Friction Force Calculator with Steps | F = μN, Any Units

A friction force calculator multiplies the coefficient of friction by the normal force: F = μN. On a flat floor N = mg, so a 20.0 kg crate with μ_k = 0.250 feels 0.250 × 20.0 × 9.81 = 49.1 N of friction.

Rule of thumb: static friction can be anything up to μ_sN; it only matches the push. Once the object slides, friction is μ_kN, usually 10 to 30% less.

A friction force calculator works out the force that resists one surface sliding over another. Friction is a contact force, measured in newtons (N), that acts along the surface and against the motion or the attempted motion. The coefficient of friction (μ) is a number with no unit that describes how grippy a pair of surfaces is, and the normal force (N) is how hard the surfaces press together.

The basic formula is F = μN. To calculate the friction force, find the normal force first, then multiply by μ. On a level floor with no other vertical forces, N equals the weight, mg. On a slope, N = mg cos θ. With a rope pulling at an angle above the horizontal, N = mg − P sin θ.

A worked one-liner: a 20.0 kg crate slides across a warehouse floor with μ_k = 0.250. N = 20.0 × 9.81 = 196.2 N, so F = 0.250 × 196.2 = 49.1 N. To keep the crate moving at a steady speed you must push with 49.1 N.

Who uses it: students in CBSE and ICSE Class 9 to 12, IGCSE, GCSE and A level, AP Physics 1, JEE and NEET, and first-year engineering. Engineers use friction to size brakes, belts, conveyors and clutches, and road safety experts use it to work out skid speeds. Leonardo da Vinci studied friction around 1500, Guillaume Amontons stated its laws in 1699, and Charles-Augustin de Coulomb separated static and kinetic friction in 1785.

Pick a situation, choose what to solve for, then type your values in any units.

4.1 What the friction force means

Friction is the surface pushing sideways on the object. It always points along the surface and against the sliding, or against the direction the object would slide if friction vanished. A friction force of 49.1 N on the crate means you must push with 49.1 N just to keep it moving at a steady speed. Any push beyond that makes it speed up.

At a microscopic level, surfaces touch only at tiny high points called asperities. The real contact area is a small fraction of the visible area, and it grows in proportion to the load. That is why friction depends on the normal force but not on the size of the contact patch.

4.2 Static friction is a "match", not a fixed number

This is the idea students most often miss. A box at rest feels exactly as much static friction as it needs to stay at rest, up to a maximum of μ_sN. Push a 25.0 kg box with 20 N and friction is 20 N. Push with 100 N and friction is 100 N. Only when the push passes μ_sN = 110 N does it start to slide. So "F = μN" gives static friction only at the point of slipping.

4.3 Is it a sensible size?

For most dry surfaces μ lies between about 0.2 and 1.0, so friction is between a fifth of the weight and the full weight. Ice on ice is about 0.03 to 0.1; rubber on dry concrete is about 0.6 to 1.0. If your answer is more than the weight on a level floor, check μ. If it is 1000 times too big or too small, check whether the mass was in grams.

4.4 How it scales

Friction is proportional to the normal force. Double the load and you double the friction. It does not depend on the contact area: a brick slides with the same friction on its wide face or its narrow face. Kinetic friction is roughly independent of speed for most dry surfaces, which is why a sliding crate slows down at a steady rate.

4.5 How the normal force changes

• Flat floor: N = mg.
• Pull at angle θ above the floor: N = mg − P sin θ. Lifting reduces friction.
• Push at angle θ below the horizontal: N = mg + P sin θ. Pressing down increases friction.
• Incline at angle θ: N = mg cos θ.
• Lift accelerating up at a: N = m(g + a).

4.6 What the charts show

For a push, chart 1 is the classic friction graph: a straight rising line while static friction matches the push, a peak at μ_sN, then a drop to the flat kinetic level μ_kN once it slides. For a ramp, chart 1 rises as mg sin θ until the slip angle, then falls as μ_kmg cos θ. The point marked on each chart is your problem.

4.7 Where the simple model breaks down

Coulomb friction (F = μN) is an excellent model for dry, clean surfaces at everyday speeds and loads. It breaks down for very smooth or very clean metals, where adhesion makes μ large; for rubber, where friction depends on speed and temperature; for lubricated surfaces, where a fluid film carries the load; and for very high speeds, where surfaces heat up. Published μ values can vary by 20% or more for the "same" pair of materials.

