A force 𝐅1 of magnitude 6.00 units acts on an object at the...

A force $\overrightarrow{\mathbf{F}}_{1}$ of magnitude 6.00 units acts on an object at the origin in a direction $\theta=30.0^{\circ}$ above the positive $x$ axis (Fig. P3.10). A second force $\overrightarrow{\mathbf{F}}_{2}$ of magnitude 5.00 units acts on the object in the direction of the positive $y$ axis. Find graphically the magnitude and direction of the resultant force $\overrightarrow{\mathbf{F}}_{1}+\overrightarrow{\mathbf{F}}_{2}.$

Key Concepts

Graphical Vector Addition

This technique involves representing vectors as arrows in a coordinate system and visually arranging them tip-to-tail. The resultant vector is then drawn from the start of the first vector to the end of the last vector, providing an intuitive understanding of the vector sum and serving as a check against analytical methods.

Trigonometric Analysis for Direction

Determining the direction of a resultant vector involves using trigonometric functions, particularly the inverse tangent function. This step calculates the angle of the resultant vector relative to a reference axis based on the ratio of its vertical to horizontal components.

Vector Decomposition

This concept involves breaking a vector into its horizontal and vertical (or generally orthogonal) components. It is a fundamental step in analyzing forces or any vector quantity because it simplifies complex vector operations such as addition and subtraction by allowing them to be performed component-wise.

Component-wise Vector Addition

Once vectors are decomposed into their components, the addition of vectors is performed by summing their corresponding components. This allows for the systematic computation of the resultant vector's components, which can then be used to determine the overall magnitude and direction.

Resultant Vector Magnitude

After determining the component sums, the magnitude of the resultant vector is calculated using the Pythagorean theorem. This process combines the horizontal and vertical components into a single scalar value representing the overall size of the vector.

Transcript

00:02 Okay, for this problem, we have two vectors.

00:05 The first vector has a magnitude of 6 newtons at a direction of 30 degrees above the positive x -axis.

00:17 The second vector is pointed in the same direction as the positive y -axis and has a magnitude of 5 newtons.

00:28 And the question is asking us to find graphically the magnitude and direction of the resultant force when we add those two vectors together.

00:37 And all we have to do, if we've drawn them like this, is to draw a resultant vector starting from the tail of the first to the tip of the last vector.

00:51 And that is, in fact, the answer to the question.

00:54 That's graphically the result of adding those two vectors together.

00:58 If you wanted to find the numerical answer, though, it's going to take a little bit of math.

01:04 So what we're going to do first is we're going to break this first vector into x and y components, and then we'll break the second vector into x and y components in first.

01:15 So for x, we're looking at six cosine of 30 degrees since we're down here looking for the adjacent side.

01:29 And when we do that math, we get a value of 5 .19 newton's.

01:36 For that component in the x direction.

01:39 In the y direction, it's very similar.

01:42 It is six times the sign of 30 degrees.

01:45 Hopefully we can remember that the sign of 30 degrees is one half, and so we can do this without a calculator if we wanted, but we get an answer of 3 newton's...

Please give Ace some feedback