In the shown figure a triangular wedge and a cylinder each having equal mass 'm' are released from rest. The vertical wall and horizontal surface are frictionless and between the wedge and the cylinder friction is sufficient to allow cylinder to roll over it. Choose the correct option(s), just after system is released from rest: [Take $: g = 10 m / s^{2}$ ]

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Acceleration of wedge is $g/ 3$

Minimum value of coefficient of friction between wedge and cylinder for rolling without slipping is $1/2$

Minimum value of coefficient of friction between wedge and cylinder for rolling without slipping is $1/3$

If $m = 3 \sqrt{20} k g$ , the contact force between cylinder and wedge is $100 N$ .

answer is A, B, D.

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Detailed Solution

FBD of cylinder
Question Image
$m g - f_{s} sin 45^{\circ} - N_{1} cos 45^{\circ} = m a_{1}$ ……….(i)
$\frac{m R^{2}}{2} α = f_{s} R$ ……….(ii)
FBD for wedge $N_{1} sin 45^{\circ} - f_{s} cos 45^{\circ} = mA$ ……….(iii)
Condition for rolling    $R α - a_{1} sin 45^{\circ} = A cos 45^{\circ}$ ……….(iv)
and    $A sin 45^{\circ} = a_{1} cos 45^{\circ}$ ……….(v)
also    $f_{s} \leq μ N_{1} μ_{m i n} = f_{s} / N_{1}$
Contact force $F_{c} = \sqrt{N_{1}^{2} + f_{s}^{2}}$