Laws of Motion 2 Question 6
6. A uniform rope of length $L$ and mass $M$ lying on a smooth table is pulled by a constant force $F$. What is the tension in the rope at a distance $l$ from the end where the force is applied?
(1978)
(a) $0.4 ms^{-2}$
(b) $3.2 ms^{-2}$
(c) $0.8 ms^{-2}$
(d) $0 ms^{-2}$
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Answer:
Correct Answer: 6. (c, d)
Solution:
- $\mu _2$ can never be zero for equilibrium.
When $\mu _1=0$, we have
$$ \begin{aligned} N _1 & =\mu _2 N _2 \\ & N _2=m g \\ \tau _B & =0 \\ \Rightarrow \quad & m g \frac{L}{2} \cos \theta=N _1 L \sin \theta \\ \Rightarrow \quad & N _1=\frac{m g \cot \theta}{2} \Rightarrow N _1 \tan \theta=\frac{m g}{2} \end{aligned} $$
When, $\mu _1 \neq 0$ we have
$$ \begin{aligned} \mu _1 N _1+N _2 & =m g \\ \mu _2 N _2 & =N _1 \\ \Rightarrow \quad N _2 & =\frac{m g}{1+\mu _1 \mu _2} \end{aligned} $$
