M012 · Newtonian contact mechanics

Inclined Plane & Friction

At zero speed, will the block stick or reverse? It sticks if static friction can hold it. At the moment it stops, work out the friction needed to balance gravity along the slope and any applied force. If that is within the static limit, the block stays put; if not, it slides off in the direction of the net force.

Drag a block on a rough incline, change the two friction coefficients and apply an uphill or downhill force. Compare stopping, sticking and reversal with the release-from-rest map and the energy dissipated along the path.

Recorded time $0.372\,\mathrm{s}$

Distance down the face $1.6\,\mathrm{m}$

Velocity relative to the face $1.1\,\mathrm{m\,s^{-1}}$

Acceleration along the face $2.95\,\mathrm{m\,s^{-2}}$

Normal force on block $8.32\,\mathrm{N}$

Contact regime $\text{Sliding downhill}$

Signed friction down the face $-2.25\,\mathrm{N}$

Margin for rest feasibility $-1.45\,\mathrm{N}$

Effective sliding coefficient $0.27$

Block kinetic energy $0.602\,\mathrm{J}$

Gravitational energy change $-1.06\,\mathrm{J}$

Applied-force work $0\,\mathrm{J}$

Friction work $-0.458\,\mathrm{J}$

Mechanical energy change $-0.458\,\mathrm{J}$

Record end time $1.33\,\mathrm{s}$

Energy-balance residual $5.55\times10^{-17}\,\mathrm{J}$

Predict, then release and check

At zero speed, will the block stick or reverse?

Physical forcesForce arrows keep one fixed scale for each trial.Release slotMarks the starting coordinate on the moving slope face.Contact reaction on baseShows the opposite normal force that acts on the base at the contact.Slope angle

\theta

32.00\,{}^{\circ}

Block mass

m

1.00\,\mathrm{kg}

Release fraction along the face

s_0/\ell

0.35\,

Initial velocity down the face

\dot s_0

0.00\,\mathrm{m/s}

Force down the face

F

0.00\,\mathrm{N}

Static friction coefficient

\mu_s

0.45\,

Sliding friction fraction

\eta

0.60\,

Kinetic friction coefficient

\mu_k

0.27\,

Inspect recorded time

t/t_{\mathrm{end}}

0.00\,\%

Drag the block or use the sliders to set a new trial. Downhill force and velocity are positive. At zero relative speed, static friction supplies the demand up to its limit; sliding friction opposes relative motion. The release map predicts motion from rest, not the state of an already moving block. The two coefficient controls keep kinetic friction no greater than static friction. Curves are analytic model predictions; the run stops at the first face endpoint or six seconds, before impact or detachment.

Model assumptions

Friction follows the ideal Coulomb law with constant coefficients, and the incline is rigid and fixed. The kinetic coefficient is constrained not to exceed the static coefficient. The track is finite, and the record ends before the block hits anything. Temperature, deformation and speed-dependent friction are left out.

Start with the essentials

Focus question: At zero speed, will the block stay or reverse?
One-sentence intuition: Stopping gives static friction a chance to hold. The block stays only when the force it needs is within the static limit.

Core mathematical model

Gravity and the applied force

N=mg\cos\theta,\qquad D=mg\sin\theta+F

Distance and force are positive downhill. The applied force acts along the face, so it adds no normal loading.

The force needed to remain at rest

f_s=-D,\qquad |D|\le\mu_sN

Static friction adjusts to hold the block at rest, up to its maximum. The margin subtracts the required holding force from the largest static friction; equality is included in the ideal sticking condition.

Sliding friction

\begin{aligned}m\ddot s&=D-\mu_kN\operatorname{sgn}(\dot s)\\0&\le\mu_k\le\mu_s\end{aligned}

The first coefficient slider sets static friction, and the second sets kinetic friction. Lowering the static coefficient below the kinetic value lowers the kinetic value to match; this keeps the stated coefficient ordering valid. At zero speed, the static condition is checked again.

Motion between stops and endpoints

\begin{aligned}s&=s_a+v_a(t-t_a)+\tfrac12a(t-t_a)^2\\v&=v_a+a(t-t_a)\end{aligned}

Forces are constant within each segment. The first track-end or future zero-velocity event changes what happens next. The track endpoint is shown before a collision, detachment or tipping is modeled.

Energy loss follows the full path

\Delta K+\Delta U=F(s-s_0)-\mu_kN\int_0^t|\dot s|\,\mathrm dt

Friction dissipates energy on both sides of a reversal. The energy-loss plot therefore uses travelled distance, not the net displacement. Temperature, deformation and velocity-dependent coefficients are outside this Coulomb model.

Common difficulties

Static friction can be below its maximum

Typical misconceptionStatic friction always equals its maximum value.

Better mental modelChoose Static hold. The actual friction balances gravity and the applied force along the face. Its maximum is larger.

A stop can be a turning point

Typical misconceptionZero speed means the block will stay there.

Better mental modelCompare Stop and stick with Stop and reverse at their zero-speed event. Only the first can be held by static friction.

The green region predicts release from rest

Typical misconceptionA moving block must stop as soon as its settings lie in the rest region.

Better mental modelThe map asks what happens when released with zero relative velocity. An already sliding block can keep moving inside that region.

Run the experiment

01
Watch the default slide
Watch the block, its amber friction arrow and the growing energy-loss curve.
What to observe: Friction points uphill while the block moves downhill.
02
Let static friction hold it
Choose Static hold.
What to observe: The friction arrow balances the force pulling downhill, and the block remains still.
03
Compare two stops
Choose Stop and stick, then Stop and reverse. Open Work, constraints and diagnostics and press Exact zero speed.
What to observe: Both reach zero velocity. One stays, and the other begins sliding downhill.
04
Pull uphill
Choose Pull uphill.
What to observe: A sufficiently strong uphill force changes the direction of motion and the friction arrow.
05
Read the release map
Change the slope or applied force and follow the marker in the release-from-rest map.
What to observe: The marker crosses between uphill, possible-rest and downhill regions. A nonzero initial velocity is a separate condition.
06
Reset and place the block
Press Reset, then drag the block along the face or use the Release slider in the folded group.
What to observe: A new trial begins from the chosen release coordinate. Replay repeats the current trial.

Core mathematical model

Gravity and the applied force

The force needed to remain at rest

Sliding friction

Motion between stops and endpoints

Energy loss follows the full path

Common difficulties

Static friction can be below its maximum

A stop can be a turning point

The green region predicts release from rest

Run the experiment

Watch the default slide

Let static friction hold it

Compare two stops

Pull uphill

Read the release map

Reset and place the block