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Question: A right triangular plate ABC of mass \(m\)is free to rotate in the vertical plane about a fixed hori...

A right triangular plate ABC of mass mmis free to rotate in the vertical plane about a fixed horizontal axis through A. It is supported by a string such that the side AB is horizontal. The reaction at the support A is:

(A) mg3\dfrac{{mg}}{3}
(B) 2mg3\dfrac{{2mg}}{3}
(C) mg2\dfrac{{mg}}{2}
(D) mgmg

Explanation

Solution

In order to solve this question, we need to use the concept of centroid as it is the point at which the entire mass of the system is balanced. Now we need to find the force experienced at point A. Now, Taking moment of force around point B, we can obtain the required solution.

Complete Step-By-Step solution:
We know that the length AB given to us is ll.
We know that, in the case of a right angled triangle, the centroid lies at a distance of 23\dfrac{2}{3}od the base, in this case it is ll.
Now, let us consider that at any point in the given system the torque is conserved. Suppose we consider the torque to be conserved at point B.
Now, the tension due to the string acts at point B as well.
Now, we know that torque is the product of force and its perpendicular distance. Thus, at point B torque is 00as the perpendicular distance is00.
We know that point A acts as a support, therefore, an upward normal force must exist acting at the distancell.
Let us consider this force asNN.
Now, this normal force is opposed by the weight of the body which acts from the centroid.
Thus, the total torque at A is given by:
N×l−mg×2l3N \times l - mg \times \dfrac{{2l}}{3}
Where, mmis the mass of the plate
Now, since the total torque is conserved, we can write:
N×l−mg×2l3=0N \times l - mg \times \dfrac{{2l}}{3} = 0
On solving the equation and cancelling the equal terms, we get:
N=2mg3N = \dfrac{{2mg}}{3}
This is our required solution.

Hence, option (B) is correct.

Note:
Centroid of a triangle implies the point where three medians intersect. This can be computed using the coordinates of three points that makes the triangle. The centre of mass of a triangle lies at its centroid. This is thus the point where the entire weight of the triangle can be balanced.