4.8 Where this leads next

Friction feeds straight into Newton’s second law: net force = applied force − friction = m a. It sets stopping distances, the steepest road a car can climb, how fast you can take a bend, and how much energy is lost as heat. The work done against kinetic friction, f × d, is energy turned into heat in the surfaces.

▶ Press Calculate above to fill all five write-ups with your own numbers: the full GRESS working, every unit, an uncertainty, a mark scheme and a plain-text version.

This flowchart shows how to calculate the friction force step by step, from a word problem to a checked answer. Every box uses the same worked example as the Excel, R and Python sections: a 20.0 kg crate slides across a level floor with μ_k = 0.250. Find the friction force.

Check any box yourself: the Excel sheet in Section 9, the R script in Section 10 and the Python script in Section 11 all return 49.1 N for this crate. Section 14 shows the same method written out by hand.

1. Choose the situation. Pick Friction F = μN for the basic formula, Push or pull for static vs kinetic friction at any angle, Friction on an incline for ramps, or Without μ to work friction out from motion data.
2. Choose what to solve for. Click a Solve for chip, for example friction from mass, μ from force and mass, or the normal force. That field turns into a dashed answer box.
3. Type the known values. Plain numbers, decimals and scientific notation all work: 0.250, 20.0 or 6.2e-3.
4. Pick the unit beside each value. Grams, pounds, kN, lbf, kgf, km/h, mph and degrees or radians are converted for you.
5. Enter both coefficients when you have them. μ_s decides whether the object starts to move; μ_k gives the friction once it slides.
6. Set g for your syllabus. Use 9.81 for IGCSE, A level and most US courses, 9.8 for CBSE and AP, or 10 for quick estimates.
7. Press Calculate. Nothing is worked out until you do, and changing any input clears the old answer so you never read stale numbers.
8. Read the answer box and the badges. They show the friction with its units, whether it is static or kinetic, Units check ✓ and Verified.
9. Follow the working, the free-body diagram and the charts. The GRESS table, numbered steps, results table, diagram and four charts explain the answer.
10. Copy or download the working. Section 5 gives five ready-to-paste write-ups, and Download Doc or Download PDF saves the whole solution.

Excel has no friction function, and it does not need one. F = μN is one multiplication, so a sheet with one problem per row can handle a whole class set. The two habits that matter are working out the normal force in its own column, and keeping g in a single named cell so you can switch between 9.81, 9.8 and 10.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases

• CONVERT strings are case-sensitive. Force uses "N", "dyn" and "lbf"; mass in pounds is "lbm". A wrong string returns #N/A.
• Static friction is not a formula on its own. Use IF to return the push when the object stays still, and μ_kN when it slides.
• Solve backwards with Goal Seek. For the force needed to start an angled pull, set up the friction test and use Data → What-If Analysis → Goal Seek.
• Precision: Excel keeps about 15 significant digits. Round only in the final display with =ROUND(E2,1) or the number format.

9.4 Why Excel might disagree with R, Python or this page

So a different last digit is usually a different g. Note 49.05 itself: Excel and most calculators round it half away from zero to 49.1, while R’s round() and Python’s round() round half to even and can give 49.0. Formatting with sprintf("%.3g") or f"{x:.3g}" avoids the surprise in most cases.

9.5 Charting it

1. Put normal force in one column and measured friction in the next.
2. Insert → Charts → Scatter. Do not use the Line chart type, which spaces x values evenly.
3. Click a point → Add Trendline → Linear → tick Set Intercept = 0 and Display Equation on chart.
4. The gradient is μ. A line that misses the origin by a lot points to a zero error in the force meter.
5. Label the axes with units: Normal force N (N) and Friction force F (N).

9.6 Errors you will hit

R works out friction in one line, F <- mu * m * g. The script below does more: it finds the normal force, both static and kinetic friction, μ and N by rearranging, an angled pull, a ramp and a skidding car. It prints every answer with its unit, runs a substitution check, and saves one 300 dpi figure, the classic graph of friction against applied force. It uses base R only.

