The Plus/Minus sign on Forces in a Cartesian coordinate system

Question

I have been struggling with Forces in a Cartesian Coordinate System and whether to understand what signs to put on to solve simple problems in the view of mathematics.

Let's make a simple one dimensional problem, in the y-axis.

I have an increasing $+yî$ direction up and we have gravity force that points down in the frame of reference. Most books will give it $\vec{F}_yî=m(-\vec{g})î$. Why is the minus sign only for the gravitational acceleration in the expression? Is it always opposite sign in the algebraic expressions in situations like this? Meaning $\vec{F}_yî$ has not a minus sign but the acceleration on the other side of the equation does. Even thou the force and acceleration is pointing down.

Does the unit vectors have any impact to tell in what direction the forces point and then declares it signs? Please help!

Answer 1

In a Cartesian coordinate system, the sign convention for forces and accelerations is crucial to ensure consistency and accurate calculations.

In your problem, you have an increasing $+y$ direction up, and gravity points down. To maintain a consistent sign convention, we typically define the following conventions:

1. Positive and Negative Directions:

   • Positive direction: Upward along the $+y$ axis.
   • Negative direction: Downward along the $-y$ axis.

2. Gravitational Acceleration:

   • The acceleration due to gravity ($\vec{g}$) is typically considered positive when pointing in the negative $y$ direction. This is because gravity always acts downward.

3. Forces:

   • When you write the equation for the force acting on an object near the Earth's surface, such as $\vec{F}_y = m(-\vec{g})\hat{i}$, the negative sign in front of $\vec{g}$ ensures that the force is correctly directed in the opposite direction of the positive $y$ axis. It indicates that the force is acting downward, consistent with the convention that positive forces act in the positive direction.