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A viscous clutch as shown in figure tran...


A viscous clutch as shown in figure tranmits torque. Radius of each clutch plate is R and separation between the plates is a and is completely filled with liquid of coefficient of viscous `mu`. If `omega_(i)` and `omega_(0)` are angular velocities of plates connected to input and output respectively.
The torque transmitted is

A

`(pimu(omega_(i)^(2)-omega_(0)^(2))R^(4))/(omega_(i)a)`

B

`(pimu(omega_(i)^(2)-omega_(0)^(2))R^(4))/(omega_(0)a)`

C

`(pimu(omega_(i)^(2)-omega_(0)^(2))R^(4))/(2omega_(0)a)`

D

`(pimu(omega_(i)-omega_(0))R^(4))/(2a)`

Text Solution

Verified by Experts

The correct Answer is:
D

Let's consider a circular element of radius r and width dr, co-axial with the system.
Velocity difference across the element
`dv=r(omega_(i)-omega_(0))`
Hence `dF_(v)=muxx2pirdrxx(r(omega_(i)-omega_(0)))/(a)`
These force are tangential and hence
`dtau_(v)=dF_(v)xxr=(2pimu(omega_(i)-omega_(0)))/(a)r^(3)dr`
Hence `tau=int_(0)^(R)dtau_(v)=(pimu(omega_(i)-omega_(0))R^(4))/(2a)`
Power (P) `=tauomega`
Hence efficiency `=(tauomega_(0))/(tauomega_(i))=(omega_(o))/(omega_(i))`
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