A plastic circular disc of radius R is placed on a thin oil film, spread over a flat horizontal surface. The torque required to spin the disc about its central vertical axis with a constant angular velocity is proportional to

Engineering

Physics

Viscosity and Terminal Velocity New

Torque and Equilibrium of Rigid Bodies

Forces and Types of Forces

Question

A plastic circular disc of radius R is placed on a thin oil film, spread over a flat horizontal surface. The torque required to spin the disc about its central vertical axis with a constant angular velocity is proportional to

R³

R²

R⁶

R⁴

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Let coefficient of viscosity & thickness of film → η & x respectively
Viscous force on element ring is

$dF = η (2 πrdr) [\frac{wr - 0}{x}]$

dτ = (dF) r

$∴ τ = \frac{η 2 πw}{t} \int_{0}^{R} r^{3} . dr = (\frac{η (2 π) w}{4 x}) R^{4}$

$∴ τ \propto R^{4}$