Rotation 3 Question 23
29. A man pushes a cylinder of mass $m _1$ with the help of a plank of mass $m _2$ as shown. There is no slipping at any contact. The horizontal component of the force applied by the man is $F$. Find
$(1999,10 M)$
(a) the accelerations of the plank and the centre of mass of the cylinder and
(b) the magnitudes and directions of frictional forces at contact points.
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Solution:
- We can choose any arbitrary directions of frictional forces at different contacts.
In the final answer the negative values will show the opposite directions.
Let $f _1=$ friction between plank and cylinder
$f _2=$ friction between cylinder and ground
$a _1=$ acceleration of plank
$a _2=$ acceleration of centre of mass of cylinder
and $\alpha=$ angular acceleration of cylinder about its CM.
Since, there is no slipping anywhere
$\therefore \quad a _1=2 a _2$
$a _1=\frac{F-f _1}{m _2}$
$a _2=\frac{f _1+f _2}{m _1}$
$\alpha=\frac{\left(f _1-f _2\right) R}{I}=\frac{\left(f _1-f _2\right) R}{\frac{1}{2} m _1 R^{2}}$
$\alpha=\frac{2\left(f _1-f _2\right)}{m _1 R}$
$a _2=R \alpha=\frac{2\left(f _1-f _2\right)}{m _1}$
(a) Solving Eqs. (i) to (v), we get
(b)
$$ \begin{aligned} a _1 & =\frac{8 F}{3 m _1+8 m _2} \text { and } a _2=\frac{4 F}{3 m _1+8 m _2} \\ f _1 & =\frac{3 m _1 F}{3 m _1+8 m _2} ; \\ f _2 & =\frac{m _1 F}{3 m _1+8 m _2} \end{aligned} $$
Since, all quantities are positive, they are correctly shown in figures.
