Question

Assertion: A ladder is more apt to slip, when you are high up on it, than when you just begin to climb.
Reason: At the high up on a ladder, the torque is large and on climbing down, the torque is small.
A) Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
B) Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
C) Assertion is correct but Reason is incorrect
D) Both Assertion and Reason are incorrect

Answer 1

Torque on an object is defined as the moment of force of that object, when we consider rotational motion of the object. A ladder leaning on a vertical wall is one of the best examples to understand torque. On each point considered on the ladder, the torque is different.

Formula used:
$\tau =Fr\sin \theta$

Complete step-by-step answer:
The force that tends to rotate an object about a pivot point is termed as the torque on that object. It is also defined as the moment of force of an object, when we consider the rotational motion of the object.
To understand torque, let us take the help of a diagram as shown below.

Let us consider an object placed at $B$. The object is considered to be rotating with respect to a pivot point $A$. Position of the object with respect to the pivot point is given by $AB=r$ and the rotational force acting on the object is denoted as $F$. If $\theta$ is the angle between rotational force on the object and the position of the object with respect to the pivot point, torque on the object is given by
$\tau =Fr\sin \theta$
where
$\tau$ is the torque acting on the object at $B$
$F$ is the rotational force acting on the object
$r$ is the position of the object with respect to pivot point $A$
$\theta$ is the angle between the position of the object and the rotational force acting on the object
Let this be equation 1.
A ladder leaning against a wall can be considered as one of the best examples to explain torque. Torque acting on different points on the ladder can be determined from its position from the floor and the force acting on that considered point.
Now, let us consider a man standing on the ladder at a point just above the point of contact between the ladder and the floor, as shown in the following diagram.

Clearly, from equation 1, torque at point $B$ is given by
${{\tau }_{1}}=F{{r}_{1}}\sin \theta$
where
${{\tau }_{1}}$ is the torque acting at point $B$
$F$ is the force acting on the ladder due the weight of the man
${{r}_{1}}$ is the position of man with respect to point $A$
$\theta$ is the angle between the position of man and the force on ladder
Let this be equation 2.

Similarly, let us consider the same man, standing at the top of the ladder, as shown in the following figure.

Clearly, from equation 1, torque at the point $D$ is given by
${{\tau }_{2}}=F{{r}_{2}}\sin \theta$
where
${{\tau }_{2}}$ is the torque acting at point $D$
$F$ is the force acting on the ladder due the weight of the man
${{r}_{2}}$ is the position of man with respect to point $A$
$\theta$ is the angle between the position of man and the force on ladder
Let this be equation 3.

Carefully observing both the diagrams above, it can be inferred that
${{r}_{1}}<{{r}_{2}}\Rightarrow {{\tau }_{1}}<{{\tau }_{2}}$
where
${{r}_{1}}$ is the position of man on the ladder just above the floor (at point $B$)
${{r}_{2}}$ is the position of man on the ladder at its top (at point $D$)
${{\tau }_{1}}$ is the torque at point $B$
${{\tau }_{2}}$ is the torque at the point $D$
Let this be equation 4.
We know that when torque acting at a point on the ladder is more, the chances of the ladder to slip is more. Hence, the given assertion is correct.
Also, from equation 2, equation 3 and equation 4, it is clear that torque increases as we go high up the ladder. Hence, the given reason is also correct.
Since the given reason is the correct explanation for the given assertion, the correct option to be marked is A.
So, the correct answer is “Option A”.

Note: Students should clearly understand that the angle between the force on the ladder and the position of the ladder stays the same, even when two different points on the ladder are considered. The same is the case with force applied on the ladder too. The same man is moving from the bottom of the ladder to the top of the ladder. It also needs to be understood that this force acting on the ladder when a man climbs on it is nothing but the weight of that man, acting downwards.