A force of 125 N pulls due west on a body, and a second force pulls N 28.7^∘ W The resultant for

A force of 125 N pulls due west on a body, and a second...

Key Concepts

Direction Angles

This concept involves computing the angle at which a vector points relative to a reference direction, typically using inverse trigonometric functions. Determining the direction is key to fully describing the vector's influence and orientation.

Vector Decomposition

This concept refers to breaking a vector into its horizontal and vertical components. It is essential in physics for analyzing forces acting at different angles, allowing for the use of trigonometric functions to determine the contributions along the coordinate axes.

Vector Addition

This involves summing the respective components of different vectors to obtain a resultant vector. It's a fundamental procedure in mechanics, where the net effect of multiple forces is determined by adding their components along common axes.

Resultant Force

The resultant force is the single equivalent force that represents the combined effect of two or more individual forces acting on a body. Its magnitude and direction are calculated through vector addition, taking into account all component contributions.

Trigonometric Methods

Trigonometric functions such as sine, cosine, and tangent are used to translate between the magnitude and direction of a force and its components along predefined axes. These methods are crucial for resolving vectors and finding unknown variables in force problems.

Transcript

00:01 In this question, problem number 15 of the book technical mathematicians with calculus 6th, chapter 8, section 6, we are given that a force of 125 newtons acts in the west direction on the body.

00:24 So, f1 equals 125 newtons in the west direction, and another force acts at the angle at the angle.

00:33 Of theta equals 28 .7 degrees from the north we are given that the resultant the resultant force has the value of 212 newton so this is f2 which is at an angle of 28 .7 degrees and we want to find out what the f2 value is and what the angle is of the resultant is.

01:11 So we'll call the resultant r.

01:17 So we have we can break the force f2 into two pieces if 2 along the y axis and after along the x -axis.

01:29 So here we break f2 into f2 cost theta along the y axis and f2 sine theta along the x -axis.

01:39 Theta is 28 .7.

01:43 So our resultant along the x -axis now is the sum of two forces, f1, which is along the x -axis, and f -2 -sign -theta, which is also along the x -axis.

01:58 So, resulted along x is 125, which is f -1, plus f -2 -sine -theta, and we don't know what f -2 is yet.

02:09 Along the y -axis if one has no component f2 has the component f2 cost theta so along the y -axis we have the value f -2 cost theta for the resultant r now the magnitude of r is r x square plus r y squared on the root of the whole thing so let me write it here magnitude of r is equal to root of rx square plus ry square using this we have r equals r x and our y values are just about this if we solve for this by taking r square and then expanding removing the under root first from here after removing the under root we expand the terms inside and we get this is a plus b whole square basically so this is a square plus b square plus 2 a b 125 square plus f2 sine square theta plus f2 core square theta i remove theta here because it's 28 .7 everywhere so we can take together f2 sine square theta and f2 cause square theta and we have f2 sine square plus core square theta sine square plus core square is 1 so we have we can ignore this 1 multiplied by f2 square is f2 square then we have a 2 a b term from here becomes 2 multiplied by 1 25 which is 250 f2 sine teta now we know the value of theta so we can we can insert all these values from our calculator we also know the value of r2 square which is 212 square so that is 44944 so r2 square equals this is here f2 is here and sign 28 .7 is 0 .48...

Please give Ace some feedback