You find yourself stranded on planet Alpha, where the gravitational field is \(5 \mathrm{~N} / \mathrm{kg}\). Which of the following statements are true? The \(10 \mathrm{~kg}\) mass you hold in your hand i. will weigh about half as much on Alpha as on Earth. ii. will weigh about twice as much on Alpha as on Earth. iii. will have the same mass on Alpha as on Earth. iv. will fall with a greater acceleration than on Earth. v. will require a lesser force to push it across a horizontal surface with a small amount of friction. (A) i and iv (B) i, iii and \(v\) (C) i, iii, iv and \(v\) (D) ii, and iii (E) ii, iii and iv

Understanding the distinction between mass and weight is fundamental in physics and crucial for solving gravitational field problems. Mass refers to the quantity of matter contained in an object and remains constant regardless of the object's location in the universe. In contrast, weight is the force exerted on an object's mass by the gravitational field of a celestial body, such as a planet or moon, and can vary depending on where the object is located.

When addressing problems involving mass and weight, it's important to recognize that weight is directly proportional to both the mass of the object and the strength of the gravitational field it is in. This relationship is expressed by the equation: \( W = mg \), where \( W \) is weight, \( m \) represents mass, and \( g \) is the gravitational acceleration. Hence, if you move from Earth to a planet with a weaker gravitational field, your mass remains unchanged, but your weight decreases since \( g \) is smaller.

Gravitational force is an invisible pull that attracts two objects with mass. The more massive an object, the stronger its gravitational pull. This fundamental force not only keeps the planets in orbit around the sun but also determines the weight of objects on the surface of these celestial bodies.

The force of gravity acting on an object at the surface of a planet is what we often refer to as the object's weight. As seen in the step-by-step solution, we calculate this force using the same formula presented earlier, \( F = mg \). Thus, an object's weight will change if either its mass changes (which it does not under regular conditions) or if the gravitational acceleration \( g \) changes, such as when moving to a planet with a different \( g \) value than Earth's.

Acceleration due to gravity, denoted as \( g \), is the rate at which objects accelerate when they fall freely in a gravitational field, and it varies depending on the celestial body. Earth's average gravitational acceleration is about \( 9.81 m/s^2 \), a value that we often round to \( 9.8 m/s^2 \) for simplicity in many calculations. However, on different planets or moons, \( g \) can be significantly more or less due to the variation in their sizes and masses.

As illustrated in the solution steps, any change in \( g \) affects the weight of an object but not its mass. For instance, on planet Alpha, with a given \( g = 5 N/kg \), objects experience less gravitational pull than on Earth, thus accelerating slower when dropped. Still, their inertia—that is, resistance to change in motion—remains consistent because that's a property of their mass, not weight.