A force of 28.7 lb makes an angle of 42° 10′ with a second for...

Question

A force of 28.7 lb makes an angle of 42° 10′ with a second force. The resultant of the two forces makes an angle of 32° 40′ with the first force. Find the magnitudes of the second force and of the resultant.

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Accepted Answer

Hello, today, we are going to be calculating the magnitudes of forces. So we are told that we have two forces A and B which are shown to be acting on a certain point. The angle between the forces A and B is 72 degrees with 30 minutes. We want to determine the magnitude of the force B and the result in magnitude. If the resultant force makes an angle of 65 degrees with 50 minutes with force A, we are also told that the magnitude of force A is £81.6. So how can we start to solve for the magnitude of B and the magnitude of the resulting force? Well, keep in mind that the methods of vectors allow us to move vectors around as long as we keep the same magnitude and the same direction. So what we can do is we can take the magnitude A and set it above the vectors B and the result in force which will allow us to create a triangle. So just to draw that picture, we are given vector B and we have the resulting force which we are going to label as R and we are given the vector A, if we move A above B and R, we can go ahead and create a triangle with the vector A. We know that the magnitude A or the side length for A is going to be £81.6. Now, we know that the angle between A and the resultant force is 65 degrees with 50 minutes. The equal opposite angle to this angle is going to be the upper right angle of the triangle. So we can label the angle with respect to the side length B to be 65 degrees with 50 minutes. Now, what we wanna do is you want to calculate the angle with respect to the side A and with respect to the side R. So let's go ahead and start by calculating the angle A. Now we know that the total angle between A and B it is 72 degrees with 30 minutes. If we subtract the angle between A and R, which is 65 degrees of 50 minutes, the result will be the angle between B and R. So we are going to take 72 degrees with 30 minutes and subtract 65 degrees with th with thir 50 minutes, this is going to give us six degrees with 40 minutes. This is going to be the angle between B and R. Now we're going to calculate the angle with respect to the side length arm in order to calculate this angle. We are going to use the sum angle property of a triangle. Now, the property states that the sum of all three angles within a triangle should equal to 180 degrees. So we're going to take angle B which is 65 degrees with 50 minutes at six degrees with 40 minutes. And at the angle R and the sum of three, these three numbers should equal to 180 degrees. We can use this equation to solve for the angle R 65 degrees with 50 minutes plus six degrees with 40 minutes is going to give us a sum of 72 degrees with 30 minutes. So we have 72 degrees with 30 minutes plus the angle R is equal to 180 degrees. Subtracting 72 degrees with 30 minutes to both sides of the equation will give us the angle R is equal to 107 degrees with 30 minutes. This is going to be the angle with respect to the side R. So now that we have all three angles and we have one side length. How can we continue to solve for the magnitude of B and the magnitude of R? Well, what we can do is we can use the law of science. Now, the law of signs states that we have 33 equivalent ratios, we have the sine of angle A divided by the side length A is equal to sine of angle B divided by the si length B is equal to sine of angle R divided by the side length R. What we are going to do is we are going to pick any two of the three ratios, set them equal to each other. And use that to solve for the magnitudes of B and R. Now looking back at the triangle, we created, we know the magnitude of A is £81.6 and we know that the angle A is six degrees with 40 minutes. So we're going to take the ratio with respect to A and we're going to set it equal to the ratio with respect to B to solve for the side length or the magnitude of B since we have the angle B which is 65 degrees of 50 minutes. So let's start by solving for B. So we have sign of A divided by S length A equal to sine of angle B divided by side length B. We can solve for the S length B by cross multiplying both of the ratios that will give a side length B multiplied by sine of angle A equal to side length. A multiplied by sine of angle B dividing sine of A to both sides of the equation will give a side length B is equal to a multiplied by sine of angle B divided by sine of angle A. Now if we plug in the numbers that we sold for earlier, we have the side length A is equal to £81.61. I'm sorry, £81.6. Then we have this number multiplied by sine of angle B which is 65 degrees with 50 minutes divided by sine of angle A which is six degrees with 40 minutes. At this point. I would recommend plugging in this value into a calculator. And when doing so, you should end up with the magnitude of £641.3 for the force of B. Now, we're going to repeat this process in order to solve for the magnitude of R. Now, just like B, we're going to take the ratio with respect to A and set it equal to the ratio with respect to R. Since we have the angle R again, the angle R is equal to 100 and seven degrees with 30 minutes. So we have sine of A divided by sine length, A equal to sine of R divided by the side length are we're going to cross multiply in order to solve for R and that will leave us with R multiplied by sine of A is equal to a multiplied by sine of R. We will divide both sides of this equation and we will have the side length R is equal to the side length. A multiplied by sine of angle R divided by sine of angle A if we plug in the numbers, we will have 81.6 multiplied by sign of 107 degrees with 30 minutes divided by sign of six degrees with 40 minutes. After plugging this into a calculator, you should end up with a magnitude of £670.4. This is going to be the magnitude of the angle or of the resulting force. So the magnitude of four B is £641.3 and the magnitude of force R or the resulting force is £670 670.4 pounds. With that being said, the answer to this problem is going to be c so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

A force of 28.7 lb makes an angle of 42° 10′ with a second force. The resultant of the two forces makes an angle of 32° 40′ with the first force. Find the magnitudes of the second force and of the resultant.

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Key Concepts

Vector Addition of Forces

Law of Cosines in Vector Problems

Angle Relationships Between Forces and Resultant

Watch next

A force of 28.7 lb makes an angle of 42° 10′ with a second force. The resultant of the two forces makes an angle of 32° 40′ with the first force. Find the magnitudes of the second force and of the resultant.<IMAGE>

Key Concepts

Vector Addition of Forces

Law of Cosines in Vector Problems

Angle Relationships Between Forces and Resultant

Watch next

A force of 28.7 lb makes an angle of 42° 10′ with a second force. The resultant of the two forces makes an angle of 32° 40′ with the first force. Find the magnitudes of the second force and of the resultant.

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