A man pushes a cylinder of mass $m_{1}$ with the help of a plank of mass $m_{2}$ as shown in the figure. There is no slipping at any contact. The horizontal component of the force applied by the man is $F$ . The acceleration of the plank is

\frac{3 m_{1} F}{3 m_{1} + 8 m_{2}}

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\frac{m_{2} F}{3 m_{1} + 8 m_{2}}

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\frac{4 F}{3 m_{1} + 8 m_{2}}

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\frac{8 F}{3 m_{1} + 8 m_{2}}

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Solution

The correct option is D $\frac{8 F}{3 m_{1} + 8 m_{2}}$
$f 1 =$ force of friction between the two bodies and is towards right for the cylinder and left for the plank.
$f 2 =$ =force of friction between the ground and the cylinder and is towards left.
$F$ =Applied horizontal force on the plank.
$a 1 =$ =acceleration of plank towards the right.
$a =$ =acceleration of cylinder towards right.
$B$ =angular acceleration of cylinder
Then the equations o motion are
$F - f_{1} = M_{2} a_{1} f_{1} - f_{2} = M_{1} a (f_{1} + f_{2}) r = 1 β$
Condition for non slipping are $a_{1} = a + β r ⟶ (4)$ (At top contact of two bodies)
$a_{2} = β r ⟶ (5)$ (At the contact of cylinder and the ground)
Solving these equations
$a = \frac{4 F}{(8 M_{2} + 3 M_{1})} a_{1} = \frac{8 F}{(8 M_{2} + 3 M_{1})}$

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A man pushes a cylinder of mass m1 with the help of a plank of mass m2 as shown in the figure. There is no slipping at any contact. The horizontal component of the force applied by the man is F. The acceleration of the plank is

A man pushes a cylinder of mass $m_{1}$ with the help of a plank of mass $m_{2}$ as shown in the figure. There is no slipping at any contact. The horizontal component of the force applied by the man is $F$ . The acceleration of the plank is