A uniform solid cylinder of mass "m" rests on two horizontal planks as shown in figure. A thread is wound on the cylinder. The hanging end of the thread is pulled vertically up with a force F. The force F is applied at its maximum value which still does not bring about any sliding of the cylinder, if coefficient of friction between cylinder and plank is equal to μ. What is the maximum acceleration of the axis of the cylinder?

Engineering

Physics

Rolling

Question

A uniform solid cylinder of mass "m" rests on two horizontal planks as shown in figure. A thread is wound on the cylinder. The hanging end of the thread is pulled vertically up with a force F. The force F is applied at its maximum value which still does not bring about any sliding of the cylinder, if coefficient of friction between cylinder and plank is equal to μ. What is the maximum acceleration of the axis of the cylinder?

A

$\frac{3 μg}{3 + 2 μ}$

B

$\frac{2 μg}{3 + 2 μ}$

C

$\frac{3 μg}{2 + 3 μ}$

D

$\frac{2 μg}{2 + 3 μ}$

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As the force F is applied, the cylinder tends to rotate in anticlockwise direction with point of contact C tending to move to right.

τ = Iα

$(F - f) r = (\frac{{mr}^{2}}{2}) α$ .... (1)

f = ma ... (2)

a = αr ...(3)

(1) / eqn. (2)

$\frac{F - f}{f} = \frac{r . α}{2 a} = \frac{1}{2}$

$F = \frac{3}{2} f \Rightarrow F_{max} = \frac{3}{2} f_{\max}$

f_max = μN = μ(mg – F)

$F_{max} = \frac{3}{2} . μ (mg - F_{\max})$

$F_{max} = \frac{3 μmg}{2 + 3 μ}$

$a_{max} = \frac{f_{\max}}{m} = \frac{2}{3 m} F_{\max}$

$a_{max} = \frac{2 . μg}{2 + 3 μ}$