M011 · Choose the system, then find its forces

Free-Body Diagram Builder

Which forces belong in an object's free-body diagram? Only external forces acting on the chosen system belong in its diagram. An interaction's two forces act on different bodies.

Build force diagrams for a table, slope, cable, spring or two-block system. Choose the boundary and directions, then replay the resulting motion.

Contact state $\text{Sliding}$

Model net force $20.3\,\mathrm{N}$

Model acceleration $10.2\,\mathrm{m\,s^{-2}}$

System mass $2\,\mathrm{kg}$

Choose the body, then its forces

What makes the block accelerate?

Drive or cable tension

F

24.00\,N

Crate mass

m

2.00\,kg

Lower mass

m_2

4.00\,kg

Inter-block static friction

\mu_{s,1}

0.45\,

Surface static friction

\mu_{s,0}

0.25\,

Slope angle

\theta

25.00\,{}^\circ

Initial spring extension

e_0

0.80\,m

Spring stiffness

k

16.00\,N\,m^{-1}

Kinetic / static coefficient ratio

r

0.75\,

Gravity

g

9.81\,m\,s^{-2}

Apparatus

q

0.00\,

Driven block

b

0.00\,

Each parameter change starts a separate release from rest. Static friction takes the value needed to prevent slipping, up to its limit; equality stays stuck. Each kinetic coefficient is its static coefficient multiplied by the chosen ratio.

Ideal guides prevent rotation. The upper block is 0.8 metres wide and the lower 2.4 metres. A stacked run ends at 0.8 metres of relative travel, 2.4 metres of lower travel or two seconds. Tipping, partial support and falling are outside this model.

The cable is a constant-tension winch with changing length. Its run ends after one metre of travel. The spring has a two-metre natural length and follows Hooke's law; its exact first monotone excursion ends at the first turn, 2.4 metres of travel or two seconds. A replay repeats that finite segment.

All apparatus lengths share one scale. Force arrows are translated to the selected system centre and share one gain, capped at 0.03 metres per newton and fixed for a whole release. They show force vectors, without pressure distributions or torque application points.

The example supplies consistent forces. In the exercise, identify the nonzero external forces and set their initial directions within five degrees; the model supplies magnitudes. Candidate contact assumptions are separate from the selected physical response. Charts and CSV contain simulated model values.

Model assumptions

Ideal guides keep the blocks from rotating. Contact forces and spring forces follow the displayed model, with a finite release from rest.

Start with the essentials

Focus question: When two blocks become one system, which forces disappear from its diagram?
One-sentence intuition: Forces between the two included blocks are internal. Their equal and opposite contributions cancel in the total force, while forces from the floor and Earth remain external.

Core mathematical model

External force and system acceleration

\sum\mathbf F_{\mathrm{ext}}=M\mathbf a_{\mathrm{CM}}

For one crate, the centre of mass moves with the crate. For two sliding blocks, the centre-of-mass acceleration is mass weighted; the blocks can have different accelerations.

An interaction has two receivers

\mathbf F_{12}=-\mathbf F_{21}

These forces act on different bodies. The force on the other body is not an extra force on the one you selected.

A spring changes its pull with extension

F_s=-ke

Extension is measured from the spring's natural length. A stretched spring pulls back; a compressed spring pushes outward.

Friction depends on the contact state

\begin{aligned}|f_s|&\leq\mu_sN,\\ |f_k|&=\mu_kN.\end{aligned}

Static friction supplies the required value within its limit. Sliding friction opposes relative slip. A stationary contact does not automatically use maximum friction.

Common difficulties

Balanced arrows can still name the wrong interactions

Typical misconceptionAny arrows that add to zero make a valid equilibrium diagram.

Better mental modelEvery force must have an agent outside the selected system and a receiver inside it. Two arrows can cancel numerically while one actually acts on another body.

Run the experiment

01
Compare the ready examples
Try the slope, cable and spring. Pause and move Selected time.
What to observe: Weight remains vertical. The support is normal to the surface, cable tension follows the cable, and spring force changes with extension.
02
Build a release diagram
Choose Build it yourself. Drop nonzero force cards into the diagram, then drag their arrowheads or turn them with the keyboard. Check before playing.
What to observe: A missing force, a force on the wrong body or an incorrect direction keeps motion locked. Magnitudes are supplied by the model; directions are checked within five degrees.
03
Change the boundary
Choose the two-block example, then switch between Upper, Lower and Both blocks.
What to observe: The contact forces reappear on individual blocks and cancel in the combined system. The combined net force follows the centre-of-mass acceleration, even when the blocks slip.

Physical model, approximation limits and sources

Positions are in metres, time in seconds and forces in newtons. Right and up are positive; on a slope, displacement is positive uphill. Ideal guides allow translation and suppress rotation. Contact pressure and tipping are outside this model.

Every control change defines a new release from rest. Surfaces stay stuck when the required friction lies within the static limit. Force comparisons use a tolerance of one billionth of a newton; equality selects sticking. Kinetic coefficients are the corresponding static coefficients multiplied by the chosen ratio.

The default shows a worked example. Building a diagram checks nonzero external interactions and their initial directions within five degrees; valid force magnitudes come from the model. Reaction forces on excluded bodies and velocity cards do not unlock replay. A verified spring diagram follows the changing physical force during motion.

The blocks are 0.8 and 2.4 metres wide. The finite stacked record ends at 0.8 metres of relative travel, 2.4 metres of lower travel or two seconds. It does not continue into partial support, edge departure or collision. The cable winch holds a nonnegative tension while changing length and stops the record after one metre of travel.

The spring has a two-metre natural length. Extension is the rendered endpoint separation minus that length, and its force is computed from the same extension: $F_s=-ke$ Coil turns and wire dimensions are schematic; stiffness is an assumed parameter. The exact first monotone excursion ends at its first turn, 2.4 metres of travel or two seconds. Terminal forces and acceleration are incoming branch limits; the model does not solve the next slip or stick event.

Apparatus lengths share one scale. Force vectors are translated to the selected system centre, with one display gain fixed for the full release and every system boundary and capped at 0.03 metres per newton. The separate force diagram has its own common plot scale. CSV contains 101 model samples; residual roundoff is retained. These are simulated values, not measurements or fitted material properties.