10.1 The complete script

R, base only, saves friction_force_plot.png

# ============================================================
# Friction force calculator in R, with one 300 dpi figure
# StatsUnlock - https://statsunlock.com/friction-force-calculator/
# Base R only. No packages to install.
# ============================================================

# ---- 1. Your values (SI units) -----------------------------
m    <- 20.0     # mass in kg
mu_k <- 0.250    # coefficient of kinetic friction (no unit)
mu_s <- 0.300    # coefficient of static friction (no unit)
g    <- 9.81     # m/s^2; use 9.80665, 9.8 or 10 to match your book

# ---- 2. Friction on a flat surface -------------------------
N  <- m * g                  # normal force, newtons
Fk <- mu_k * N               # kinetic friction while sliding, N
Fs_max <- mu_s * N           # largest static friction before it slips, N

# ---- 3. Solve for any variable -----------------------------
mu_back <- Fk / N            # coefficient from force and normal force
N_back  <- Fk / mu_k         # normal force from friction and mu

# ---- 4. Angled pull, incline and "without mu" --------------
P <- 150; theta <- 35 * pi / 180          # 150 N pull at 35 degrees (radians!)
N_pull <- 30.0 * g - P * sin(theta)       # 30.0 kg sled: N = mg - P sin(theta)
f_pull <- 0.15 * N_pull                   # kinetic friction on the sled, N
slope  <- 30 * pi / 180                   # 5.0 kg block on a 30 degree ramp
f_ramp <- 0.30 * 5.0 * g * cos(slope)     # kinetic friction on the ramp, N
v <- 72 / 3.6; d <- 40                    # car skids from 72 km/h in 40 m
f_car  <- 1200 * v^2 / (2 * d)            # friction from the stopping distance, N

# ---- 5. Print the answers with units -----------------------
cat(sprintf("Normal force N      = %.4g N\n", N))
cat(sprintf("Kinetic friction    = %.3g N\n", Fk))
cat(sprintf("Max static friction = %.3g N\n", Fs_max))
cat(sprintf("check mu = %.3f, check N = %.4g N\n", mu_back, N_back))
cat(sprintf("Sled friction       = %.3g N (N = %.4g N)\n", f_pull, N_pull))
cat(sprintf("Ramp friction       = %.3g N\n", f_ramp))
cat(sprintf("Car friction        = %.0f N, mu_k = %.2f\n", f_car, f_car / (1200 * g)))
stopifnot(abs(mu_k * N - Fk) < 1e-12)     # substitution check
cat("Units: (no unit) x kg x m/s^2 = kg m/s^2 = N\n")

# ---- 6. One figure: friction against applied force ---------
P_app <- seq(0, 2 * Fs_max, length.out = 200)
f_app <- ifelse(P_app <= Fs_max, P_app, Fk)   # static rises with the push, then kinetic
png("friction_force_plot.png", width = 7, height = 4.5, units = "in", res = 300)
plot(P_app, f_app, type = "l", lwd = 3, col = "#ea580c",
     xlab = "Applied force P (N)", ylab = "Friction force f (N)",
     main = sprintf("20.0 kg crate: static up to %.3g N, then kinetic %.3g N", Fs_max, Fk))
abline(h = Fk, lty = 2, col = "#64748b")
legend("bottomright", c("Friction", "Kinetic level"), col = c("#ea580c", "#64748b"),
       lwd = c(3, 1), lty = c(1, 2), bty = "n")
invisible(dev.off())

Output when run: Normal force N = 196.2 N, Kinetic friction = 49.1 N, Max static friction = 58.9 N, Sled friction = 31.2 N, Ramp friction = 12.7 N, Car friction = 6000 N, mu_k = 0.51. These match Sections 3, 7, 9 and 11.

10.2 What each part does

R traps: trig functions take radians; round(49.05, 1) may give 49 because of half-to-even rounding and binary storage; -2^2 is −4 in R. Keep SI units inside the calculation and convert only when you print.

10.3 What the figure shows

The orange line rises at 45° while the crate stays still, because static friction exactly matches the push. At 58.9 N (μ_sN) the crate breaks free and friction drops to the flat kinetic level of 49.1 N (μ_kN). That drop is why it is harder to start a heavy box moving than to keep it moving.

