Topic 2.2 Forces and Free-Body Diagrams: identify the forces acting on a chosen object, represent them on a free-body diagram, and resolve them into components on chosen axes to find the net force.

What this topic is asking

The College Board (Topic 2.2) wants you to identify the forces acting on a chosen object, represent them on a free-body diagram, and resolve them into components on convenient axes to find the net force. The free-body diagram is the single most important habit in the course: nearly every dynamics problem, from inclines to pulleys to circular motion, begins with one, and AP graders award points for it directly.

Contact and field forces

The forces in AP Physics C: Mechanics fall into two kinds. Field forces act at a distance, chiefly gravity (weight $mg$). Contact forces act where objects touch: the normal force perpendicular to a surface, friction along it, tension in ropes, the spring force, and any applied push or pull. Drag (a resistive force) also acts on objects moving through a fluid. Every force on a free-body diagram must have an identifiable source, another object that exerts it.

Drawing a correct free-body diagram

The discipline is to ask, for each candidate arrow, "what object exerts this force on my object?" If you cannot name a source, the arrow does not belong. In AP Physics C the convention is to draw each force as a single straight arrow from the dot, not broken into components on the diagram itself; you resolve into components in the equations afterwards. Getting this right prevents the most common error in the course, inventing a forward "force of motion."

Choosing axes and resolving

Once the diagram is drawn, pick a coordinate system and resolve every force onto it. The smart choice often is not horizontal and vertical: on an incline, tilt the axes so one points along the slope and the other perpendicular to it, so the normal force and the acceleration each lie along an axis. With angle $\theta$, the weight resolves into $mg\sin\theta$ along the slope (down it) and $mg\cos\theta$ perpendicular (into it). Then the net force on each axis is the algebraic sum of the components:

These net components feed straight into Newton's second law, $F_{net,x} = ma_x$ and $F_{net,y} = ma_y$, in the next topic.

Try this

Q1. List every force on a book resting on a horizontal table, and name the object that exerts each. [2 points]

• Cue. Weight (Earth, down) and normal force (table, up). With no horizontal push there is no friction.

Q2. A $5.0$ kg sign hangs from two symmetric cables each at $30^\circ$ from the horizontal. Calculate the tension in each cable ($g = 9.8$ m/s squared). [3 points]

• Cue. Vertical balance: $2T\sin 30^\circ = mg$, so $T = \dfrac{(5.0)(9.8)}{2(0.5)} = 49$ N.