AP PHYSICS C: MECHANICS › UNIT 2, FORCE AND TRANSLATIONAL DYNAMICS › TOPIC 2.4
Newton’s First Law
The simplest case in the unit, and the one that took longest to believe: when the arrows add to nothing, the motion does not change. Not stop, not slow, not settle. Continue exactly as it was.
2.4.A Describe the conditions under which the velocity of a system stays constant, add forces as vectors to find the net force, and recognize translational equilibrium.
The whole topic on one sheet, the same one handed out in class. Click it to open the full-size version, which prints cleanly on a single page.
1. Add the arrows, and add them as vectors
The free-body diagram gives you a collection of arrows. The net force is what you get by adding them all, head to tail, as vectors. Not the biggest one, not the sum of the sizes, and not the count.
ΣF = F1 + F2 + F3 + …added as vectors, not as numbers
In practice you add them one axis at a time: all the sideways parts into one sum, all the up and down parts into another. Two numbers, and the whole diagram is accounted for.
Two of the three forces are fixed. Set the size and direction of the third until both sums read zero and the dot turns green. There is exactly one answer, and the red arrow is telling you how far away you are from it.
2. Equilibrium is a statement about the sum, not the arrows
Translational equilibrium is the condition that the net force is zero. It can be reached with no forces at all, with two forces, or with seven; all that matters is that they add to nothing.
ΣF = 0in every direction at once
And then the first law: if the net force on a system is zero, the velocity of its center of mass does not change. Read that carefully. It does not say the system is at rest. Constant velocity includes zero velocity, but it also includes a spacecraft coasting at eleven kilometers a second, and the two situations have identical free-body diagrams.
A puck sliding on a surface, photographed every 0.4 seconds, with its speed graphed beneath. Two of these three situations produce the same graph, and they look nothing alike on the ice.
3. Balanced one way, unbalanced another
The two sums are independent, and a system can be in equilibrium along one axis while being accelerated along the other. A ball thrown horizontally has nothing pushing it sideways, so its horizontal velocity is constant, exactly as the first law promises; meanwhile gravity is changing its vertical velocity the whole time.
This is why the axis choice in Topic 2.2 mattered. Line one axis up with the acceleration and the other equation becomes an equilibrium statement, which is usually the one that hands you the normal force or the tension for free.
4. The frames in which any of this is true
The first law is not true everywhere. Stand a coffee cup on the dashboard, brake hard, and the cup slides forward across the dash with nothing pushing it forward. Inside that braking car the first law appears to fail.
A frame in which the first law does hold is called an inertial reference frame, and the practical test is the one just described: put something where nothing touches it and see whether it keeps a constant velocity. The ground is close enough to inertial for this whole course. An accelerating car, a turning car and a spinning carousel are not, and the invented forces people reach for inside them, the ones that push you outward or forward, are the sign that the frame is the problem rather than the physics.
5. Reading it wrong
Constant velocity means at rest. It means the velocity is not changing. Steady motion in a straight line is equilibrium just as much as sitting still, and the exam asks about the moving case far more often.
A moving object must have a force on it. Only if it is speeding up, slowing down or turning. An object going at a steady speed in a straight line needs nothing, and if something is pushing it, something else is pushing back just as hard.
Adding the sizes instead of the vectors. Two 50 N forces at right angles give 71 N, not 100 N, and if they are opposite they give nothing at all.
Check yourself
1. An elevator moves upward at a steady 3 m/s. Draw the diagram for a 70 kg passenger and find the normal force. Then say what changes if the elevator instead moves downward at a steady 3 m/s.
Steady speed means zero acceleration means equilibrium, so the normal force equals the weight, about 690 N, in both cases. Nothing changes when the direction of travel reverses, because the first law is about the velocity changing, not about which way it points. Only an accelerating elevator changes the answer.
2. Three forces act on a point: 40 N east, 30 N north, and one more that puts the point in equilibrium. Find the size and direction of the third force.
The first two sum to 50 N at 37° north of east. The third must be 50 N pointing the opposite way, that is 37° south of west. A common wrong answer is 70 N, which comes from adding the sizes rather than the vectors.
3. A crate slides across a floor at a steady speed while a worker pushes it horizontally with 120 N. What is the friction force, and what would happen if the worker pushed with 140 N instead?
Steady speed means the horizontal forces balance, so friction is 120 N backward. With 140 N the net force is 20 N forward and the crate speeds up. Note that friction did not change: it depends on the surfaces and the normal force, not on how hard the worker is pushing.
4. A book slides across a table and comes to rest. A student says this proves that motion needs a continuous force to keep it going. Correct them, and say what evidence would settle it.
The book stops because friction is pushing it backward, not because motion runs out. The evidence is to reduce the friction and watch the stopping distance grow: a puck on ice goes far, a puck on an air table goes until it hits something. Extrapolating to no friction at all gives the first law, which is why it took so long to find. Nobody can arrange the experiment perfectly on Earth.
5. A coffee cup slides forward across the dashboard when a car brakes hard. Describe what happens from the road and from inside the car, and say which description needs an invented force.
From the road: nothing pushes the cup forward, so it keeps its constant velocity while the car slows underneath it. That is the first law, unaided. From inside the car: the cup appears to accelerate forward with nothing touching it, so an observer has to invent a forward force to explain it. The invention is the signal that the car is not an inertial frame, and it is the frame that needs fixing, not the law.
Topic 2.5, Newton’s Second Law. The first law says what happens when the sum is zero. The second says exactly how much happens when it is not, and it is the equation the rest of this unit is built out of.