Friction, Springs, and Connected Systems

Driving question

How do contact forces, elastic interactions, and connection constraints shape a system's motion?

Before learning: predict before pulling

A box rests on a rough floor while a horizontal pull slowly increases from zero. Sketch friction magnitude versus applied-force magnitude from rest through slipping. Then predict what changes at the instant sliding begins.

Hook: static friction is not automatically at its maximum

Push a heavy box gently and it stays at rest because static friction matches the push. It increases only as needed, up to a limit:

Once surfaces slide, the model becomes

Friction opposes relative slipping or its tendency, not necessarily the object's velocity.

Normal force must be solved

$F_N$ is not automatically $mg$. For a horizontal pull angled upward by $\theta$ with no vertical acceleration,

so $F_N=mg-F\sin\theta$. An upward pull reduces normal force and therefore reduces the maximum or kinetic friction predicted by these models.

Worked friction example

A $10.0\text{ kg}$ crate slides on a level floor with $\mu_k=0.20$ while pulled horizontally by $35\text{ N}$. Using $g=9.8\text{ m/s}^2$,

Spring force and linear range

For an ideal spring near equilibrium,

$x$ is displacement from the spring's relaxed/equilibrium reference appropriate to the model. The negative sign indicates restoring direction. A force-extension graph has slope magnitude $k$ within the linear range.

Worked spring example

A spring with $k=120\text{ N/m}$ is stretched $0.080\text{ m}$. Its force magnitude is $9.6\text{ N}$ toward equilibrium. If attached to a $0.60\text{ kg}$ block on a frictionless surface at that instant,

This instantaneous acceleration changes as $x$ changes.

Connected systems and constraints

For an ideal light inextensible string over an ideal pulley, connected objects have related acceleration magnitudes. Tension is uniform in the ideal string, but it is not generally equal to either object's weight.

For mass $m_1$ on a frictionless table connected to hanging mass $m_2$:

Adding eliminates internal tension:

Then $T=m_1a$.

Worked connected-system example

Let $m_1=3.0\text{ kg}$ and $m_2=2.0\text{ kg}$:

Tension is less than the hanging weight because the hanging mass accelerates downward.

Add friction to a connected system

If $m_1$ slides with kinetic friction,

provided the assumed direction is correct and $f_k=\mu_kF_N$. If the calculated acceleration has the opposite sign, revisit the assumed motion and whether static friction prevents movement.

During learning: interaction audit

1. Choose the system and draw a free-body diagram for each object.
2. Decide whether contact is static, impending slip, or sliding.
3. Solve $F_N$ before using a friction model.
4. State ideal string, pulley, and spring assumptions.
5. Write constraint relations and use a combined-system equation when useful.

Misconception clinic

“Static friction always equals $\mu_sF_N$.” That is only its maximum.

“Friction always opposes velocity.” It opposes relative slip or tendency to slip.

“Normal force always equals weight.” Other perpendicular forces/acceleration matter.

“Tension equals the hanging weight.” Only when that mass has zero acceleration.

“Spring force is constant.” It changes with displacement in Hooke's-law range.

After learning: retrieval and transfer

1. Sketch friction magnitude as an applied force grows through the slipping threshold.
2. Why can pulling upward at an angle reduce kinetic friction?
3. What does the negative sign in $F_s=-kx$ mean?
4. Why does tension disappear from a combined two-block equation?
5. What idealizations make connected accelerations share a magnitude?

AP-style evidence routine

1. Define objects/system, axes, and assumed motion.
2. Draw interaction-specific free-body diagrams.
3. Select friction/spring models only within their conditions.
4. Add connection constraints.
5. Solve symbolically, then check signs, units, limits, and contact feasibility.

Key takeaway

Friction and spring forces require conditional models; connected motion adds constraints. Clear system choice turns their internal interactions into solvable equations.

Further learning and alignment

• College Board, “AP Physics 1: Algebra-Based Course,” https://apcentral.collegeboard.org/courses/ap-physics-1 (retrieved 2026-08-19). Alignment only.
• OpenStax, “5.1 Friction” and “5.3 Elasticity: Stress and Strain,” https://openstax.org/books/physics/pages/5-1-friction and https://openstax.org/books/physics/pages/5-3-elasticity-stress-and-strain (retrieved 2026-08-19). CC BY verification; no exercises copied.
• OpenStax, “6.1 Solving Problems with Newton's Laws,” https://openstax.org/books/university-physics-volume-1/pages/6-1-solving-problems-with-newtons-laws (retrieved 2026-08-19). CC BY connected-system strategy reference.