A man is dragging a trunk up the loading ramp of a mover's...

A man is dragging a trunk up the loading ramp of a mover's truck. The ramp has a slope angle of $20.0^{\circ},$ and the man pulls upward with a force $\vec{F}$ whose direction makes an angle of $30.0^{\circ}$ with the ramp (Fig. E4.4). (a) How large a force $\vec{F}$ is necessary for the component $F_{x}$ parallel to the ramp to be $90.0 \mathrm{~N} ?$ (b) How large will the component $F_{y}$ perpendicular to the ramp be then?

Key Concepts

Free-Body Diagrams

Free-body diagrams are graphical representations that illustrate all the forces acting on an isolated object. By clearly depicting the directions and magnitudes of these forces, free-body diagrams provide a visual tool for setting up equations of motion and balance, particularly useful when multiple forces act at various angles.

Inclined Plane Analysis

Inclined plane analysis involves examining forces acting on an object on a sloped surface. In problems involving inclined planes, gravitational and applied forces are typically resolved into components parallel and perpendicular to the surface, which aids in the study of motion or equilibrium along the plane.

Vector Decomposition

Vector decomposition is the process of breaking a single force vector into its constituent components along specified directions, typically along mutually perpendicular axes. This allows one to analyze the effects of each component independently, such as one component causing motion along a surface and another affecting the normal interaction with that surface.

Trigonometric Relationships

Trigonometric functions such as sine and cosine are fundamental in resolving a force vector into its components when the force is applied at an angle. The cosine function is used to determine the component adjacent to the angle, while the sine function is used for the component opposite the angle, enabling precise calculations of each directional influence.

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