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Question: Answered & Verified by Expert
As shown in Fig. the two sides of a step ladder BA and $C A$ are $1.6 \mathrm{~m}$ long and hinged at $A$. $A$ rope $D E, 0.5 \mathrm{~m}$ is tied half way up. A weight $40 \mathrm{~kg}$ is suspended from a point $\mathrm{F}, 1.2 \mathrm{~m}$ from $\mathrm{B}$ along the ladder BA. Assuming the floor to be frictionless and neglecting the weight of the ladder, find the tension in the rope and forces exerted by the floor on the ladder.
(Take $g=9.8 \mathrm{~m} / \mathrm{s}^2$ )
(Hint: Consider the equilibrium of each side of the ladder separately.)

PhysicsSystem of Particles and Rotational Motion
Solution:
2350 Upvotes Verified Answer
The forces acting on the ladder are shown in Fig. Here, $m=40 \mathrm{~kg} \Rightarrow$ weight $=40 \times 9.8 \mathrm{~N}=392 \mathrm{~N}, A B=$
$$
A C=1.6 \mathrm{~m}, B D=\frac{1}{2} \times 1.6 \mathrm{~m}=0.8 \mathrm{~m} \text {, }
$$
$B F=1.2 \mathrm{~m}$ and $D E=0.5 \mathrm{~m}$,
In the Fig. $\triangle A D E$ and $\triangle A B C$ are similar triangles, hence,
$$
\begin{aligned}
&B C=D E \times \frac{A B}{A D}=\frac{0.5 \times 1.6}{0.8}=1.0 \mathrm{~m} \\
&\therefore \quad W \times(M B)=N_C \times(C B)
\end{aligned}
$$
But $M B=\frac{K B \times B F}{B A}=\frac{0.5 \times 1.2}{1.6}=0.375 \mathrm{~m}$
Substituting this value in (i), we get
$$
N_C=\frac{W \times(M B)}{(C B)}=\frac{392 \times 0.375}{1}=147 \mathrm{~N}
$$


Again considering equilibrium at point $C$ in similar manner, we have
$W \times(M C)=N_B \times(B C)$
$\therefore \quad N_B=\frac{W \times(M C)}{(B C)}=\frac{W \times(B C-B M)}{(B C)}$
$=\frac{393 \times(1-0.375)}{1}=245 \mathrm{~N}$
Now, it can be easily shown that tension in the string $T=N_B-N_C=245-147=98 \mathrm{~N}$.

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