Coefficients of friction — measurement, material dependence

1.2.7 · Physics › Newton's Laws & Dynamics

Coefficient of friction KIYA hota hai?

Ratio KYU? Experiment dikhata hai ki friction linearly scale karta hai is baat ke saath ki surfaces kitni zyada squeeze ho rahi hain ($N$). Isse divide karne pe ek aisa number milta hai jo weight se independent hai — isliye hum kisi bhi load ke liye friction predict kar sakte hain.

$\mu$ KAISE measure karte hain? (Do classic methods — dono derive karo)

Method 1 — Inclined plane (angle of repose)

Block ko ramp pe rakho aur dheere dheere tilt karo jab tak woh just slip karna shuru kare. Us critical angle ${\theta }_c$ pe:

Incline ke along/perpendicular coordinates set up karo. Block pe forces:

• Weight $mg$ split hoti hai: along-slope $mg\sin \theta$ (sliding drive karta hai), into-slope $mg\cos \theta$ (surfaces press karta hai).
• Normal force $N$.
• Friction $f$ slope ke upar.

Perpendicular balance (surface se koi motion nahi): $N=mg\cos \theta \text{— KYU? Ramp ke perpendicular koi acceleration nahi.}$

Slipping ki verge pe, friction maxed out hai: $f={\mu }_{s}N$, aur along-slope balance abhi bhi hold karta hai: $mg\sin {\theta }_c={\mu }_{s}N={\mu }_{s}mg\cos {\theta }_c$

$mg$ cancel karo (KYU? mass-independence — wahi trick jo Galileo ke ramps ko kaam karwaati hai): $\boxed{{\mu }_s=\tan {\theta }_c}$

Khoobsurat result: mass cancel ho jaata hai, isliye sirf ek protractor chahiye — koi force meter nahi, koi scale nahi!

${\mu }_k$ ke liye: ${\theta }_c$ se thoda zyada tilt karo aur woh angle dhoondo jahan block constant velocity pe slide kare (zero acceleration). Same algebra deta hai ==${\mu }_k=\tan {\theta }_{\text{slide}}$==.

Method 2 — Horizontal pull

Spring scale se horizontally pull karo jab tak block just move kare; force $F_{\max }$ read karo. $N=mg\Rightarrow {\mu }_s=\frac{F_{\max }}{mg}$ Phir isse steadily move karte rakho (constant velocity, $a=0$): reading $F_{\text{slide}}$ deta hai ${\mu }_k=\frac{F_{\text{slide}}}{mg}.$ Constant velocity KYU? Agar $a=0$ toh applied force exactly kinetic friction ke barabar hai — reading confuse karne ke liye koi leftover net force nahi.

Material dependence — $\mu$ ko kya change karta hai?

Factors: material hardness/softness, surface roughness, contamination/lubricants, aur surface cleanliness. YEH NAHI (first order mein): apparent contact area, sliding speed, weight.

Worked examples

Flashcards

Connections

• Newton's Second Law — friction $\sum F=ma$ mein ek force term ki tarah enter karta hai.
• Static vs Kinetic Friction — do regimes yahaan summarize kiye hain.
• Inclined Plane Dynamics — same coordinate decomposition.
• Normal Force — woh $N$ jise $\mu$ multiply karta hai.
• Lubrication & Tribology — films $\mu$ kaise kam karti hain.
• Free Body Diagrams — dono measurement methods mein use kiya gaya tool.

Concept Map