10.4 Common problems and fixes

10.5 Useful one-liners

# friction for several masses at once
0.250 * c(5, 10, 20, 40) * 9.81
# mu from 49 N to slide 10 kg (g = 9.8)
49 / (10 * 9.8)
# slip angle to static coefficient
tan(26.6 * pi / 180)
# mu from a lab graph of friction against N
coef(lm(F ~ N - 1, data = data.frame(N = c(1.96, 3.92, 5.89), F = c(0.70, 1.36, 2.05))))

Python does the same job with NumPy and Matplotlib. The script solves the crate, a sled pulled at an angle, a block on a ramp and a skidding car, prints each answer with its unit, and saves one 300 dpi figure of friction against applied force.

Install once with pip install numpy matplotlib. Google Colab already has both.

11.1 The complete script

Python 3 with NumPy, saves friction_force_plot.png

# ============================================================
# Friction force calculator in Python, with one 300 dpi figure
# StatsUnlock - https://statsunlock.com/friction-force-calculator/
# pip install numpy matplotlib
# ============================================================
import numpy as np
import matplotlib.pyplot as plt

# ---- 1. Your values (SI units) -----------------------------
m = 20.0        # mass, kg
mu_k = 0.250    # kinetic coefficient (no unit)
mu_s = 0.300    # static coefficient (no unit)
g = 9.81        # m/s^2 (scipy.constants.g would give 9.80665)

# ---- 2. Friction on a flat surface -------------------------
N = m * g                    # normal force, N
Fk = mu_k * N                # kinetic friction, N
Fs_max = mu_s * N            # largest static friction, N

# ---- 3. Solve for any variable -----------------------------
mu_back = Fk / N
N_back = Fk / mu_k

# ---- 4. Angled pull, incline and "without mu" --------------
P, theta = 150, np.radians(35)            # 150 N pull at 35 degrees
N_pull = 30.0 * g - P * np.sin(theta)     # 30.0 kg sled
f_pull = 0.15 * N_pull
f_ramp = 0.30 * 5.0 * g * np.cos(np.radians(30))   # 5.0 kg on a 30 degree ramp
v, d = 72 / 3.6, 40                       # car skids from 72 km/h in 40 m
f_car = 1200 * v**2 / (2 * d)

# ---- 5. Print the answers with units -----------------------
print(f"Normal force N      = {N:.4g} N")
print(f"Kinetic friction    = {Fk:.3g} N")
print(f"Max static friction = {Fs_max:.3g} N")
print(f"check mu = {mu_back:.3f}, check N = {N_back:.4g} N")
print(f"Sled friction       = {f_pull:.3g} N (N = {N_pull:.4g} N)")
print(f"Ramp friction       = {f_ramp:.3g} N")
print(f"Car friction        = {f_car:.0f} N, mu_k = {f_car / (1200 * g):.2f}")
assert abs(mu_k * N - Fk) < 1e-12          # substitution check

# ---- 6. One figure: friction against applied force ---------
P_app = np.linspace(0, 2 * Fs_max, 200)
f_app = np.where(P_app <= Fs_max, P_app, Fk)
fig, ax = plt.subplots(figsize=(7, 4.5))
ax.plot(P_app, f_app, lw=3, color="#ea580c", label="Friction")
ax.axhline(Fk, ls="--", color="#64748b", label="Kinetic level")
ax.set_xlabel("Applied force P (N)")
ax.set_ylabel("Friction force f (N)")
ax.set_title(f"20.0 kg crate: static up to {Fs_max:.3g} N, then kinetic {Fk:.3g} N")
ax.legend(frameon=False)
fig.savefig("friction_force_plot.png", dpi=300, bbox_inches="tight")

Output when run: Normal force N = 196.2 N, Kinetic friction = 49.1 N, Max static friction = 58.9 N, Sled friction = 31.2 N (N = 208.3 N), Ramp friction = 12.7 N, Car friction = 6000 N, mu_k = 0.51.

11.2 What each part does

Python traps: math.sin(30) uses radians; ^ is not a power; round(2.5) gives 2 (half to even); scipy.constants.g is 9.80665, which changes the last digit compared with 9.81.

