Net Force Makes Acceleration, ΣF = ma

Give a force F to a box on a smooth floor with the slider. The larger F, the larger the acceleration a; force and acceleration are in direct proportion. Double F and a doubles too. As a formula, a = Fm, or turned around, F = ma. The important thing here is that force does not set the speed directly, it sets the acceleration. Apply a force and it speeds up; remove the force and it keeps going at that speed, neither speeding up nor slowing. If the force is zero the acceleration is zero, so the state, at rest or at constant velocity, is kept. This is the law of inertia and the starting point of the second law.

In reality several forces act on one object at once. Suppose a box has a force pushing it right and another pushing it left. Adjust the rightward force with the slider. What sets the acceleration is not the individual forces but the net force, the two added together. If the right is stronger, a net force remains to the right and it accelerates right; if the left is stronger, it accelerates the other way. When the two are equal the net force is zero, so the acceleration is zero, and even amid all the pushing and pulling the object stays at constant velocity or at rest. That is why the F in the second law always means the net force, the vector sum of all forces ΣF. Not each individual force, but only their sum changes the motion.

To get the net force right, you cut out the one object and draw every force acting on it, leaving none out. This is the free body diagram (FBD). Switch the situation with the buttons. A box resting on the floor has its weight mg down and the normal force N up, the two balancing vertically. A weight hanging from the ceiling has its weight mg opposed by the tension T in the string. A box being pushed adds the push and friction on top of those. The rule is simple: draw only the forces that touch the object (contact forces) or pull on it from afar (like gravity), and never the forces the object exerts on others. Once you gather all these arrows and add them by direction, you get the net force, and it equals ma.

Now fix the force and change the mass. Under the same push, a light object speeds up easily while a heavy one is sluggish. Drag the m slider and, following a = Fm, the larger the mass the smaller the acceleration, in inverse proportion. Double the mass and the acceleration for the same force is halved. So mass is not merely heaviness but the degree to which something resists a change in its state of motion, the amount of inertia. It is like an empty shopping cart that shoots off at a light push while a full one barely moves even when you put your whole weight into it. For the same net force, a different mass gives a different resulting acceleration.

Finally, a box on an incline. With no friction, the only forces acting are the weight mg pointing down and the normal force the surface pushes back with. But the acceleration happens only along the incline, so we split the weight into two directions: the component parallel to the incline, mg sinθ, and the component perpendicular to it, mg cosθ. The perpendicular component is exactly canceled by the normal force, merely pressing on the surface without producing motion, while only the parallel component remains to slide the box. Drag the angle θ. The steeper the slope, the larger sinθ, so the acceleration a = g sinθ grows, and as θ nears 90 degrees it approaches free fall g. When θ is zero so is sinθ, and on flat ground it does not slide. The key is to decompose the net force along the direction in which motion happens.