Persons A and B are standing on a board of uniform linear density that is balanced on two...

Personand are standing on a board of uniform linear density, it is balanced on two supports, as shown in the figure.

Here, L is the length, is the mass of the person A, is the mass of the person B, and x is the maximum distance from the right end of the board.

Let be the mass of a person.

Mass of the person,

Mass of the board,

Assume the board’s weight acts as its center. To determine the distance (x) from the edge of the board that person A can stand without tipping the board, balance the torques around a pivot point.

The two supports points of the board will act as the pivot points. In this case if the board tips in a counter clock wise direction then the right support will not contribute to the torque. If the board tips in a clockwise direction, the left support will not contribute to the torque.

Consider the case when the board tipping in the clockwise direction. The pivot point must be right support and the left support will not contribute.

The torque due to the weight of the board is zero because the moment arm is zero.

Thus, the counterclockwise torque is given as follows:

The maximum clockwise torque with is given as follows:

Thus, the board cannot tip in the clockwise direction.

Consider the case when the board tipping in the counterclockwise direction. The pivot point is the left support.

Thus, the counterclockwise torque due to person B is given as follows:

The clockwise torque due to person A is given as follows:

Also the clockwise torque due to the board is,

In static equilibrium, the net torque will be equal to zero.

Now convert all masses in multiple of m as follows:

Thus, the maximum distance from the right end of the board at which person A can stand without tipping the board is .