11.3 What the figure shows

The same friction graph as in R: a straight rise while static friction matches the push, a peak at 58.9 N, then a drop to the flat kinetic level of 49.1 N. If you plot your own force-meter readings against the pull, you should see the same shape, with a little noise on the kinetic plateau.

11.4 Common problems and fixes

11.5 Useful one-liners

# friction for several masses at once
import numpy as np; 0.250 * np.array([5, 10, 20, 40]) * 9.81
# mu from 49 N to slide 10 kg (g = 9.8)
49 / (10 * 9.8)
# slip angle to static coefficient
np.tan(np.radians(26.6))
# mu from a lab graph (line through the origin)
N = np.array([1.96, 3.92, 5.89]); F = np.array([0.70, 1.36, 2.05]); (N @ F) / (N @ N)

Run it in a terminal with python fr.py, in a Jupyter notebook cell, or in Google Colab without installing anything.

12.1 Typical coefficients of friction

Typical textbook ranges. Real values depend on cleanliness, roughness, humidity, temperature and speed, and published measurements for the same pair can differ by 20% or more (Rabinowicz, 1951; Urbakh et al., 2004). Measure your own surfaces when accuracy matters.

12.2 Normal force in common situations

12.3 Slip angles (θ_c = arctan μ_s)

12.4 Force unit conversions

12.5 Braking and skidding on roads

Skid distance d = v² ÷ (2 μ_k g) with g = 9.81 m/s², braking distance only, no reaction time. Doubling the speed multiplies the distance by four.

12.6 Constants

Source: NIST, CODATA 2022 recommended values (Mohr et al., 2025, reference 1); NASA Planetary Fact Sheets.

This is the written method examiners expect, worked on the same example as Sections 7, 9, 10 and 11: a 20.0 kg crate slides across a level floor with μ_k = 0.250. Take g = 9.81 m/s². Find the friction force, then the push needed to give it an acceleration of 0.50 m/s².

14.1 The written method (GRESS)

1. Given. m = 20.0 kg, μ_k = 0.250, g = 9.81 m/s², a = 0.50 m/s². Draw the free-body diagram: weight down, normal force up, push forward, friction backward.
2. Required. Friction F (N), then push P (N).
3. Equation. N = mg; F = μ_kN; P − F = ma.
4. Substitute, with units. N = (20.0 kg)(9.81 m/s²) = 196.2 N. F = 0.250 × 196.2 N. P = F + (20.0 kg)(0.50 m/s²).
5. Solve. F = 49.05 N → 49.1 N. P = 49.05 + 10.0 = 59.05 N → 59 N (2 s.f., matching a = 0.50).
6. Check the units and the size. (no unit) × N = N. Friction is a quarter of the weight, as μ = 0.250 says.
7. Write the answer with units and direction. F = 49.1 N, opposing the motion; P = 59 N forward.

14.2 On a scientific calculator

Check the screen shows D (degrees) before any slope or angled-pull problem. In R (radian) mode, cos 30 gives 0.154 and every answer is wrong. The brackets in "μ from force and mass" matter: without them, 49 ÷ 10 × 9.8 gives 48.0.

14.3 Measure it yourself: a school lab practical

N uses g = 9.81 m/s². The gradients give μ_s ≈ 0.45 and μ_k ≈ 0.35. To check a row, choose the "μ from force and N" option in the calculator, for example F = 0.70 N and N = 1.96 N.

14.4 Practice problems with answers

14.5 Mistakes that lose marks

• Multiplying μ by the mass instead of the normal force. F = μN, and N = mg on a level floor.
• Writing static friction as μ_sN when the object is not moving. It equals the push until it slips.
• Using N = mg on a slope or with an angled pull.
• Using μ_s once the object is already sliding.
• Calculator in radian mode for an angle in degrees.
• Leaving out the direction. Friction acts along the surface, against the sliding.
• Giving μ a unit. It is a ratio of two forces and has none.

F = μN is one of the most used equations in mechanics, but it has a clear range where it works.

Use F = μN when:

• ✓ Two dry, solid surfaces slide, or are about to slide, over each other.
• ✓ You know μ for the pair of surfaces, or you can measure it.
• ✓ You can find the normal force from a free-body diagram.
• ✓ Speeds are everyday speeds, so μ_k is roughly constant.

