Resolving Vectors into Components

We often encounter problems where a vector force is acting on an object at some angle (or multiple forces are acting at various different angles), and we want to determine how the object will move, or how strong the tension is at some point. We know that the sum of all the vector forces acting on an object is equal to the mass times the vector acceleration, and that this is true regardless of what coordinate system we choose, but in many situations, we see that the motion is constrained to occur in one particular direction, so we can often simplify the solution by choosing the `right' coordinate system and breaking the vector equation into separate equations in each coordinate direction. (The direction perpendicular to the motion ends up giving us information about the normal force, which we'll need if friction is present. If there is no friction, we can often ignore that direction and just look at the components of the forces along the direction of motion to solve the problem.)

Examples

• Simple example

• More complicated example

Suggestions

• Make sure you have included all the forces acting on the object. (Is gravity present? If the object is touching a surface, a normal force will be present.)
• Choose the right coordinate system. Is the object constrained to move along one particular direction (sliding across a floor, along an incline, etc)? Then that should be one of your coordinate axes. The other axis must be perpendicular to that one.
• Make sure to convert ALL the forces acting on this object into this new coordinate system. (Some of the forces may already be aligned with your new X or Y axis, but some will probably not be. If the direction of motion is tilted (i.e. not vertical or horizontal), then the weight (the force of gravity) needs to be converted to your new axes as well.)
• When converting a force into it's new X and Y components, make sure to draw the bounding rectangle carefully. It's sides must be in line with the new X and Y axes, it's corners are right angles, and the vector you are converting should run from one corner of this rectangle to the opposite corner. The more carefully you do this step, the easier it is to see what the angles are, and which trig function you'll need to solve for the components.