Question

A 5-ounce magnetic marble is pushed upwards by a powerful fan so that the marble hovers, neither rising nor falling. What is the force of the wind on the magnetic marble?

Short answer

The force of the wind on the marble is approximately 1.389 N.

Step-by-step solution

Understanding the Problem

To solve this exercise, we need to identify the forces acting on the marble which is hovering in mid-air. The problem states the marble is neither rising nor falling, which implies the net force on it is zero.

Identify Relevant Forces

Two primary forces act on the marble: the gravitational force that pulls it downward and the force of the wind (from the fan) that pushes it upward. These forces must be equal in magnitude for the marble to hover.

Calculate Gravitational Force

The gravitational force can be calculated using the formula: \[ F_{ ext{gravity}} = m imes g \] where \( m \) is the mass in kilograms and \( g \) is the acceleration due to gravity (approximately \( 9.8 \text{ m/s}^2 \) on Earth). First, convert the mass from ounces to kilograms using the conversion factor \( 1 \text{ ounce} = 0.0283495 \text{ kg} \).

Convert Mass to Kilograms

The mass of the marble is 5 ounces. Converting, we have: \[ m = 5 imes 0.0283495 = 0.1417475 \text{ kg} \]

Calculate Gravitational Force with Converted Mass

Now that we have the mass in kilograms, substitute into the gravitational force formula:\[ F_{ ext{gravity}} = 0.1417475 \times 9.8 \approx 1.389 \text{ N} \]

Determine Force of the Wind

Since the marble is hovering in equilibrium, the force of the wind must be equal in magnitude and opposite in direction to the gravitational force. Thus, the force of the wind is also approximately 1.389 N.

Key concepts

Gravitational Force

Gravitational force is a fundamental force of nature. It is the force by which the Earth attracts any object towards its center. For any object on the surface of the Earth, this force can be calculated using the formula \( F_{\text{gravity}} = m \times g \), where:

• \( m \) is the mass of the object in kilograms.
• \( g \) is the acceleration due to gravity, approximately \( 9.8 \text{ m/s}^2 \) on Earth.

In our example, the gravitational force acting on a 5-ounce marble was calculated after converting its mass into kilograms. Mass is essential since it describes how much matter is in the marble, while gravity gives the rate at which this mass is accelerated towards Earth.When an object is stationary or moving at a constant velocity in a gravitational field, the net forces acting on it result in no acceleration, leading to a state of equilibrium. This means forces like wind must counteract gravity for a hovering marble.

Force of Wind

The force of wind in this scenario refers to the opposite force that balances the gravitational pull on the marble. When the fan blows air upwards, it exerts a push known as wind force onto the marble. Wind force competes with gravity. To keep an object like the marble hovering, the wind force must be equal to the gravitational force. However, it acts in the opposite direction. Imagine that the air from the fan acts like a gentle hand holding the marble aloft. The strength of this force is calculated to match the weight of the marble. In this problem, we found the force to be approximately 1.389 N, which is precisely what cancels out the gravitational pull caused by the marble's weight. In physics, balancing forces like this is crucial to achieving net force equilibrium, allowing objects to remain motionless or move predictably.

Net Force

Net force is the vector sum of all forces acting on an object. When multiple forces work on an object, they can either reinforce or cancel each other, determining the object's movement or lack thereof. In our problem, the marble is hovering. This indicates that the forces acting on it are perfectly balanced. Specifically, the gravitational force (pulling it down) and the wind force (pushing it up) are equal in magnitude. When calculating net force, you simply add vector forces together. Since these forces act in opposite directions, the net result is:

• If the forces are equal, the net force is zero, indicating dynamic equilibrium.
• If not equal, the object would accelerate towards the stronger force.

Thus, for an object in equilibrium like our hovering marble, the net force must equal zero, ensuring no change in its motion state. This balance of forces keeps objects stationary or moving at a constant speed.