A uniform beam of length L whose mass is m, rest with its ends on two digital scales ( Figure 13.2). A block whose mass is M rests on the beam, its center is one-fourth away from the beam's left end. What do the scales read ?

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To determine the readings on the two digital scales, we need to calculate the weight distribution of the beam and the block.

First, let's define the variables:
L = length of the beam
m = mass of the beam
M = mass of the block

Given that the center of the block is one-fourth away from the beam's left end, we can assume that the distance from the left end of the beam to the center of the block is L/4.
This means that the distance from the center of the block to the right end of the beam is 3L/4.

To find the readings on the two digital scales, we need to consider the moments (torques) acting on the beam and the block.

1. Beam Moments:
The beam is in equilibrium, so the sum of the torques acting on it is zero.

1.1. Torque due to the weight of the beam:
The weight of the beam acts at its center (L/2), so the torque due to the weight is 0.

1.2. Torque due to the weight of the block:
The weight of the block acts at its center, which is located at L/4 away from the left end of the beam.
Therefore, the torque due to the weight of the block is M * g * (L/4), where g is the acceleration due to gravity.

2. Block Moment:
The block is also in equilibrium, so the sum of the torques acting on it is zero.

2.1. Torque due to the weight of the block:
The weight of the block acts at its center, which is located at L/4 away from the left end of the beam.
Therefore, the torque due to the weight of the block is M * g * (L/4).

Based on the moments calculated above, we can now determine the readings on the two digital scales.

The left digital scale will read the combined weight of the beam and the block. It can be calculated as follows:
Reading on the left scale = weight of the beam + weight of the block
= m * g + M * g

The right digital scale will read the weight of the beam only, as it supports the right end of the beam where the weight of the block is not acting.
Therefore, the reading on the right scale = weight of the beam = m * g

So, the left digital scale will read m * g + M * g, and the right digital scale will read m * g.