Two boys are pulling a box across a horizontal floor as shown in the figure below. vector F 1 = 50.0 N and vector F 2 = 100 N.

(a) Find the resultant (or sum) force by the graphical method.

(b) Find the resultant (or sum) force by the component method.

204.9N

To find the resultant force by the graphical method, we can use the concept of vector addition. In this method, we will draw a scaled diagram representing the given vectors and then measure the magnitude and direction of the resultant vector.

(a) Graphical Method:

1. Draw a scaled diagram: Start by drawing a horizontal line to represent the floor. Choose a convenient scale, for example, 1 cm on the diagram can represent 10 N of force. Draw a vector representing F1 with a length of 5 cm (50 N) in the direction shown in the figure. Next, draw a vector representing F2 with a length of 10 cm (100 N) in the direction shown in the figure. Make sure to keep the angle and direction consistent.

2. Complete the parallelogram: Starting from the tail of the first vector (F1), draw a line parallel to the second vector (F2), originating from the head of the first vector. Continue this line until it intersects with an imaginary line parallel to the first vector from the head of the second vector (F2). This forms a parallelogram.

3. Determine the resultant vector: Draw a line from the tail of the first vector (F1) to the point of intersection of the parallelogram. This new line represents the resultant force. Measure its length on the diagram and convert it back to real values using the scale chosen.

(b) Component Method:

1. Resolve the vectors: Break down each vector (F1 and F2) into its horizontal (x) and vertical (y) components. The x-component of F1 is F1x = F1 * cos(theta1), where theta1 is the angle of F1 with respect to the horizontal axis. Similarly, the y-component of F1 is F1y = F1 * sin(theta1). Repeat this process for vector F2 to obtain F2x and F2y.

2. Add the x-components: Sum up the x-components of the vectors. Rx = F1x + F2x. This represents the x-component of the resultant force.

3. Add the y-components: Sum up the y-components of the vectors. Ry = F1y + F2y. This represents the y-component of the resultant force.

4. Find the magnitude and direction: Calculate the magnitude of the resultant force using the Pythagorean theorem: R = sqrt(Rx^2 + Ry^2). To find the direction, use the inverse tangent function: theta = arctan(Ry/Rx).

By following either the graphical method or the component method, you can find the resultant force (sum of the forces) exerted by the two boys pulling the box across the horizontal floor.

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