#02 Free Body Diagram

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. One Drawing Before Any Equation

   

   50 kg crate · smooth ramp at 30° · rope parallel to the ramp

   Most mistakes are not in the equations — they are in one drawing made before them.

   That drawing: the free body diagram (FBD)

   Forget a force, or add one that is not there → perfect math, wrong answer

   Today: isolate the body · turn every contact into a force · forget none

   Narration transcript

   A crate of mass 50 kilograms rests on a smooth ramp inclined at 30 degrees, held by a rope parallel to the ramp. The question: how hard does the rope pull? We will solve that with numbers in the next lesson. But let me tell you a secret: in problems like this, most mistakes happen not in the equations, but in one single drawing made before any equation. That drawing is called the free body diagram. Forget one force, or add a force that is not there, and every bit of mathematics after it can be perfect and the answer will still be wrong. Today we learn to make that drawing correctly: isolate the body, turn every contact into a force, and forget none of them.

2. 2. Cut the Body Loose

   

   Free: erase the ramp, cut the rope, remove the floor — only the crate remains

   Rule: every connection you cut becomes a force

   Cut the rope → draw the force the rope was pulling with

   Erase the ramp → draw the force the ramp was pushing with

   One force without cutting: the weight — the Earth pulls without touching

   $\displaystyle W = m \cdot g, g \approx 9.81 m/s²$

   $\displaystyle 50 \times 9 = 450$

   $\displaystyle 50 \times 0.81 = 40.5$

   $\displaystyle W = 450 + 40.5 = 490.5 N$

   Drawn from the center of gravity, vertically down

   Only forces ON the body. The crate's push on the ramp belongs to the ramp's diagram.

   Narration transcript

   What does free body diagram mean? Free: cut the body loose from its surroundings. Erase the ramp, cut the rope, remove the floor; only the crate remains. But careful: everything you removed was exerting a force on the crate. That is the rule: every connection you cut becomes a force. Cut the rope, and in its place you draw the force the rope was pulling with. Erase the ramp, and in its place you draw the force the ramp was pushing with. And there is one force that comes without cutting anything: the weight. The Earth pulls the crate downward all the time; it does not touch it, but it pulls. Weight is mass times the acceleration of gravity: W equals m times g, with g about 9.81 meters per second squared. The crate has 50 kilograms: 50 times 9.81. Step by step: 50 times 9 equals 450; 50 times 0.81 equals 40.5; the total is 490.5 newtons. This force is drawn from the center of gravity, vertically downward. One more rule: only forces acting ON the body go into the diagram. The force the crate exerts on the ramp does not belong in the crate's diagram; it belongs to the ramp's diagram.

3. 3. The Catalog of Contacts

   

   Rope: pulls only, along the rope, away from the body → tension T

   Smooth surface: pushes only, perpendicular → normal force N

   Rough surface: N plus friction f along the surface, against the sliding tendency

   Spring: stretched pulls, compressed pushes · F = k·s

   Applied push / pull: P

   Trap: N is NOT always equal to W — only on a horizontal floor with no other vertical force

   On a ramp: N ⟂ ramp, W straight down — not even the same line → N < W

   Narration transcript

   Now let us build the catalog of contacts; each type of contact has a known force. One: a rope or cable. A rope can only pull, never push. The force is along the rope, pointing away from the body; we call it tension and write T. Two: a smooth surface. The surface can only push, and only perpendicular to itself. We call that the normal force, N; normal means perpendicular. Three: a rough surface. Besides the perpendicular push there is a friction force parallel to the surface; friction points against the direction the body tends to slide. Four: a spring. A stretched spring pulls the body toward itself, a compressed spring pushes; its size is the spring constant times the stretch: F equals k times s. Five: a directly applied push or pull; call it P. Now the most common trap: the normal force is not always equal to the weight. It equals the weight only on a horizontal floor with no other vertical forces. On an inclined ramp N is smaller than the weight, and one look at the directions tells you why: N is perpendicular to the ramp, the weight is straight down; they are not even along the same line.

4. 4. The Four-Step Recipe

   

   ① Isolate — outline of the body only

   ② Weight — center of gravity, straight down, W = m·g

   ③ Contacts — one catalog force per cut contact

   ④ Label — numbers, letters, angles, axes

   Check: forces = 1 + contacts (rough surface counts 2)

   More → you invented one · fewer → you forgot one

   Never draw: velocity, motion, acceleration arrows — forces only

   Narration transcript

   A free body diagram has four steps, always in the same order. Step one: choose the body and isolate it. Draw only the outline of the body; no ramp, no rope, no wall. Step two: draw the weight; from the center of gravity, vertically down, W equals m g. Step three: go to every contact you cut and put its force from the catalog: tension for a rope, normal for a surface, friction if it is rough, spring force for a spring. Step four: label. Write known magnitudes as numbers and unknowns as letters; mark the angles; draw the x and y axes. Then a small check: the number of forces must equal one plus the number of contacts. One for the weight; one more for each contact, two for a rough surface. More than that and you invented one; fewer and you forgot one. And never draw these: a velocity arrow, a motion arrow, an acceleration arrow. A free body diagram contains forces only.

