One boat pulls a barge with a force of 100 newtons. Another bo...

Question

One boat pulls a barge with a force of 100 newtons. Another boat pulls the barge at an angle of 45° to the first force, with a force of 200 newtons. Find the resultant force acting on the barge, to the nearest unit, and the angle between the resultant and the first boat, to the nearest tenth.

Accepted Answer

Hey, everyone. Here we are given a rope is strangled on a pole. One end of the rope is pulled by kid A with a force of £16. And another end of the rope is pulled by kid B with a force of £22 at an angle of 60 degrees from the first force determine the resultant force applied on the pole. Also find out the angle between the resultant and the force applied by kid A. We have four answer choice options, answer choices A through D where the angle is either 35 degrees or 25 degrees. And the resultant is either £23 or £33. To begin this problem, we want to start by creating a sketch containing our given information. So we can start by drawing a small circle to represent the pole. And we can label this point as point C. Now representing the force from kid A, we start at the left side of the pole and draw an arrow pointing downward and towards the left. And this is again our force from kid A. So we just label this arrow as £16. And we also can label the arrowhead as point A next, we want to represent Kid B. So we draw another arrow going from C now down towards the right. And this arrow is slightly longer since the force is larger. And we label this second arrow as £22 and we can label the arrowhead as point B. Next. We are also given that the angle in between these two forces is 60 degrees. Now, we need to find the resultant force. So we just want to draw this arrow going directly downward from point C. And we also need to find the angle between the resultant force and the force applied by kid A. So we can label this angle as let's say angle alpha. And now we just want to finish our parallelogram in our sketch by just drawing a dash line to connect each of the arrowheads. And now, in order to apply the parallelogram rule, we first need to find the adjacent angle of our given angle 60 degrees. And so we can label this angle which here let's choose angle A and we can label it as unknown theta. And so to find the, we just need to recall the supplementary angle property which tells us that 180 degrees minus our given angle, 60 degrees is equal to the adjacent angle theta. Now just evaluating, we find that theta is equal to 120 degrees. So now that we found our adjacent angle, we can begin sketching our triangle. And so we can start again with our point C. And now we draw the arrow going down towards the left, which is our force from kid A. So we label this arrow £16 and label the arrowhead as point A. Now, starting back at point C again, we draw another arrow this time going directly downward and this is our result in force. So we also want to label the angle between our two forces here as our angle alpha. And now going from point A to the end of our resultant force, we have the force from kid B which we sketch as an arrow and label it as £22. And now we just want to label the adjacent angle we found which is angle A as 120 degrees. So now that we have sketched our triangle, we can utilize the law of cosines to find our results in force, which is the side length CB. So we also make sure that we label that final point on our triangle as point B. And so utilizing the law, we have side length CB, the resultant force is equivalent to the square root of side length A B squared plus side length ac squared minus two multiplied by side length A B multiplied by length ac multiplied by cosine of angle A. Now just substituting in our values that we can take from our triangle we find that CB is equal to the square root of 22 squared plus 16 squared minus two, multiplied by 22 multiplied by 16, multiplied by cosine of 120 degrees. And now just evaluating, we find that length CV or the resultant force is approximately equal to 33.0454. And now rounding to the nearest whole number, we find that our resultant force, which again is length CB is approximately equal to £33. And now that we have found our missing side length CB, we can find the unknown angle alpha utilizing the law of signs. So from the law of signs, we have that length BC divided by sine of angle A is equal to length ac divided by S of angle C or in this case, we labeled angle C as alpha. Now substituting in our known information, we have 33 divided by s of 120 is equal to 22 divided by sine of alpha. And now solving for first sine of alpha, we have sine of alpha is equal to 22 multiplied by sine of 120 divided by 33. Now solving for alpha, we just take the sine inverse of both sides. And so we have that alpha is equal to sine inverse of 22 multiplied by sine of 120 divided by 33. Now just evaluating, we find that alpha is approximately equal to 35 0.2087 degrees or simplifying to the nearest whole degree. We have that alpha, our unknown angle is approximately equal to 35 degrees. And with this, we have found our resultant force as well as the angle between the resultant and the force applied by kid a. And so with these values, we are left with answer choice C where again, our angle is 35 degrees and our resultant force is £33. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

One boat pulls a barge with a force of 100 newtons. Another boat pulls the barge at an angle of 45° to the first force, with a force of 200 newtons. Find the resultant force acting on the barge, to the nearest unit, and the angle between the resultant and the first boat, to the nearest tenth.

Key Concepts

Vector Addition

Trigonometric Functions

Resultant Force and Angle Calculation

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