CONCEPTUAL PHYSICS › CHAPTER 4, NEWTON’S SECOND LAW
Why Things Speed Up, and What Actually Decides How Much
Chapter 3 described motion without ever asking what caused it. This chapter answers that, and the answer is shorter than anyone expects: acceleration is set by the net force, divided by how much stuff you are trying to move.
4.A Find the net force on an object and use it to predict the acceleration, in words and with numbers.
4.B Explain why heavy and light objects fall together, and what changes when air resistance is not negligible.
4.C Tell the difference between a force, a net force, and the motion that results.
1. Net force is the only force that matters
Forces come in crowds. A book on a table is pulled down by gravity and pushed up by the table. A pushed crate is shoved forward by you and dragged back by friction. What decides the motion is never any one of these. It is the net force, the whole collection added up with directions taken seriously.
Add a force pointing one way to an equal force pointing the other way and you get nothing. Not a small motion, not a slow motion. Nothing changes at all. This is why a tug of war can involve enormous effort and no movement whatever, and why the rope is not confused about it.
2. The law itself
Acceleration is directly proportional to the net force and inversely proportional to the mass. Both halves of that sentence do real work.
a = Fnet / m
Double the net force, double the acceleration. Push twice as hard and the thing gains speed twice as fast. That half is the one people guess correctly.
Double the mass, halve the acceleration. The same push on twice the stuff produces half the result. Mass is the resistance to being sped up, which is why it is also called inertia, and it is not the same thing as weight.
Notice what the equation does not say. It does not say force causes motion. It says force causes a change in motion. A puck sliding on ice with no net force keeps going, and needs no force to do so.
Push it and see
A cart on a low-friction track. Set the push and the mass, then start it and watch. The dots are laid down once a second, so their spacing is the speed, and the way the spacing grows is the acceleration. Try doubling the force, then instead doubling the mass, and compare what happens to the dots.
3. Free fall, and the thing everybody gets wrong
Drop a bowling ball and a marble together and they land together. Almost nobody believes this the first time, because the bowling ball is obviously pulled harder. That part is true. It is just not the whole calculation.
The bowling ball is pulled harder, and it also has more mass to move. Both the top and the bottom of the fraction grow by the same factor, and the ratio does not budge.
a = Fnet / m = mg / m = g
The mass cancels. That cancellation is the entire content of the demonstration, and it is why every object in free fall near the earth accelerates at the same rate no matter what it is made of or how much of it there is.
4. Air resistance, which is where the intuition came from
Everyday experience says a feather falls slower than a coin, and everyday experience is not lying. It is describing a case with a second force in it.
Air resistance grows with speed and with the area you present to the air. So a falling object starts with only gravity acting, speeds up, and as it speeds up the air pushes back harder. The net force shrinks. The acceleration shrinks with it. Eventually the air resistance equals the weight, the net force is zero, and the object stops speeding up entirely. That final speed is terminal velocity.
A feather reaches its terminal velocity almost at once, because it presents a lot of area for very little weight. A coin takes far longer. Crumple the feather into a tight ball and it falls much more like the coin, which tells you the difference was never about being light.
Two falls, with and without air
Two balls of the same size and shape, so at any given speed the air pushes back on them equally hard. The only difference is that one is five times heavier. Drag the slider to add air. At zero they fall together, which is the result from section 3. Turn it up and watch which one is stopped first, and ask yourself why the size being equal matters.
Check yourself
1. You push a 4 kg cart with 12 N and it accelerates at 3 m/s². You then load it to 8 kg and push with the same 12 N. Predict the acceleration, and say what push you would need to get 3 m/s² back.
Twice the mass with the same force gives half the acceleration: 12 / 8 = 1.5 m/s². To recover 3 m/s² you need F = ma = 8 × 3 = 24 N, twice the original push. Doubling the mass and doubling the force leaves the acceleration alone, which is the same cancellation that makes everything fall at the same rate.
2. A crate slides across a floor at a steady 2 m/s while you push it. What is the net force on it? What happens the instant you stop pushing, and why is that not a contradiction?
Steady speed means no acceleration, so the net force is zero: your push and friction are equal and opposite. Stop pushing and friction is suddenly unopposed, so the net force is backward and the crate slows down. Nothing contradicts anything. Constant velocity needs zero net force, not zero force, and there were two forces the whole time.
3. A skydiver reaches terminal velocity and then pulls the ripcord. Describe what happens to the air resistance, the net force, the acceleration and the speed, in that order.
The parachute suddenly presents far more area, so the air resistance jumps well above the weight. The net force is now upward, so the acceleration is upward while the motion is still downward, which means slowing down. As she slows, the air resistance falls back until it again equals the weight, the net force returns to zero, and she descends at a new and much smaller terminal velocity. Note that the acceleration points opposite to the motion for a while, which is allowed and is exactly what slowing down means.
4. On the moon there is no air. A hammer and a feather are released together. What happens, and what is different about the moon that matters here? Is it that gravity is weaker?
They land together. The relevant difference is the absence of air, not the weaker gravity. Weaker gravity means both fall more slowly, at about 1.6 m/s² instead of 9.8, but they still fall together, because the mass cancels there exactly as it does here. On earth the same experiment works in a vacuum chamber, and it has been done.
5. Two students argue. One says a heavy object falls faster because gravity pulls it harder. The other says gravity pulls all objects the same. Both are wrong. Fix each of them.
The first is right that gravity pulls the heavy object harder and wrong to stop there: the heavy object also resists being sped up more, in exactly the same proportion. The second has overcorrected: gravity does not pull all objects with the same force, it pulls with a force proportional to mass. What is the same for all objects is the acceleration, because force and mass grow together and cancel. Getting the right answer from the wrong reason is the trap here.
Chapter 5, Newton’s Third Law. This page treated forces as things that act on an object. The next one asks where they come from, and the answer is that a force is always one half of an interaction between two things.