Your neighbor Paul has rented a truck with a loading ramp. The ra...

Question

Your neighbor Paul has rented a truck with a loading ramp. The ramp is tilted upward at 25°, and Paul is pulling a large crate up the ramp with a rope that angles 10° above the ramp. If Paul pulls with a force of 550 N, what are the horizontal and vertical components of his force? (Force is measured in newtons, abbreviated N.)

Accepted Answer

Hi, everyone in this particular practice problem, we are asked to calculate the force in the X and Y horizontal and vertical components of the applied force where we will have a trolley loaded with water bottles, pushed up an incline, making an angle of 15 degrees above the horizontal. So that's the incline and the force applied has a magnitude of 1 Newton. And the force itself will make an angle of 15° counterclockwise with respect to the incline itself. So I'm gonna start us off with basically making an illustration of a diagram that we have in this system. So we'll have an incline right here. The incline will have a data of 15° above the horizontal. And in this case, we will have a trolley. I'm just gonna indicate the trolley with trying my best to draw it here. We will have a trolley and the trolley will actually have a force applied to it. Well, the force itself is actually going to be 15° making an angle of 15° counterclockwise with respect to dis incline. So I am going to first make a pretty arbitrary access for our incline here And essentially our forces going to make a 15° angle from there. So like this is going to be our force here. So this angle right here is going to be 15°. So now I am going to make uh horizontal and a vertical X and Y axis here with respect to the trolley itself. So I'm gonna move this all the way to the trolley here and from there, we will actually be able to then do our projection. Okay. So we have the X and the Y access, we have the F here and we have the degrees incline or ramp incline here. Okay. So now what we wanna indicate here is we know what this data is, which is 15°. So we want to know what the total angle here is. If we look at our system, we can determine that this angle here will be exactly the same with our inclined angle because as we can see this arbitrary access and our rep is parallel and we have two lines also parallel, which is the like the ground here and our access, those two are also parallel. So two parallel lines will both give us here the same angle which this is also going to be theta which is 15 degrees which corresponds to this. And this total angle right here will actually be 30 degrees. And I'm going to indicate that with just a fi I guess to make it easier for you guys 30° equals data. I'm gonna name this data one and this data too. So this is going to be data one plus data to just like. So, so that's the effect of the incline itself from the ramp and the incline from the force applied to it. So the trolley in this case is essentially modeled as a particle like object and the force F makes an angle of five here in total, which is data, one plus data to 15 degrees plus 15 degrees equals to degrees above the horizontal above the X axis here. Okay. So now that we have pretty much solved our problem angle problem, we will then want to actually do our projection. So our projection here, as always, we will have an X direction of our force. This is going to be F X and at the same time, we will also have a wide direction of our force here. And this is going to be our F Y okay. And then we can pretty much kind of decompose this and put it all the way to decide to create a right angle triangle here. And our triangle will essentially just be oops that is not a good triangle. So our triangle here is going to be having an F here and F X here. And our fy here with a total of five here equals 30 degrees. And this will be our right angle here. So we'll have a right angle triangle. So we can pretty much use R cosine and sine to do our decomposition. So first, you want to recall that using this triangle here, co sign of FI will equal to B F X over F. So rearranging this F X will then be F multiplied by cosine of phi, we know what F S and we know what the fight is. So we can actually solve this F is 1 20 Newton co sign of fires co sine of 30 degrees. And F X will then be 104 Newton. Similarly, we can use our sine sine of Phi to determine our fy here because sine of Phi will equal to whatever is opposite over the hypothalamus, which is F Y over F and that will be after rearranging F Y will equals to F multiplied by sine of Phi and F is still 1 20 Newton and sign off 30 degrees. And that will make our fy value to correspond to Newton. And that will be the answer of our practice problem with an X value of 104 Newton and F F F Y valley of Newton that will correspond to option C that we have here. An option C is going to be the answer to this particular practice problem and I'll be all for this video because have any sort of confusion. Please make sure to check out our other lesson videos similar to this topic and that will be all. Thank you.

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