Use something else when:

• ✗ The object is at rest and the push is below μ_sN: friction just equals the push.
• ✗ Wheels roll without skidding: use rolling resistance, F = C_rrN, with C_rr about 0.01.
• ✗ The object moves through air or water: use drag, which grows with v².
• ✗ The surfaces are well lubricated: friction then depends on speed and the oil film (hydrodynamic friction).
• ✗ The material is soft rubber at high speed: μ changes with speed and temperature.

Real-world examples:

1. Moving furniture: the force to start a sofa is μ_smg; once it slides you need only μ_kmg.
2. Car braking: with locked wheels, friction μ_kmg sets the stopping distance, d = v² ÷ (2μ_kg).
3. Conveyor belts and ramps: a parcel stays put on an inclined belt while tan θ ≤ μ_s.
4. Sledging and skiing: pulling a sled with an upward-angled rope lowers N and the friction.
5. Accident investigation: skid length and μ_k give the speed before braking.
6. Animal and human movement: walking needs static friction between foot and ground; on ice μ_s is too low to push off.

Assumptions

1. Coulomb friction: F is proportional to N. Check: dry, solid surfaces at moderate loads. For soft materials such as rubber, friction grows more slowly than N, so F = μN overestimates friction at high loads.
2. μ_k does not depend on speed. Check: everyday sliding speeds. For many surfaces μ_k falls a little as speed rises, so at high speed the formula overestimates friction.
3. Friction does not depend on contact area. Check: rigid blocks. For tyres and soft pads, area matters, and the simple model can be off in either direction.
4. μ is a fixed property of the two surfaces. Check: clean, dry surfaces. Dust, moisture or oil usually lowers μ, so a textbook value overestimates friction on dirty or wet surfaces.
5. Rigid block, no rolling or tipping. Check: the block is wide compared with its height. A tall object pushed high up can tip before it slides, and the friction answer no longer applies.
6. No air resistance. Check: speeds below a few m/s. Drag adds to friction, so ignoring it underestimates the total resistance and overestimates stopping distances in the Without μ mode only if drag is large.
7. Constant friction while stopping (Without μ mode). Check: a steady, straight skid. If braking varies, the result is the average friction over the stop.
8. Uniform g. g is taken as constant. Over the surface of the Earth it varies from 9.78 to 9.83 m/s², an effect of about 0.5%.

Limitations

• It does not model rolling resistance; wheels need a rolling coefficient, not μ.
• It does not model stick-slip (squeaks, juddering brakes), where friction jumps between static and kinetic many times a second.
• It does not include lubricated or fluid friction, which depends on viscosity and speed.
• It treats a single block. Stacked blocks, pulleys and connected masses need one free-body diagram per object.
• Typical μ values in Section 12 are guides only; measured values for the same pair often differ by 20% or more.

A friction force calculator rests on one short formula, F = μN, yet friction problems lose more marks than almost any other topic in mechanics. The multiplication is easy. The hard parts are finding the right normal force and knowing whether friction is static or kinetic.

Three habits fix nearly every error. First, draw the free-body diagram and find N before you touch μ: mg on a level floor, mg cos θ on a slope, mg − P sin θ with an upward pull. Second, decide whether the object is moving. At rest, friction only matches the push, up to μ_sN; sliding, it is μ_kN. Third, check the units: grams instead of kilograms and degrees in radian mode cause most wrong answers.

You do not always need μ. If you know the push and the acceleration, f = P − ma; if an object slides to rest, f = mv² ÷ (2d). Then μ_k = f ÷ mg. That is how road investigators and school labs measure friction for real.

Pick your situation, type your values in any units, and let this friction force calculator show every step, the static or kinetic verdict, a units check and a free-body diagram drawn from your numbers.

These peer-reviewed articles support the friction laws, the static and kinetic model, the lab methods and the common mistakes covered by this friction force calculator. Every DOI was checked in October 2026.

