Final answer:
The acceleration 'a' of the descending block that exerts one-fourth of its weight on the floor is three-fourths of the acceleration due to gravity (a = (3/4)g).
Step-by-step explanation:
If a block of mass M placed inside a box descends vertically with acceleration 'a' and exerts a force equal to one-fourth of its weight on the floor of the box, we can find the acceleration 'a' by applying Newton's second law of motion, which relates the net force exerting on a body to its mass multiplied by its acceleration (F = ma).
The weight (W) of the block is equal to mg, where 'g' is the acceleration due to gravity. The net force (Fnet) on the block is the difference between the weight and the normal force exerted by the floor of the box. Since the normal force is one-fourth of the block's weight, Fnet = W - (1/4)W = (3/4)W. Substituting W with mg, we get Fnet = (3/4)mg.
Applying Newton's second law, we set Fnet = ma, where 'm' is the mass of the block. Hence, (3/4)mg = ma. By dividing both sides by m, we get a = (3/4)g. Therefore, the acceleration 'a' of the block is three-fourths of the acceleration due to gravity.