A viscous clutch as shown in figure transmits torque. Radius of each clutch plate is R and separation between the plates is a and is completely filled with liquid of coefficient of viscosity μ. If ωᵢ and ωₒ are angular velocities of plates connected to input and output respectively, the torque transmitted is πμ(ωᵢ − ωₒ)R⁴/(xa). Find x.
Physics · Mixed Chapter
JEE Mains

Question

A viscous clutch as shown in figure transmits torque. Radius of each clutch plate is R and separation between the plates is a and is completely filled with liquid of coefficient of viscosity μ. If ωᵢ and ωₒ are angular velocities of plates connected to input and output respectively, the torque transmitted is πμ(ωᵢ − ωₒ)R⁴/(xa). Find x.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: At radius r, relative tangential speed of plates is Δv=(ωᵢ−ωₒ)r. The viscous shear stress is τ_shear=μΔv/a. Step 2: For an annulus of area dA=2πrdr, force dF=τ_shear dA and torque dT=r dF=2πμ(ωᵢ−ωₒ)r³dr/a. Step 3: Integrate r=0 to R: T=2πμΔω/a×(R⁴/4)=πμΔωR⁴/(2a). Step 4: Compare denominators to get x=2.