Question

With what acceleration (a) should a box descend so that a block of mass \(\mathrm{M}\) placed in it exerts a force \((\mathrm{Mg} / 4)\) on the floor of the box? (A) \((4 \mathrm{~g} / 3)\) (B) \((3 \mathrm{~g} / 4)\) (C) \(\mathrm{g} / 4\) (D) \(3 \mathrm{~g}\)

Short answer

The correct acceleration (a) for the box to descend such that the block of mass M inside the box exerts a force of \(\frac{Mg}{4}\) on the floor is (B) \(\frac{3g}{4}\).

Step-by-step solution

Identify the forces acting on the block

Inside the descending box, there are two forces acting on the block of mass M: the gravitational force acting downward (Mg) and the normal force (N) acting upward from the floor of the box. The normal force is responsible for the force exerted on the floor of the box.

Write down Newton's second law for the vertical forces

Since the box is descending, we write Newton's second law of motion to describe the vertical forces acting on the block: \[ \sum F_y = Ma_y \] Where \(\sum F_y\) is the total force in the vertical direction and \(a_y\) is the acceleration in the vertical direction. The forces acting are the normal force (N) and the gravitational force (Mg).

Calculate the net force in the vertical direction

Calculate the net force in the vertical direction by the difference between the gravitational force and the normal force: \[ \sum F_y = Mg - N \]

Use the given force exerted on the floor

Since the force exerted on the floor by the block is given as \(Mg / 4\), we know that the normal force, N, is equal to \(Mg / 4\).

Substitute the normal force in the Newton's second law equation

Now, substitute the normal force in the equation from step 3: \[ \sum F_y = Mg - \left( \frac{Mg}{4} \right) = \frac{3Mg}{4} \]

Substitute the net force in the Newton's second law equation

Substitute the net force in the equation from step 2: \[ Ma_y = \frac{3Mg}{4} \]

Solve the equation for the acceleration in the vertical direction

Divide both sides of the equation by M to isolate the acceleration in the vertical direction: \[ a_y = \frac{3g}{4} \] Based on our calculations, the correct option is: (B) \(\frac{3g}{4}\)