Vectors and Free-Body Diagrams
Key Takeaways
- Vector questions require both magnitude and direction, so signs, arrows, angles, and components carry meaning.
- The 2025 Physics Reference Tables give component relationships using cosine, sine, and tangent, but the diagram determines which component is adjacent or opposite.
- A free-body diagram should show only the forces acting on the selected object, not forces exerted by that object on something else.
- Weight points toward Earth, normal force is perpendicular to the contact surface, and friction acts along the surface opposing relative motion or impending motion.
- Third-law force pairs act on different objects, so they do not both appear as forces on the same free-body diagram.
Vectors Are Physics Claims
A vector is not just a number with an arrow added later. It is a claim about magnitude and direction together. On the Physical Science: Physics Regents, vectors appear in displacement, velocity, acceleration, force, momentum, impulse, and fields. If direction changes, the vector changes even when the size stays the same.
The reference tables include vector component relationships. If a vector A is measured from the positive horizontal axis, the horizontal component is Ax = A cos theta and the vertical component is Ay = A sin theta. The tangent relationship connects the components and the angle.
Those equations do not replace a diagram. The angle in the prompt must match the angle in your component triangle. If the angle is measured from the vertical instead of the horizontal, sine and cosine switch roles for horizontal and vertical components.
Component Thinking
Components let you analyze perpendicular directions separately. Horizontal forces affect horizontal acceleration. Vertical forces affect vertical acceleration. In projectile motion, horizontal and vertical velocity components follow different rules when air resistance is ignored.
| Situation | Useful component idea |
|---|---|
| Pulling a sled at an angle | split the pull into horizontal and vertical parts |
| Object on a ramp | split weight into parallel and perpendicular parts |
| Projectile launch | separate initial velocity into horizontal and vertical components |
| Two-dimensional displacement | combine east-west and north-south parts |
A component can be negative if it points opposite the chosen positive direction. The sign is not an error. It is the direction information written mathematically.
Building a Free-Body Diagram
A free-body diagram isolates one object and shows all external forces acting on that object. It does not show velocity, acceleration, path, or the forces the object exerts on other objects. Those may be important, but they are not forces on the selected body.
Use this checklist:
- Name the object being analyzed.
- Replace the object with a dot or simple box.
- Draw weight straight downward toward Earth.
- Draw normal force perpendicular to the contact surface.
- Draw tension along a rope or string.
- Draw friction along the surface opposite sliding or possible sliding.
- Draw applied forces in the direction they act.
- Label each force with a symbol or description.
If the object is a box on a ramp, the normal force is not vertical. It is perpendicular to the ramp surface. Weight is still vertical downward. Static friction on a resting box on an incline points up the ramp if the box would otherwise slide down.
Real Forces, Not Motion Arrows
Students often draw an arrow in the direction of motion and call it force. That is only correct if a real interaction is pushing or pulling that way. A puck sliding right on smooth ice can move right even when no rightward force acts on it. Motion does not require a force in the direction of motion.
Likewise, acceleration is not a separate force. Acceleration is the result of the net force. If a diagram already shows gravity, normal force, friction, and tension, do not add an acceleration arrow as though it were another interaction.
Weight, Normal, and Friction
Weight is the gravitational force Fg = mg. Near Earth, the reference tables give g = 9.8 N/kg for gravitational field strength and 9.8 m/s^2 for free-fall acceleration. The same symbol is used in related but distinct ways.
Normal force is a contact force perpendicular to a surface. It is not automatically equal to weight. It equals weight for a simple object at rest on a horizontal surface with no other vertical forces. On a ramp or with an angled pull, the normal force changes.
Friction is also a contact force. Kinetic friction acts opposite sliding motion. Static friction acts opposite the tendency to slide and can adjust up to a maximum value. In Regents explanations, describe what motion or possible motion the friction is opposing.
Third-Law Pairs and Free-Body Diagrams
Newton's third-law pairs are equal in magnitude and opposite in direction, but they act on different objects. If a student pushes a wall, the force of the student on the wall and the force of the wall on the student are a pair. A free-body diagram of the student includes the wall's force on the student, not the student's force on the wall.
This matters for net force. Third-law pairs do not cancel when analyzing one object because they are not both acting on that one object. Forces cancel only when they act on the same object in opposite directions with equal magnitude.
Resultants and Equilibrium
The resultant is the vector sum. In a force problem, the resultant force is the net force. If the net force is zero, the object is in translational equilibrium: it may be at rest or moving with constant velocity. Zero net force does not always mean zero velocity.
For two perpendicular components, the magnitude of the resultant can be found with the Pythagorean relationship shown in the vector table. The angle can be found with tangent. Always state the direction relative to a clear reference, such as north of east or above the horizontal.
Diagram Quality on Constructed Response
A clean diagram can earn credit even when the arithmetic is unfinished. Use a ruler when a scaled vector is requested. Put arrowheads at the ends. Label forces with names or symbols, not vague letters that the scorer has to guess.
If the prompt asks for a free-body diagram, do not add extra scene details. The diagram should be simple enough that the net force is visible. A correct simple diagram is better than a realistic picture with unlabeled arrows.
The best habit is to draw first and calculate second. Once the forces and components are organized, the equation usually becomes obvious.
A box is at rest on a rough ramp and would slide down the ramp if the surface were smooth. Which set of force directions belongs on the free-body diagram of the box?