A disc of mass m and radius R lies flat on a smooth horizontal table. A mass less string runs halfway around it as shown in figure. One end of the string is attached to a small body of mass m and the other end is being pulled with a force F. The circumference of the disc is sufficiently rough so that the string does not slip over it. Find acceleration of the small body.

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$\frac{F}{2 m}$

$\frac{2 F}{3 m}$

$\frac{2 F}{m}$

$\frac{F}{4 m}$

answer is A.

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Detailed Solution

For no slipping condition of thread acceleration of point A on the circumference of the disc = acceleration of the small body
$\Rightarrow Rα = a_{0} = a$

Other equations are
T = ma ………(ii)

F +T = ma₀ ……..(iii)

And $(F - T) R = \frac{1}{2} {mR}^{2} . α \Rightarrow F - T = \frac{1}{2} mRo$ . …..(iv)
Using (i) in (iv) $F - T = \frac{1}{2} m (a + a_{0})$
Using (ii) in this $F = \frac{3}{2} ma + \frac{1}{2} ma$

From (ii) and (iii) F = ma₀ -ma ……(vi)
Solving (v) and (vi) $a = \frac{F}{4 m}$

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