Two forces of 128 lb and 253 lb act on a point. The resultant ...

Question

Two forces of 128 lb and 253 lb act on a point. The resultant force is 320 lb. Find the angle between the forces.

Accepted Answer

Hey, everyone here we are given two forces, £428 and £729 are concurrent or their lines of action pass through a common point. If the resultant force is £986 what is the angle between the two forces? We have four answer choice options. Answer a 114.4 degrees. B 111.9 degrees C 68.1 degrees and D 65.6 degrees. So to begin this problem, we need to start by creating a sketch containing all of our given information. So we can start by representing our first force as an arrow pointing slightly downward. And so we will label this arrow as £428. Next drawing in our second force, we will draw a second arrow which has its tail connecting to the tail of the first force. And this arrow will be pointing slightly upward. It will also be slightly longer than the first force since the second force is larger. And so we will label this second arrow as £729. Now again, we are asked to find the angle between these two forces. So we can go ahead and label the angle between our two arrows as angle. So now we are ready to draw in our results in force. So we just need to draw in a third arrow in between our two other arrows. And so this third arrow we can also notice will be the largest or longest arrow since it is the largest force. And so now we just label this arrow £986. Next, we just want to connect each of the heads of each arrow by a dash line. And you can notice we form a parallelogram. And so we just want to label the topmost angle as angle theta. Now with our angle, theta, we can notice that we can actually simplify our sketch of this parallelogram as just its upper triangle. And so drawing our triangle. Now we again have this slightly upward pointing arrow, which is the £729 force. Now connecting to the tail of this arrow, we have our resultant force arrow which again is £986. And now for the last side of the triangle, we can notice on the parallelogram, we just have a dash line. However, if you go to the opposite side, we notice that this dash line is the same length as the £428 force. So we will draw a downward facing arrow as the last side of our triangle. And we can label it now as £428. And so, lastly here, we just want to carry over our angle theta. So the top angle of the triangle is angle theta. Now, before we can find angle alpha, we need to find angle theta, utilizing our drawing of the triangle. And so we can notice with our triangle, we have the values for each side identified. So to find the angle, we need to recall the law of cosines. And this law tells us that C squared is equivalent to A squared plus B squared minus two multiplied by a multiplied by B multiplied by cosine of capital B. And so now we just want to substitute in our given values into this formula to find data. And so we have 986 squared is equal to 729 squared plus 428 squared minus two multiplied by 729 multiplied by 428 multiplied by cosine of theta. And now we just want to take our current equation and solve it for theta. So the first step here is to just rearrange the equation so that cosine of theta is alone. And so rearranging, we have two multiplied by 729 multiplied by 428 multiplied by cosine of theta is equivalent to 729 squared plus 428 squared minus 986 squared. Now, continuing solving four theta, we now need to divide both sides of the equation by the coefficient of cosine. And so doing this, we have cosine of theta is equal to 729 squared plus 428 squared minus 986 squared. All divided by two multiplied by 729 multiplied by 428. Continuing solving for theta, we now just need to take the cosine inverse of both sides. And so we have theta is equal to cosine inverse of 729 squared plus 428 squared minus 986 squared, all divided by two multiplied by 729 multiplied by 428. Now just evaluating this expression, we find that theta is equivalent to 114.378 degrees. And so now that we have found our angle theta, we can refer back to the sketch of our parallelogram and we noticed that theta is adjacent to alpha. So to find alpha, we need to recall that adjacent angles of a parallelogram are supplementary. So this tells us that alpha beta is equivalent to 180 degrees. Now just solving this expression for alpha, we have that alpha is equal to 180 degrees minus theta. And now just substituting in our value for theta, we have alpha is equal to 180 degrees minus 114.378 degrees. Now, just evaluating, we have that alpha is equal to 65.622 degrees. Lastly, we just need to round this value to the nearest 10th. And so we find that alpha is equal to 65.6 degrees. And with this final rounded answer, we are left with answer choice D where again, we found that the angle between our two forces is 65.6 degrees. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Two forces of 128 lb and 253 lb act on a point. The resultant force is 320 lb. Find the angle between the forces.

Key Concepts

Vector Addition

Law of Cosines

Trigonometric Functions

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