Draw forces on one object without double counting

1. Orient: Draw forces on one object without double counting

A free-body diagram isolates one object or an explicitly stated system. Each arrow names an interaction acting on that object: Earth on cart, track on cart, or string on cart. Forces the cart exerts on something else belong on a different diagram. Velocity and acceleration can be shown nearby but are not additional force arrows.

2. Explain and derive

Choose axes that make constraints simple. For a cart on a horizontal track with no vertical acceleration, vertical force components sum to zero. If a pull F makes angle θ above horizontal, its components are F cosθ to the right and F sinθ upward. The normal force is then N=mg−F sinθ, not automatically mg.

In the horizontal direction of the frictionless model, F cosθ=maₓ. Increasing the pull angle while holding its magnitude fixed decreases the horizontal component and decreases the normal force. The pull is a single force: drawing it and also counting both of its components would count the same interaction twice.

A normal contact force cannot pull the cart down onto the surface. If the calculation required N<0, the assumed contact would fail and a new motion model would be needed. The bounded settings here keep N positive. On an incline, rotating axes along and perpendicular to the surface makes the gravitational components mg sinθ and mg cosθ; the normal still follows the perpendicular constraint, not a memorized equality.

3. Work an example

For m=4 kg, g=10 m/s² and a 20 N pull at 30°, the horizontal pull is 20 cos 30°≈17.32 N and the upward part is 10 N. Vertical balance gives N=40−10=30 N. Horizontal acceleration is 17.32/4≈4.33 m/s². Do not add a separate “net force” arrow to these interactions.

4. Predict and investigate

Predict first: At fixed pull magnitude, does a steeper upward angle increase or decrease horizontal acceleration and normal force?

Compare pull angles 0°,30° and 60° at 20 N. Draw the three actual interaction arrows before using the ledger, then resolve components in equations. Check the vertical sum and the horizontal sum separately. Predict what would break if an unbounded pull lifted the cart off the track. Use the lesson controls and record the settings with your evidence.

5. Transfer and explain

Try a new practice case

A 5 kg cart is pulled horizontally by 24 N on a frictionless track. Find horizontal acceleration.

Your numerical result (m/s²) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

For a frictionless inclined plane with angle 30°, derive acceleration from mg sinθ=ma. Explain why changing mass does not change this ideal acceleration.

Retain your first answer, force diagram, equations, units, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Supplied synthetic outputs do not certify actual laboratory skill.

Contact mechanism and an exam-style dot diagram

Normal and friction forces are macroscopic descriptions of microscopic electromagnetic interactions at contacting surfaces; gravity acts without contact. A force must still name an external agent and a recipient. A body cannot produce a net force on itself by adding its own internal interactions. In the force analysis here, electrical microscopic detail is explanatory context, not an additional electric-force calculation.

Represent the selected cart by a dot and draw each actual force as a straight arrow originating there. Do not draw component arrows in addition to their parent force, or an extra net-force arrow. Two distinct forces pointing in the same direction must remain distinguishable side by side rather than overlap. Choose an axis along an incline when helpful, while the gravity arrow stays vertical. Changed representation: two strings pull a dot rightward with 3 and 5 N while a third pulls leftward with 4 N. Draw three arrows with a declared common scale, then calculate net force +4 N. For a 2 kg cart the acceleration is +2 m/s². Replacing the two rightward strings by one 8 N string preserves the net acceleration but changes the interaction inventory.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 2 topic mapping: 2.2; 2.5. Checked 30 October 2026. This is a content cross-reference, not AP authorization.