind, we need to remember that force is a vector quantity. And these are two non parallel forces, which means we need to decompose these vectors and then recompose them in order to get the resultant. So let us do the decomposition first. Right? So if we're talking about F. F. Of the Black bowl, well, it is only an X. Component. It has only an X. Component because it is parallel to the X. Axis. So we could say that uh the horizontal component of the uh force of the black hole is simply adjust the force of the Black bowl. However, for the White bull, there is both an X. And a Y. Component here. So let's break that down and let's do that in blue. So using our trick, no metric identities. We know that the for the white bull in the X direction, it will be the force multiplied into cosine Theta which is 8 newtons into cosign. 40. Kassian. Oops. Sorry, I went away from the mic. It is 8:16 Newtons multiplied by Kassian 40°. Which gives us 658.8 Newtons. We can do the same for the Y direction. Using our trigger metric identities. We get this the Y direction would be F. W. Into sign data Which is just 860 newtons into sign 40 Which gives us a value of 552.8 newtons. So the net force in both the X and Y directions can be redefined as F W. X. Sorry, F in the X direction would simply be uh F. B of X plus F W of X, which is equal to 7:50 Newtons plus 658 0.8 newtons, which gives 408.8 Newtons. And for F Y. Well, it could simply be The uh y component of the White Bull, which is simply 552. newtons. So now that we have both the horizontal and vertical components, we can now determine the magnitude and direction and we can use some of our vector composition formulas to help us do so. So, and I'm just gonna scroll down a little bit. So I have a little more space. So when trying to find the magnitude of the force, we can use the formula and let's write this in red. We can use the formula for magnitude. That states that the force. And let's just say total or the resultant, I should say let's rewrite that the result in force equal to the square root of the X four squared plus the vertical component squared. Which in this case go to the square root of 1408.8 Squared plus Plus 552.8 Squared, Which will give us 1,513 newtons. Similarly to find the angle of a resultant vector to find the angle we can use another angle or vector composition formula that states that data of a resultant or in regards to X I should say would make more sense is equal to the inverse tangent of the horizontal component divided by the oops of the force of the horizontal component. My Ben divide ability forces the vertical component of the force divided by the horizontal component of the force. My tongue is getting all jumbled. Anyways solving this, we get the inverse tangent of Uh 552. Divided by 1408. which gives us a value of 21.4°. However, that doesn't mean that our answer is 21.4 degrees. We want to find, if you recall from the question stem, we want to find the direction of the resultant force measured from whites rope, which means since we're dealing with the angle in respect in respect to the in blacks rope, which is in the X axis, we have to take theta Which will be 40° -21.4°. And she called 18.6. So the direction of the the direction of the resultant force Will be 18.6° and the force itself will have a magnitude of 1,513 Newtons, which corresponds with answer choice C. I hope this helps, and I look forward to seeing you all in the next one.