Friction, losses, and mechanical efficiency

• Use simple static and kinetic friction models within their stated limits.
• Calculate transmission efficiency from input and output power and isolate likely loss sources.

• forces-free-body
• torque-gearing

Friction is useful when a tire pushes a toy car, a belt grips a pulley, or a screw joint resists slip. It is wasteful when a misaligned axle scrubs its bushing. The design question is not “remove friction,” but put traction where force transfer is needed and reduce parasitic resistance elsewhere.

A simple dry-contact model bounds static friction by |F_t| ≤ μ_sN. Sliding friction is often approximated as F_k = μ_kN opposite relative motion. These coefficients summarize a particular pair of surfaces and conditions; they are not fundamental constants. Rolling resistance, bearing seals, lubricant shear, gear tooth loss, air drag, cable flex, and material hysteresis need different models. A vacuum blower, for example, spends shaft power moving air as well as overcoming bearing and aerodynamic losses; blocking flow does not imply a simple universal reduction in motor load.

Efficiency is η = P_out/P_in. For steady rotation, P = τω; for translation, P = Fv. Efficiencies multiply through stages: three stages each at 90% yield 0.9³ ≈ 73%, not 70% or 90%. This informs a concrete build decision: before buying a larger motor, disconnect transmission stages and measure no-load current or hand resistance to find a bent shaft, overtight belt, rubbing wheel, or misaligned bearing.

A common misconception is that static friction always equals μ_sN. That value is a maximum; actual static friction takes the value required to prevent slip up to the limit. Another failure is estimating efficiency from force alone while input and output speeds differ. Heat, sound, vibration, and wear are clues to loss, but they are qualitative until measured. Any efficiency over 100% signals inconsistent boundaries, units, timing, or stored energy—not a miraculous gearbox.

Problem

A gearbox receives 0.050 N·m at 200 rad/s and delivers 0.42 N·m at 18 rad/s. Find power, efficiency, and loss.

Steps

1. Input power is 0.050 N·m × 200 rad/s = 10.0 W.
2. Output power is 0.42 N·m × 18 rad/s = 7.56 W.
3. Efficiency is 7.56 W ÷ 10.0 W = 0.756 = 75.6%; loss is 10.0 - 7.56 = 2.44 W.

Result

The gearbox is 75.6% efficient at this operating point and dissipates 2.44 W.

Sanity check

Output power is below input power and the implied speed reduction of 11.1:1 gives less than the ideal 11.1-fold torque gain.

A box rests on a level surface. The maximum static friction is 12 N, and you push horizontally with 5 N without motion. What is the friction force magnitude?

A. 0 N B. 5 N C. 12 N D. 17 N

Your answer: ______________________________

Reasoning:

Goal

Separate rolling resistance, alignment loss, and drivetrain drag on a small cart.

Equipment

• Toy car or simple cart
• Spring scale
• Tape measure
• Small removable mass
• Stopwatch

Procedure

1. Predict how added normal load will change steady pull force on the same surface.
2. Pull the unpowered cart slowly at roughly constant speed and record actual force over a marked distance, then repeat with added mass.
3. Disconnect or lift one drivetrain element if accessible without damage and compare pull force.
4. Deliberate fault: skew one wheel slightly; identify the fault from increased pull, curved travel, and local rubbing, then restore alignment.

What to observe

• Expected: a skewed wheel raises resistance and causes drift; added load often raises contact losses.
• Measure: actual force range, mass, distance, and configuration; hand-pull variation limits precision.

Transfer challenge

Create a loss budget for a printer-style belt axis and identify the first isolation test if its motor runs hot.

Cautions

• Perform tests unpowered and keep hair, cords, and fingers clear of wheels and belts.

Learner: ____________________ Date: ____________________

Evidence artifact or file: ____________________________________________

Notes: