AP PHYSICS 1 AND 2 › UNIT 2, FORCE AND TRANSLATIONAL DYNAMICS › TOPIC 2.2
Forces and Free-Body Diagrams
A force is not a property an object carries around. It is an interaction between two objects, and the free-body diagram is the picture that keeps you honest about which two. Draw it correctly and the algebra is bookkeeping. Draw it carelessly and no amount of algebra will save it.
2.2.A Describe a force as an interaction between two objects, and say why an object cannot exert a net force on itself.
2.2.B Draw a free-body diagram for a chosen object, and use it to write the equations that describe the situation.
1. A force is a sentence with two nouns
A force is never something an object has. It is something that happens between two objects, and you can test any arrow you have drawn by trying to finish this sentence: the ——— pushes or pulls on the ———. If you cannot fill both blanks with objects, the arrow does not belong on the page.
The consequences are immediate. A ball in flight has no arrow for the throw, because the hand stopped touching it. A crate sliding across a floor has no forward arrow, because nothing is in front of it doing any pushing. And nothing can exert a net force on itself: your muscles pull on your bones as hard as your bones pull back, and the pair contributes nothing to how you move as a whole. That is Topic 2.1 again, one level down.
2. Touching, and reaching across a gap
Almost every force in this course is a contact force: normal, tension, friction, the push of a hand, the pull of a spring. They exist only where two surfaces are actually in contact, and what they really are is the enormous number of electric interactions between the atoms on either side, added up until only the average survives.
Weight is the exception. Gravity reaches across empty space, which is why a falling object still has one arrow on its diagram and a resting one has two. When a surface stops touching an object, its arrow leaves the diagram at once, and forgetting to remove it is one of the two or three errors that cost the most marks in this unit.
3. One dot, one arrow, no components
The free-body diagram is the whole method compressed into a picture. The object becomes a single dot, because you have already agreed to treat the system as one object at its center of mass. Every force exerted on it by something else becomes one straight arrow starting at the dot and pointing the way the force points. Nothing else goes on the diagram: no velocity, no acceleration, no arrows for forces the object exerts on other things, and no dashed component arrows. Components belong on a separate sketch if you want them at all.
If two forces point the same way, draw them side by side rather than on top of one another, so that a reader can count them.
A crate on a smooth floor, pulled by a rope at an angle. The picture is on the left and its free-body diagram on the right: one dot, one straight arrow per force. Raise the angle and the normal arrow shrinks, because a floor only has to supply what the rope does not. Push far enough and the crate leaves the floor, and the diagram loses an arrow entirely.
4. Tilt the axes to the motion
The diagram is drawn. Turning it into algebra means choosing axes, and the choice is free, so choose well: put one axis along the direction the object is accelerating. Then one equation carries the motion and the other carries the balance, and neither of them mixes the two.
On an incline that means tilting the axes to lie along and across the surface, rather than keeping them level and upright. The block accelerates along the slope and not at all across it, so the across-slope equation immediately gives the normal force and the along-slope equation gives the acceleration. Keep the axes horizontal and vertical instead and you get two equations, both with two unknowns, describing the same simple slide.
along the acceleration: ΣF = maacross it: ΣF = 0
5. The arrows that are not there
Most diagrams that fail do not fail by leaving something out. They fail by adding an arrow with no second noun. Three of them account for nearly all of it: a leftover force from whatever set the object moving, a force belonging on the other object in a pair, and an invented force of motion.
Three situations, each with the diagram that is correct. The last button adds the arrow that students most often draw, in red, with the reason it does not belong. Read the label on each real arrow and check that it names the other object; then look at the red one and try to.
Check yourself
1. A ball is thrown straight up. At the highest point it is momentarily at rest. Draw its free-body diagram and explain, in terms of the two-noun test, why there is no upward arrow.
One arrow: weight, downward, from the Earth. Nothing is touching the ball, so there is no contact force to draw, and the hand that threw it stopped interacting with it at release. Being momentarily at rest says nothing about the forces; it is a fact about the velocity, and the diagram is about the acceleration.
2. A book sits on a table. A student draws three arrows: the weight, the normal force from the table, and the force of the book on the table. Which one does not belong, and where does it belong instead?
The third. It is a real force, but it is exerted on the table, not on the book, and a free-body diagram carries only the forces exerted on the object it is drawn for. It belongs on the free-body diagram of the table. Two arrows remain on the book, equal and opposite, and the book stays put.
3. A crate of mass 10 kg is pulled by a rope at 30° above the horizontal with a force of 60 N. Find the normal force, and state the angle at which the crate would leave the floor.
Vertically the floor supplies whatever the rope does not: N = 98 − 60 sin 30° = 98 − 30 = 68 N. The crate lifts when the upward part of the pull reaches the full weight, which needs sin θ = 98/60, greater than one, so at this pull it never lifts however you angle it. A pull of at least 98 N is needed before any angle will do it.
4. A block slides down a rough incline at a steady speed. Say which axes you would choose and why, and name every arrow on the diagram.
Axes along and across the surface, because the acceleration is along the slope and is zero across it. Three arrows: the weight straight down from the Earth, the normal force perpendicular to the surface from the incline, and friction up the slope from the incline. Steady speed means the along-slope forces balance as well, which is a fact about this particular case and not about inclines in general.
5. Two students argue about a car speeding up along a straight road. One says there must be a forward arrow labeled the force of the engine. Settle it by naming the objects involved.
The engine is inside the car, so any force it exerts is internal and cannot accelerate the car as a whole. The forward arrow on the diagram is friction from the road on the tires: the road is the other object, and it is the only thing in contact with the car that can push it forward. The engine turns the wheels, and the road does the pushing, which is why a car on frictionless ice goes nowhere with the engine screaming.
Topic 2.3, Newton’s Third Law. Every arrow you have just learned to draw has a twin somewhere else, on the diagram of the other object, equal in size and opposite in direction. The next topic is about where that twin lives and why it never cancels the first one.