Newton's second law explained

Set the force F on a box on a frictionless floor with the slider; it is the size of the gold arrow. The larger F, the larger the acceleration a shown by the orange arrow above the box, and since the two are directly proportional, doubling F doubles a. In symbols a = Fm, or, turned around, the familiar F = ma. The important point is that force determines acceleration, not velocity. Apply a force and the box speeds up; remove it and the box neither speeds up nor slows down but keeps going at that speed. With zero force, 0 acceleration, so rest or uniform motion simply continues; that is the law of inertia and the starting point of the second law.

In reality several forces act on an object at once. The box in the figure has a fixed blue force pushing it left, and as you adjust the gold force pushing right with the slider, it becomes clear that the acceleration is set not by either force alone but by their sum, the net force. When the right force is stronger, a net force to the right remains and the green acceleration arrow points right; when the left one wins, the arrow turns red and points left. When the two are equal, the net force is 0 and so is the acceleration: even while being pushed from both sides, the object coasts at constant velocity or stays at rest. So the F in the second law, ΣF, always means the vector sum of all the forces; only that sum, not any single force, changes the motion.

To find the net force correctly, cut one object free and draw every force acting on it; this is a free-body diagram (FBD). Switch situations with the buttons and the arrows change from case to case. A box resting on the floor has its weight mg pointing down and the normal force N pointing up, balancing each other. A weight hanging from the ceiling has its weight mg opposed by the string's tension T, and a box being pushed adds the pushing force and friction. The rule is to draw only forces applied directly to that object by contact and forces that act at a distance, such as gravity, and never the forces the object exerts on other things. Add up the collected arrows direction by direction and you get the net force; set it equal to mass times acceleration, ma, and you have the equation of motion.

Now hold the force fixed and change the mass. Push with the same force and a light object speeds up easily while a heavy one is sluggish: drag the mass slider m and the box grows while the orange acceleration arrow shrinks in inverse proportion, as a = Fm says. Double the mass and the acceleration under the same force halves. So mass is not mere heaviness but resistance to a change in the state of motion: inertia. An empty shopping cart zips off at a light push while a full one barely moves under a hard shove; the same net force on different masses produces different accelerations.

Last comes a box on a frictionless incline. The only forces on it are its weight mg pointing down and the normal force from the surface, yet the acceleration happens only along the slope. So split the weight into a component along the surface, mg sinθ, and one perpendicular to it, mg cosθ. The perpendicular component cancels exactly against the normal force and only presses on the surface, while the parallel component is left over and slides the box down. Drag the incline angle θ to steepen the slope and sinθ grows, lengthening the orange acceleration arrow a = g sinθ; as θ approaches 90 degrees it tends to the free-fall acceleration g. At θ of 0, sinθ is 0 too, so nothing slides on level ground. The key is to resolve the net force along the direction in which the motion happens.