Module 05
Force and Newton's laws
Carry a bowling ball and a coin to a balcony. Set the air aside; give it a still day and a short drop. Ask anyone which object the earth pulls harder, and they answer correctly: the bowling ball, by a mile. You can feel the difference in your two hands.
Drop them together anyway. They land together. Not approximately together: together, to the limit of any clock you bring.
Sit with how strange that is. One object is pulled toward the ground with hundreds of times the strength, and the extra pull buys it nothing at all. A harder pull should mean a faster fall; every intuition built by pushing furniture says so. So why does doubling the pull not double the fall?
The takeaway
A force is a push or a pull; mass is the stubbornness that resists it; acceleration is their ratio. The bowling ball is pulled harder and is exactly that much more stubborn, and the ratio cancels to the last decimal.
The rule, in three sentences
Everything module 04 left unexplained is supplied by three sentences, published together in 1687 and load-bearing ever since.
First: left alone, velocity keeps. A body with no pushes on it holds its velocity, whether that velocity is zero or enormous. Rest needs no explanation and neither does cruising; only change does. This flips the everyday picture, where things "naturally slow down." They do not. The slowing was friction all along, a push in disguise, and module 01's coasting Rule R was closer to nature than the furniture ever suggested.
Second: when pushes act, they buy acceleration, at a price. The total push on a body, a vector called force, produces acceleration in the force's direction. How much acceleration one unit of force buys depends on the body: that exchange rate is its mass. Big mass, expensive acceleration. This is the rule the demon was handed for free in module 04, now with a supplier: forces are where acceleration comes from.
Third: pushes come in pairs. Press the wall and the wall presses back, equally, oppositely, always. There is no such thing as a one-sided push. This one looks like a curiosity today; in module 06 it quietly becomes the most powerful of the three.
Now the balcony resolves. The earth's pull on a body, its weight, happens to scale with the very same mass that sets the body's stubbornness. Four times the mass means four times the pull and four times the resistance to being pulled. The two effects are a fraction whose top and bottom grow together, and every body, bowling ball or coin, falls at the same . Whether those two roles of mass, the pulled thing and the stubborn thing, had to be the same is a question classical mechanics never answers. It measures that they are, to exquisite precision, and moves on. Four volumes from now, that coincidence turns out to be the doorway to general relativity.
The working tool this module hands you is the force inventory, drawn as a free-body diagram: isolate one body, draw every push acting on it, and add the arrows like the vectors they are. The bookkeeping analogy is exact enough to state with its crack: a free-body diagram is an account ledger for pushes, and it breaks the way ledgers break, by omission. Forget one entry, a hidden friction, a forgotten normal force, and the books balance to the wrong answer without complaint.
The math
The second law, as an equation. The sum of forces on a body equals its mass times its acceleration:
It is a vector equation: one copy per component. Forces in newtons; one newton accelerates one kilogram at one meter per second squared.
Read it solved for the quantity the whole course has been chasing:
State in, forces summed, acceleration out, integrate as in module 04, repeat. That loop is classical mechanics.
Apply it to the balcony. Weight is the one force acting, and experiment says its size is , proportional to mass:
The cancels. Nothing about the body survives into its fall. That cancellation is the entire cold open.
One warning the equation itself gives you: force determines acceleration, never velocity. A body can move left while pushed right; it is slowing, not disobeying. The push in your hand and the motion of the thing are different columns of the ledger, related only through change.
Play with it
One block, every force on it drawn live. Push it, load it, switch the floor's friction on. Watch the ledger balance itself, and find the setting where the push is real, the block moves, and the acceleration is still zero.
Check yourself
Problem 1. A 2 kg block on a frictionless floor takes a steady 10 N push. What is its acceleration?
Reveal the solution
a = F/m = 10/2 = 5 m/s², in the direction of the push.
Problem 2. Same block, but the floor now drags on it with 4 N of friction. What is the acceleration under the 10 N push? And what happens under a 4 N push?
Reveal the solution
Net force: 10 - 4 = 6 N, so a = 6/2 = 3 m/s².
Under a 4 N push the net force is zero, so a = 0: if the block is already moving it keeps moving at constant velocity. It does not stop. Zero net force preserves velocity; it does not erase it.
Problem 3. A 70 kg person stands on a scale in an elevator accelerating upward at 2 m/s². Take g = 9.8. What does the scale read?
Reveal the solution
Two forces act: the scale pushing up with N, weight mg pulling down. The second law along the vertical: N - mg = ma, so N = m(g + a) = 70 × 11.8 = 826 N. The scale reads the push it supplies, and the pair rule says the person presses down on it with the same 826 N.
What this did to the demon
The demon finally holds the full algorithm: read the state, sum the forces, divide by mass, integrate, repeat forever. In principle it is finished. Then hand it a fireworks burst, ten thousand fragments, every fragment pulling on every other, and watch the ledger explode into fifty million entries per tick. Module 06 discovers the first great shortcut: one number the whole mess cannot change.
Many particles