With what acceleration should a box descend so that a block of mass M, placed in it exerts force mg/4 on thefloor of the box ?

Question

Accepted Answer

To determine the correct acceleration with which the box should descend so that a block of mass $ M $ exerts a force of $ \frac{mg}{4} $ on the floor of the box, we need to analyze the forces acting on the block.

### Explanation for Option 3 (Correct Answer: $ \frac{3g}{4} $)

1. **Understanding the Forces:**
   - When the box is descending with acceleration $ a $, the apparent weight of the block is reduced.
   - The apparent weight $ W' $ of the block is given by the equation: 
     $$
     W' = M(g - a)
     $$
   - According to the problem, the block exerts a force of $ \frac{mg}{4} $ on the floor. Therefore, we set:
     $$
     M(g - a) = \frac{mg}{4}
     $$

2. **Solving for $ a $:**
   - Rearrange the equation:
     $$
     g - a = \frac{g}{4}
     $$
   - Solve for $ a $:
     $$
     a = g - \frac{g}{4} = \frac{4g}{4} - \frac{g}{4} = \frac{3g}{4}
     $$

Thus, the correct acceleration $ a $ with which the box should descend is $ \frac{3g}{4} $, which corresponds to Option 3.

### Explanation for Why Other Options are Incorrect:

1. **Option 1 ($ \frac{g}{4} $):**
   - If the box descends with an acceleration of $ \frac{g}{4} $, the apparent weight would be:
     $$
     M(g - \frac{g}{4}) = M \times \frac{3g}{4}
     $$
   - This is not equal to $ \frac{mg}{4} $, hence this option is incorrect.

2. **Option 2 ($ \frac{g}{2} $):**
   - If the box descends with an acceleration of $ \frac{g}{2} $, the apparent weight would be:
     $$
     M(g - \frac{g}{2}) = M \times \frac{g}{2}
     $$
   - This is not equal to $ \frac{mg}{4} $, hence this option is incorrect.

3. **Option 4 ($ \frac{4g}{5} $):**
   - If the box descends with an acceleration of $ \frac{4g}{5} $, the apparent weight would be:
     $$
     M(g - \frac{4g}{5}) = M \times \frac{g}{5}
     $$
   - This is not equal to $ \frac{mg}{4} $, hence this option is incorrect.

In conclusion, the correct acceleration for the box to descend so that the block exerts a force of $ \frac{mg}{4} $ on the floor is $ \frac{3g}{4} $, which is Option 3.