A disc of mass \(m\) and radius \(R\) placed on a smooth horizontal table as shown in figure. A light ideal…

A disc of mass \(m\) and radius \(R\) placed on a smooth horizontal table as shown in figure. A light ideal string passes through the disc such that the 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.

1. \(\frac{F}{4 m}\)
2. \(\frac{F}{m}\)
3. \(\frac{F}{2 m}\)
4. \(\frac{F}{3 m}\)

Solution

\(\Rightarrow R \alpha=a_{0}+a\) ...(i)
Other equations are
\(\begin{array}{l}
T=m a \quad \text{...(ii)}\\
F+T=m a_{0}\quad \text{...(iii)}
\end{array}\)
\(\begin{array}{l}
\text { and }(F-T) R=\frac{1}{2} m R^{2} \cdot \alpha \\
\Rightarrow \quad F-T=\frac{1}{2} m R \alpha \quad \text{...(iv)}
\end{array}\)
Using (i) in (iv) \(F-T=\frac{1}{2} m\left(a+a_{0}\right)\)
Using (ii) in this \(F=\frac{3}{2} m a+\frac{1}{2} m a_{0}\) ...(v)
From (ii) and (iii) \(F=m a_{0}-m a\) ...(vi)
\(\ldots .(\mathrm{vi})\)
Solving \((\mathrm{v})\) and (vi) \(a=\frac{F}{4 m}\)

Asked in: JEE Mains - Rotational Motion - Chapter Test

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