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Sandbox Physics

M012 · Newtonian contact mechanics

Inclined Plane & Friction

A rigid incline with a signed applied force separates sticking, sliding uphill and sliding downhill. Change slope and friction, release with a kick, catch exact zero-speed events and compare the recorded trajectory with a separate frictionless reference.

Interactive modelInclined Plane & Friction
Recorded time0 s0\,\mathrm{s}
Distance down the face1.4 m1.4\,\mathrm{m}
Signed velocity down the face0 m/s0\,\mathrm{m/s}
Acceleration along the face0 m/s20\,\mathrm{m/s^2}
Normal force on block0 N0\,\mathrm{N}
Contact regimeSticking\text{Sticking}
Signed friction down the face0 N0\,\mathrm{N}
Margin for rest feasibility0 N0\,\mathrm{N}
Effective sliding coefficient0.30.3
Block kinetic energy0 J0\,\mathrm{J}
Gravitational energy change0 J0\,\mathrm{J}
Applied-force work0 J0\,\mathrm{J}
Friction work0 J0\,\mathrm{J}
Mechanical energy change0 J0\,\mathrm{J}
Record end time0 s0\,\mathrm{s}
Energy-ledger residual0 J0\,\mathrm{J}

Physics tutorial

Friction changes its rule at zero speed

BackgroundA block resting on a rough slope can remain still even while gravity pulls downhill. Once sliding starts, friction opposes its relative velocity. A stop requires a new decision.

Why it mattersOpenStax University Physics 6.2 distinguishes the inequality for static friction from the sliding law. This workbench resolves those regimes and their transitions.

Start with the essentials

Focus question
After an uphill release, does zero speed mean sticking or reversal?
One-sentence intuition
At zero velocity the required balancing friction must fit within its static capacity. Otherwise the block restarts downhill or uphill.

Core mathematical model

Resolve physical forces

N=mgcos⁡θ,D=mgsin⁡θ+FN=mg\cos\theta,\qquad D=mg\sin\theta+F

The coordinate points down the face; applied force is signed in that direction. The support prevents normal acceleration.

Rest is a constraint

fs=−D,∣D∣≤μsNf_s=-D,\qquad |D|\le\mu_sN

Static friction supplies exactly the force required for rest. The margin readout is the capacity minus the magnitude required; the envelope shows this condition across slopes and applied forces.

Sliding and zero-speed events

μk=ημs,0≤η≤1,ms¨=D−μkNsgn⁡(s˙)\mu_k=\eta\mu_s,\quad 0\le\eta\le1,\qquad m\ddot s=D-\mu_kN\operatorname{sgn}(\dot s)

Sliding friction opposes relative velocity. At zero speed the static inequality is tested again: the block either stays put or restarts in the direction of the unbalanced drive. The initial zero-speed state is not a future stop event.

Exact segment and event timing

s=sa+va(t−ta)+12a(t−ta)2,v=va+a(t−ta)s=s_a+v_a(t-t_a)+\tfrac12a(t-t_a)^2,\qquad v=v_a+a(t-t_a)

Forces are constant between events. Positive quadratic roots find the first track edge; a future velocity root finds a stop. No time-step overshoot or impact clamp is used.

Work uses distance, not signed displacement

ΔK+ΔU=F(s−s0)−μkN∫0t∣s˙∣ dt\Delta K+\Delta U=F(s-s_0)-\mu_kN\int_0^t|\dot s|\,\mathrm dt

A reversing path dissipates energy on both legs. Signed displacement would cancel some travel and give the wrong loss. Constant friction is an empirical teaching approximation.

Common difficulties

Capacity is not the actual force

Typical misconceptionStatic friction always equals its maximum.

Better mental modelInspect the static-hold preset: the actual friction balances the downhill drive and is below capacity.

Zero speed does not settle the question

Typical misconceptionEvery turning point is a permanent stop.

Better mental modelCompare stop-and-stick with stop-and-reverse at the exact event. Only the first has enough static capacity.

The rest envelope is conditional

Typical misconceptionA point inside the envelope cannot slide.

Better mental modelThe envelope says rest is possible at zero relative speed. A block with a nonzero release velocity can still slide inside it.

Run the experiment

  1. 01

    Predict the hold

    Change slope or applied force from the static preset.

    What to observe: Compare the required static force with capacity and find the envelope boundary.
  2. 02

    Catch the stop

    Choose stop-and-stick, then use exact zero speed and step one millisecond.

    What to observe: The friction changes to the force required for rest, not a kinetic value at an arbitrarily small speed.
  3. 03

    Reverse and account

    Choose stop-and-reverse and complete the record.

    What to observe: The path-length friction work closes the mechanical-energy ledger across both legs.
  4. 04

    Test the ideal limit

    Set static friction to zero.

    What to observe: The recorded velocity coincides with the separate frictionless reference.