Resolving and Adding Forces Acting on an Object on a Frictionless Surface

In this article, we will delve into the process of resolving and adding forces acting on an object on a frictionless surface. Specifically, we will calculate the overall net force when two forces are applied to a box on such a surface.

Problem Context

A box is subjected to two forces on a frictionless surface: one force of 40 N downward [S], and another force of 70 N at 30° west of north [N30°W]. Our goal is to determine the overall net force acting on the box.

Force Resolution into Components

To find the overall net force, we first need to resolve the two given forces into their x and y components. We will then sum these components to determine the net force.

Force 1: 40 N [S]

This force acts vertically downward.

Horizontal component (F1x): 0 N Vertical component (F1y): -40 N (negative because it is directed south)

Force 2: 70 N [N30°W]

Force 2 is applied at an angle of 30° west of north.

Horizontal component (F2x): -70 cos 30° N (negative because it is directed west) Vertical component (F2y): 70 sin 30° N (positive because it is directed north)

Using the trigonometric functions:

cos 30° √3/2 ≈ 0.866 sin 30° 1/2 0.5

Calculating the components:

F2x -70 × 0.866 ≈ -60.62 N F2y 70 × 0.5 35 N

Summing the Components

Next, we sum the x and y components of the forces to find the net force.

Net Force in the x-direction (Fnetx): F1x F2x 0 (-60.62) -60.62 N Net Force in the y-direction (Fnety): F1y F2y -40 35 -5 N

Calculating the Magnitude and Direction of the Net Force

To determine the overall net force, we calculate its magnitude and direction.

Magnitude (Fnet): Fnet √(Fnetx2 Fnety2) √((-60.62)2 (-5)2) ≈ √(3675.0644 25) ≈ √3700.0644 ≈ 60.83 N Direction: The angle (θ) can be found using the tangent function. Since both components are negative, the direction is in the third quadrant (southwest). θ tan-1(Fnety/Fnetx) tan-1(-5/-60.62) ≈ tan-1(0.0826) ≈ 4.73°

Conclusion

The overall net force acting on the box is approximately 60.83 N at an angle of 4.73° south of west.

Alternative Calculation Using a Graph

Alternatively, the net force can be computed using the graphical method:

Net force to the North (Fnet in y-direction): 70 cos 30° - 40 N ≈ 20.6 N Net force to the West (Fnet in x-direction): 70 sin 30° N ≈ 35 N Overall net force: √(20.62 352) ≈ 40.6 N Direction of net force: N arctan(20.6/35) W ≈ N 30.5° W

By following these steps, we have successfully resolved and added the forces to find the overall net force acting on the box.