1. Mohr, P. J., Newell, D. B., Taylor, B. N., & Tiesinga, E. (2025). CODATA recommended values of the fundamental physical constants: 2022. Reviews of Modern Physics, 97(2), 025002. https://doi.org/10.1103/RevModPhys.97.025002
2. Popova, E., & Popov, V. L. (2015). The research works of Coulomb and Amontons and generalized laws of friction. Friction, 3(2), 183–190. https://doi.org/10.1007/s40544-015-0074-6
3. Hutchings, I. M. (2016). Leonardo da Vinci’s studies of friction. Wear, 360–361, 51–66. https://doi.org/10.1016/j.wear.2016.04.019
4. Bowden, F. P., & Tabor, D. (1939). The area of contact between stationary and between moving surfaces. Proceedings of the Royal Society of London. Series A, 169(938), 391–413. https://doi.org/10.1098/rspa.1939.0005
5. Rabinowicz, E. (1951). The nature of the static and kinetic coefficients of friction. Journal of Applied Physics, 22(11), 1373–1379. https://doi.org/10.1063/1.1699869
6. Dieterich, J. H. (1979). Modeling of rock friction: 1. Experimental results and constitutive equations. Journal of Geophysical Research: Solid Earth, 84(B5), 2161–2168. https://doi.org/10.1029/JB084iB05p02161
7. Ruina, A. (1983). Slip instability and state variable friction laws. Journal of Geophysical Research: Solid Earth, 88(B12), 10359–10370. https://doi.org/10.1029/JB088iB12p10359
8. Krim, J. (2002). Surface science and the atomic-scale origins of friction: What once was old is new again. Surface Science, 500(1–3), 741–758. https://doi.org/10.1016/S0039-6028(01)01529-1
9. Ringlein, J., & Robbins, M. O. (2004). Understanding and illustrating the atomic origins of friction. American Journal of Physics, 72(7), 884–891. https://doi.org/10.1119/1.1715107
10. Urbakh, M., Klafter, J., Gourdon, D., & Israelachvili, J. (2004). The nonlinear nature of friction. Nature, 430(6999), 525–528. https://doi.org/10.1038/nature02750
11. Gao, J., Luedtke, W. D., Gourdon, D., Ruths, M., Israelachvili, J. N., & Landman, U. (2004). Frictional forces and Amontons’ law: From the molecular to the macroscopic scale. The Journal of Physical Chemistry B, 108(11), 3410–3425. https://doi.org/10.1021/jp036362l
12. Persson, B. N. J. (2001). Theory of rubber friction and contact mechanics. The Journal of Chemical Physics, 115(8), 3840–3861. https://doi.org/10.1063/1.1388626
13. Besson, U., Borghi, L., De Ambrosis, A., & Mascheretti, P. (2007). How to teach friction: Experiments and models. American Journal of Physics, 75(12), 1106–1113. https://doi.org/10.1119/1.2779881
14. Sherwood, B. A., & Bernard, W. H. (1984). Work and heat transfer in the presence of sliding friction. American Journal of Physics, 52(11), 1001–1007. https://doi.org/10.1119/1.13775
15. Puttharugsa, C., Wicharn, S., Plaipichit, S., & Khemmani, S. (2021). Determining the coefficient of kinetic friction using smartphone sensors. The Physics Teacher, 59(6), 504–505. https://doi.org/10.1119/10.0006145
16. Baldock, C., & Johnson, R. (2016). Investigation of kinetic friction using an iPhone. Physics Education, 51(6), 065005. https://doi.org/10.1088/0031-9120/51/6/065005
17. Staacks, S., Hütz, S., Heinke, H., & Stampfer, C. (2018). Advanced tools for smartphone-based experiments: phyphox. Physics Education, 53(4), 045009. https://doi.org/10.1088/1361-6552/aac05e
18. Hestenes, D., Wells, M., & Swackhamer, G. (1992). Force concept inventory. The Physics Teacher, 30(3), 141–158. https://doi.org/10.1119/1.2343497
19. Thornton, R. K., & Sokoloff, D. R. (1998). Assessing student learning of Newton’s laws: The Force and Motion Conceptual Evaluation and the evaluation of active learning laboratory and lecture curricula. American Journal of Physics, 66(4), 338–352. https://doi.org/10.1119/1.18863
20. Hake, R. R. (1998). Interactive-engagement versus traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses. American Journal of Physics, 66(1), 64–74. https://doi.org/10.1119/1.18809