Question

Two uniform equal ladders AB and AC, each of weight W leans against each other and a string is tied between B and C. They stand on a smooth horizontal surface. Then

A) The force exerted by one ladder on the other at A is equal to the tension T in the string.
B) \[T=\dfrac{W}{2}\cot \theta \]
C) The normal reaction at B and C are equal.
D) Both (A) and (C) are correct.

Answer 1

Hint: We have given a diagram in which two ladders are inclined towards each other and it is tied with a string. Now the diagram shows the direction of tension and weight. It is also given that the weight of both the ladders are the same. We have to find which of the given options are correct. With the use of diagrams itself we will solve the question.

Complete step by step answer:
Here we have given two uniform equal ladders i.e. ladders of same length and they are not having a motion. Also they are inclined towards each other and the point of contact is A.
Now according to option A, the force exerted by one ladder on the other ladder A is equal to the tension in string. We know the third law of motion which says that every action has an equal and opposite reaction. Therefore both ladders will apply the same force on each other just the direction will be opposite. Now the magnitude of force will be equal to tension in the string because only force acting the ladders is due to the tension in the string. As the weight or gravitational force is always balanced by the normal reaction force. And the force will be applied horizontally and only horizontal force here is tension.
Hence option A is correct.
The string is common in between ladder AB and AC, therefore we can find tension by considering one of the ladder. Now the weight, tension and θ can be shown as

Now using trigonometry, we can write
\[\begin{align}
  & \tan \theta =\dfrac{W}{T} \\
 & \Rightarrow T=\dfrac{W}{\tan \theta } \\
 & \Rightarrow T=W\cot \theta \\
\end{align}\]
From the above equation we can conclude that option B is wrong.
Now consider the third option which says that normal reaction is at B and C. Then we know that normal reaction is always in the opposite direction of the weight and the magnitude of normal reaction force is equal to weight. As weight for both ladders are the same therefore their normal reaction will also be the same.
So option C is also correct.

Hence the actual correct option will be D which says that option A and option C is correct.

Note:
Here one can consider that the tension in the string is twice as the tension due to AB and AC. And the given diagram shows T twice but in actuality the tension will be the one which we calculated above as it is attached to the same string therefore there won’t be different components of tension due to AC and AB.