5. 5. Worked Example — Crate on a Ramp

   

   ① isolate → outline only

   ② W ↓ from the center

   $\displaystyle W = 50 \times 9.81 = 490.5 N$

   ③ contacts: ramp + rope = 2

   ramp (smooth) → N ⟂ ramp, pushes

   N is 30° from the vertical

   rope → T ∥ ramp, up-slope

   ④ labels: W = 490.5 N; N, T unknown

   axes x′ ∥ ramp, y′ ⟂ ramp

   check: 1 + 2 = 3 forces ✓ (W, N, T)

   Diagram complete → next lesson: equations

   Narration transcript

   Now let us apply it to the crate on the ramp. Step one: isolate the crate; only the outline of the box remains. Step two: the weight. An arrow from the center of gravity, vertically down; its size is 490.5 newtons, which we computed a moment ago. Step three: the contacts. The crate has two contacts: the ramp surface and the rope. The ramp is smooth; the catalog says there is only a normal force. N is perpendicular to the ramp surface and pushes the crate, so it points away from the ramp. Because the ramp is inclined at 30 degrees, N makes 30 degrees with the vertical. Why? If the surface is rotated 30 degrees from the horizontal, the line perpendicular to it is rotated 30 degrees from the vertical. The rope: parallel to the ramp, pulling the crate up the slope. The tension T points up along the ramp. Step four: label. W equals 490.5 newtons; N and T are unknown, so they stay as letters. Angles: the ramp is 30 degrees from the horizontal, N is 30 degrees from the vertical. Axes: in problems like this, choosing the axes parallel and perpendicular to the ramp makes life much easier, because then N and T lie on the axes and only the weight needs to be resolved into components. Check: two contacts; the number of forces is one plus two, three. W, N, T. Done. The diagram is complete; in the next lesson we turn these three arrows into equations.

6. 6. Two More Diagrams and Five Classic Mistakes

   

   Lamp on two cables: W + T₁ + T₂ = 3 forces

   Each cable pulls away from the lamp, along the cable; label the angles

   Pushed crate, rough floor: W + N + f + P = 4 forces

   P tilted 20° below horizontal; f points backward

   ① Forgetting the weight

   ② Saying N = W on a ramp

   ③ Drawing the crate's force on the ramp (action–reaction mix-up)

   ④ Drawing a motion arrow — motion is not a force

   ⑤ A rope that pushes — ropes only pull

   Narration transcript

   Two more examples, quickly. A lamp hangs from the ceiling by two cables. Choose the body: the lamp. Weight down. Contacts: two cables; each pulls away from the lamp along its own cable: T1 and T2. Three forces. The cables may be at different angles; do not forget to label the angles. Second example: on a rough floor you push a crate, pressing downward at 20 degrees below the horizontal. Choose the body: the crate. Weight down. Contacts: the floor and your hand. The floor is rough: normal force up, friction backward. Your hand: the push P, tilted 20 degrees. Four forces. The classic mistakes: One, forgetting the weight; the weight is always there. Two, saying N equals W on the ramp; next lesson we will see that for the crate on the ramp N comes out as 424.8 newtons, not 490.5. Three, putting the force the crate exerts on the ramp into the diagram; that force acts on the ramp, not on the crate. Four, drawing the direction of motion as an arrow; motion is not a force. Five, drawing a rope that pushes; a rope only pulls.

7. 7. What We Gathered

   

   FBD = isolate the body, draw all forces on it — only those

   Every cut connection → a force: rope pulls, surface pushes ⟂, roughness adds f, spring pulls/pushes

   Weight always: W = m·g, from the center, down

   Four steps: isolate · weight · contacts · label

   Check: forces = 1 + contacts · no motion arrows · forces on the body only

   Next: ΣF = 0 on these drawings → T and N with numbers

   Narration transcript

   Let us gather what we have. A free body diagram means isolating the body and drawing all the forces acting on it, and only those. Every connection you cut becomes a force: a rope pulls, a surface pushes perpendicularly, roughness adds friction, a spring pulls or pushes. The weight is always there: W equals m g, from the center downward. Four steps: isolate, weight, contacts, labels. Check: the number of forces is one plus the number of contacts; no motion arrows; only forces on the body. In the next lesson we add one sentence to these drawings: on a body at rest, the forces add up to zero. With that sentence we will find the rope tension and the normal force with numbers. See you there.

Source video: Free Body Diagram (9:05)