M015 · Which of two contacts slips first?

Stacked Blocks & Friction

When you push one of two stacked blocks, which contact slips first? Test the friction needed at each contact against its static limit. Where you push matters: inter-block friction can accelerate the other block.

Push either block and find out which contact slips first. Look at one block alone or at the pair: taken together, the friction between the blocks cancels, and the work you put in equals the kinetic energy gained plus the heat lost to friction.

Which contact slips? $\text{Together}$

Upper acceleration $2.83\,\mathrm{m\,s^{-2}}$

Lower acceleration $2.83\,\mathrm{m\,s^{-2}}$

Relative displacement $0\,\mathrm{m}$

Two contacts, two possible slips

Which block carries the other?

Signed horizontal drive

F

28.00\,N

Upper mass

m_1

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 upper contact supports the upper weight; the floor supports both weights. The normal forces follow those vertical balances. Relative sliding, not motion of one body alone, determines interfacial heat.

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.

All nine signed contact candidates retain the forces required by their assumptions. Static limits and slip directions are tested separately; the selected response is identified. Charts and CSV contain simulated model values.

Model assumptions

The blocks start at rest and slide along a guide. The run stops before either block overhangs an edge, so tipping and falling are not modeled.

Start with the essentials

Focus question: Which contact first leaves rest when the drive increases?
One-sentence intuition: Assume a contact sticks, calculate the friction that assumption needs, then compare it with the static limit. If it slips, use kinetic friction and check the resulting direction of relative motion.

Core mathematical model

Balance each block

\begin{aligned}m_1a_1&=F_1+f_{21},\\ m_2a_2&=F_2-f_{21}+f_0.\end{aligned}

The inter-block friction has opposite signs on the two bodies. Only one body receives the external drive.

Two independent static limits

\begin{aligned}|f_{21}|&\leq\mu_{s,1}m_1g,\\ |f_0|&\leq\mu_{s,0}(m_1+m_2)g.\end{aligned}

These inequalities apply when the corresponding contact sticks. Sliding contacts instead use their kinetic coefficient.

The first departure from complete rest

\begin{aligned}C_1&=\mu_{s,1}m_1g,\\ C_0&=\mu_{s,0}(m_1+m_2)g,\\ F_{\mathrm{crit,upper}}&=\min(C_1,C_0),\\ F_{\mathrm{crit,lower}}&=C_0.\end{aligned}

These thresholds compare separate releases from rest as the magnitude of the load increases. The subsequent sliding response must be solved again, using kinetic friction where appropriate.

Heat follows relative sliding

\begin{aligned}E_{\mathrm{diss}}={}&|f_{21}|\,|\Delta x_1-\Delta x_2|\\ &+|f_0|\,|\Delta x_2|.\end{aligned}

Within each displayed monotone release, a sticking interface has zero relative travel and produces no frictional heat. The two-body work balance still includes the energy passed from one block to the other.

Common difficulties

Moving together still requires friction

Typical misconceptionIf the blocks do not slip against each other, their friction must be zero.

Better mental modelThe block you are not pushing needs a horizontal force to accelerate. Static friction can provide that force while its two contact points move together, transferring energy without interfacial heat.

Run the experiment

01
Watch friction carry a block
Start with Carried by friction. Select Upper, Lower and Both blocks.
What to observe: Inter-block friction accelerates the upper block and acts oppositely on the lower. It cancels from the combined system's external forces.
02
Try the other block
Switch to Upper slips, drag that block to change its drive, and vary the two friction coefficients.
What to observe: The upper can slide while the lower stays still, or both contacts can slide. The two contact strips show actual force and the static limit separately.
03
Inspect each rejected assumption
Open Contact tests, work and energy, then Test all contact assumptions.
What to observe: All nine signed candidates retain their required forces. A row can fail because a static limit is exceeded or because its predicted slip direction contradicts the assumed one.

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.

There are four contact assumptions and nine signed candidates. A static candidate retains the force it would need; a sliding candidate must also predict its assumed direction. The selected response follows the stated sticking preference. Every plotted force value represents a separate release, not a continuously ramped load.

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.

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.