Forces and Free-Body Diagrams

Learning Goals

After this lesson, you should be able to:

• describe a force by naming both interacting objects;
• distinguish contact forces from forces that act across a distance;
• identify the external forces acting on a chosen system;
• construct and evaluate a free-body diagram;
• use arrow direction and relative length to communicate force vectors; and
• convert a diagram into consistent horizontal and vertical component equations.

1. A Force Is an Interaction

A force never appears without an interaction partner. Instead of writing only “normal force,” use an agent–receiver name such as “force of the table on the book.” This naming test exposes invented forces and prevents a force exerted by the system from being placed on its diagram.

Force is a vector: its magnitude is measured in newtons and its direction is part of the physical information.

2. Contact and Long-Range Interactions

The table helps identify possible interactions; it does not mean every listed force acts in every situation. Include a force only when its agent actually interacts with the selected system.

3. Interaction Diagram Before Free-Body Diagram

When a situation is complicated, first place the chosen system at the center and connect it to every interacting object in the environment. For a book resting on a table, the book interacts with Earth and the table. Those two links become the gravitational and normal-force arrows on the book's free-body diagram.

An interaction diagram identifies pairs of objects. A free-body diagram then shows only the force vectors acting on the one selected system.

4. A Reliable Drawing Procedure

1. Name the system. State exactly which object or collection is isolated.
2. List the external agents. Ask what outside the boundary touches or acts at a distance on the system.
3. Replace the system with a point or simple outline.
4. Draw one arrow per force. Start each arrow at the system and point it in the force direction.
5. Label agent and receiver. For example, \(\vec F_{\text{floor on crate}}\).
6. Show relative magnitudes when known. A larger force should have a longer arrow.
7. Choose axes. Use axes that make the component equations simple.

6. What Does Not Belong on the Diagram?

• Velocity and acceleration: they describe motion, not interactions.
• “Centripetal force” as an extra arrow: inward net force is supplied by real interactions.
• The net force as another force: it is the vector sum of the arrows already drawn.
• A force exerted by the system: the diagram contains forces exerted on the system.
• Internal forces: interactions between parts inside the chosen boundary are not external forces on the whole system.

7. Diagram Geometry Is Vector Geometry

Force arrows begin at the isolated object. Their orientations show direction and their relative lengths show relative magnitude when that information is known. Arrows drawn at different angles cannot be compared by their vertical height alone.

Do not infer arrow lengths merely from whether the object is moving. Constant velocity requires zero net force, whereas motion itself does not require a force in the direction of velocity.

8. From Vectors to Component Equations

After the free-body diagram is complete, choose axes and write one force-sum equation per direction:

A zero vertical acceleration gives \(\sum F_y=0\); it does not mean that every vertical force is zero. Signs come from the chosen axes, not from memorized rules.

9. Worked Example: Suspended Sign

Question: A \(12.0\,\mathrm{kg}\) sign hangs motionless from one vertical cable. Draw the free-body diagram and find the cable tension using \(g=9.8\,\mathrm{m\,s^{-2}}\).

The sign interacts with Earth and the cable, so its diagram has downward \(\vec F_g\) and upward \(\vec T\). With upward positive and \(a_y=0\):

Answer: The cable exerts approximately \(118\,\mathrm N\) upward on the sign. Equal arrow lengths are justified by zero acceleration, not merely by the sign's position.

10. Worked Example: Angled Pull

Question: A \(4.0\,\mathrm{kg}\) crate on a horizontal floor is pulled by a \(20\,\mathrm N\) rope at \(30^\circ\) above horizontal. The floor exerts \(6.0\,\mathrm N\) of friction. Find the horizontal acceleration.

The horizontal component of tension is \(T_x=T\cos\theta\). The normal force and weight affect the vertical equation but have no horizontal component.

Answer: The crate accelerates horizontally at about \(2.8\,\mathrm{m\,s^{-2}}\). The final equation uses the tension component, not both \(T\) and \(T_x\).

11. Choosing Axes on an Incline

For an object on a ramp, axes parallel and perpendicular to the surface usually minimize the number of resolved forces. The normal force is perpendicular to the ramp, while gravity remains vertically downward and may be resolved as

Axis rotation does not rotate gravity. It only changes how the same gravitational vector is described by components.

12. Comparing Different System Choices

Consider two boxes touching while a hand pushes the left box.

• Left box only: include the hand's force and the contact force of the right box on the left box.
• Right box only: include the contact force of the left box on the right box.
• Both boxes together: the box–box contact interaction is internal and does not appear on the whole-system free-body diagram.

The system choice changes the diagram but not the underlying event.

13. Experimental Check: Force Vectors on a Cart

Attach two spring scales or force sensors to a low-friction cart so they pull in different horizontal directions.

1. Record each force magnitude and define positive direction.
2. Predict the net force by signed vector addition.
3. Sketch the cart's free-body diagram before releasing it.
4. Measure acceleration and compare \(m a_x\) with the predicted \(\sum F_x\).
5. Repeat with equal opposing forces and predict the motion before observing it.

Uncertainty should be considered when deciding whether measured forces are balanced.

14. Common Traps and Corrections

• Trap: drawing a force in the direction of motion. Correction: name the external agent responsible for every arrow.
• Trap: assuming \(N=mg\). Correction: derive the relationship from the perpendicular force sum.
• Trap: drawing both a vector and its components as separate forces. Correction: use one representation at a time in the sum.
• Trap: placing both members of a third-law pair on one diagram. Correction: each member acts on a different receiver.
• Trap: using unequal arrows for a stationary object without justification. Correction: test whether the stated acceleration is zero.
• Trap: forgetting Earth because it is not pictured. Correction: include relevant long-range interactions.

15. Key Takeaways

• Every force is a vector interaction between an agent and a receiver.
• A free-body diagram contains all and only the external forces acting on one selected system.
• Motion arrows, net force, and forces exerted by the system are not extra force arrows.
• Arrow geometry communicates vector direction and relative magnitude.
• Component equations must follow the chosen axes and the completed diagram.
• Changing the system boundary can change which interactions appear as external forces.