Question

Figure shows a vertical force applied tangentially to a uniform cylinder of weight `F_(g)`. The coefficient static friction between the cylinder and both surface is `0.500`. In terms of `F_(g)`, find the maximum force `P` that can be applied that does not cause the cylinder to rotate.
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Answer 1

Correct Answer - `(3)/(8) F_(g)`

`W_(t) = F_(g)`
`mu = 0.5`
The cylinder is in both translational and rotational equilibrium.
:. Force is balance
`P + f_(2) + N_(1) = F_(g)` …(1)
`N_(2) = f_(1)` ….(2)
Torque is also balance about `COM`
`f_(1) xx R + f_(2) xx R - P xx R = 0`
`f_(1) + f_(2) = P`
as force is maximum
`mu N_(1) + mu N_(2) = P`
`mu (N_(1) + N_(2)) = P`
From (2)
`mu(N_(1) +mu N_(1)) = P` or `N_(1) = (P)/(mu(1 +mu))`....(3)
from (1) and (2) `P + mu N_(2) + N_(1) = F_(g)`
`P + mu xx mu N_(1) + N_(1) = F_(g)`
`(mu^(2) + 1) N_(1) = F_(g) - P` ....(4)
from (3) and (4) `(P)/(mu(1 + mu)) = (F_(g) -P)/((mu^(2) + 1))`
Put `mu = 0.5`, we gate `P = (3)/(8) F_(g)`.