Question 737: Centre of Mass and Rotational Motion

Question

Two identical ladders, each of mass M and 
length L are resting on the rough horizontal 
surface as shown in the figure. A block of 
mass $$m$$ hangs from P. If the system is in 
equilibrium, find the magnitude and the 
direction of frictional force at A and B.

Accepted Answer

Let the block of mass $m$ hangs from the point $P$ by a string attached to the hinges of the two ladders.
Newton's second law gives the tension $T$ in the string as $T=m g$. By symmetry, string tension pulls down each ladder by a force
$$ T / 2=m g / 2 . $$
By Newton's third law, the reaction forces acting on the two ladders at the hinge point $P$ are equal and opposite. These are shown by $R_1$ and $R_2$ in the figure. By symmetry, the normal reaction $N$ at $A$ and $B$ are equal, the friction forces $f$ at $A$ and $B$ are equal in magnitude but opposite in direction. Another force acting on both the ladders is their weight $M g$ which pass through their centre of mass. In equilibrium, the resultant forces on the two ladders are zero i.e.,
$$ \begin{aligned} & N+R_2-M g-m g / 2=0, ........(1)\ & f=R_1, .........(2)\ & N-R_2-M g-m g / 2=0 .........(3) \end{aligned} $$
The equations (1) and (3) give $R_2=0$ (as expected!) and $N=(M+m / 2) g$. In equilibrium, the net torque about any point is zero. Thus, the torque about the point $P$ for the left ladder is
$$ M g(L / 2) \cos 	heta+f L \sin 	heta-N L \cos 	heta=0 $$ ..........(4)
Substitute $N=(M+m / 2) g$ in equation (4) and simplify to get
$$ f=\left(\frac{M+m}{2}ight) g \cot 	heta . $$

Answer

$$f=\left[\frac{\mathbf{M}+m}{2}ight] g \cot 	heta$$

Answer

$$f=\left[\frac{\mathbf{M}+m}{3}ight] g \cot 	heta$$

Answer

$$f=\left[\frac{\mathbf{M}+m}{2}ight] 5g \cot 	heta$$

Answer

$$f=\left[\frac{\mathbf{M}+m}{4}ight] g \cot 	heta$$

Centre of Mass and Rotational Motion

Two identical ladders, each of mass M and length L are resting on the rough horizontal surface as shown in the figure. A block of mass $m$ hangs from P. If the system is in equilibrium, find the magnitude and the direction of frictional force at A and B.

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