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) \(\frac{3g}{4}\)

Key concepts

Normal Force

Normal force is a crucial concept in physics, especially when studying motion and how objects interact with surfaces. Think of normal force as the support force. It acts perpendicular to the surface that an object is in contact with.
In the case of our exercise, where a block is descending in a box, the normal force is the push back exerted by the box's floor on the block. This is often why objects remain at rest or move with frictionless surfaces.

Key points about normal force:

• Normal force acts perpendicular to surfaces of contact.
• It adjusts based on the situation: it can vary depending on whether the surface is inclined, or if there is additional vertical motion.
• For the block in the exercise, the normal force is specified as a fraction of the gravitational force: \( N = \frac{Mg}{4} \).

Understanding how the normal force works helps us figure out the net forces acting on objects, which is necessary for calculating acceleration as we see in the exercise.

Gravitational Force

Gravitational force is simply the attraction force between two masses. Here, on Earth, it gives us the weight of an object. That's why every object with mass is pulled towards the Earth’s center. The gravitational force acting on an object can be calculated as the product of its mass (M) and the gravitational acceleration (g), which is approximately 9.8 m/s² on Earth's surface.
For the block in our problem, the gravitational force is portrayed as:\[F_g = Mg\]
This calculation allows us to understand that there is always a persistent force acting downward on objects due to gravity.

In summary:

• Gravitational force causes objects to accelerate towards the surface of the Earth with an acceleration of \( g \).
• This forms the base force that must be counteracted or supplemented by other forces, like the normal force, depending on the motion.

This knowledge is integral when applying Newton's laws, as it affects the net force used for calculating acceleration.

Vertical Motion

Vertical motion refers to any movement in which the predominant forces are acting perpendicular to the horizontal surface, either upward or downward. In our exercise, the block is accelerating downward, making it crucial to focus on the vertical forces.
The vertical forces involved in this motion are:

• The gravitational force pulling the block downwards.
• The normal force pushing upward against the block.

Newton's Second Law of Motion helps express the state of vertical motion:\[\sum F_y = Ma_y\]This indicates that the net force acting on the block in the vertical direction equals the mass times the vertical acceleration of the block. In the exercise, the calculation of these forces leads us to find the acceleration at which the box should descend to alter the normal force to a specific value. Understanding how forces affect vertical motion lays the groundwork for solving many physics problems involving trajectory and force dynamics.

Acceleration Calculations

Calculating acceleration involves understanding the total forces acting on an object and using Newton's Second Law of Motion. In our exercise, we want to find the acceleration (\( a_y \)) such that a block exerts a certain force on the box's floor.
Here's how:

• First, identify the forces acting on the object, in this case, gravitational and normal forces.
• Second, apply Newton's Second Law, which states \( \sum F = Ma \).
• Third, substitute known values from the problem: the normal force (\( N = Mg/4 \)) and gravitational force (\( F_g = Mg \)).
• Lastly, solve for the desired acceleration by substituting these values into the equation and simplifying. \[ a_y = \frac{3g}{4} \]

Acceleration calculations are simplified by keeping track of the forces involved and recognizing that the net force results in a change of motion. It’s crucial to comprehend both the forces in action and Newton's Laws to manipulate the equations and find the necessary acceleration for any motion.