Problem 20 - College Physics

Question

At the surface of Jupiter's moon Io, the acceleration due to gravity is 1.81 \(\mathrm{m} / \mathrm{s}^{2} .\) If a piece of ice weighs 44.0 \(\mathrm{N}\) at the surface of the earth, (a) what is its mass on the earth's surface? (b) What are its mass and weight on the surface of Io?

Step-by-Step Solution

Answer

Mass is 4.49 kg; weight is 8.12 N on Io.

Step 1: Find the Mass of the Ice on Earth's Surface

The weight of an object is calculated by the formula: \( W = mg \), where \( W \) is the weight, \( m \) is the mass, and \( g \) is the acceleration due to gravity. On Earth, \( g = 9.81 \ \mathrm{m/s^2} \).Given that the weight of the ice is 44.0 N, we can rearrange the formula to find the mass: \( m = \frac{W}{g} \).Substitute the known values: \( m = \frac{44.0 \ \mathrm{N}}{9.81 \ \mathrm{m/s^2}} \approx 4.49 \ \mathrm{kg} \).

Step 2: Determine the Mass on Io's Surface

Mass is an intrinsic property and does not change based on location. Therefore, the mass of the ice on Io is the same as on Earth, which is approximately 4.49 kg.

Step 3: Calculate the Weight on Io's Surface

The weight of the ice on Io can be calculated using the same weight formula: \( W = mg \) but this time using Io's gravitational acceleration, which is \( g = 1.81 \ \mathrm{m/s^2} \).Using the mass from Step 2, substitute into the formula: \( W = 4.49 \ \mathrm{kg} \times 1.81 \ \mathrm{m/s^2} \approx 8.12 \ \mathrm{N} \).

Key Concepts

Acceleration Due to Gravity

The acceleration due to gravity is a force that pulls objects toward the center of a massive body like a planet or moon. It is a key concept to understand how objects fall and behave in different gravitational fields. On Earth, the standard acceleration due to gravity is approximately 9.81 \(\mathrm{m/s^2}\). This value tells us that any object will increase its speed by 9.81 meters per second every second it falls when no other forces, like air resistance, affect it. Each celestial body has its own gravity based on its mass and size. For instance, on Io, one of Jupiter's moons, the acceleration due to gravity is much less, at 1.81 \(\mathrm{m/s^2}\). This lower gravitational pull on Io means that objects will fall more slowly compared to on Earth. Understanding these differences in gravitational pull is essential when studying the physics of different planets and moons.

Mass and Weight Calculation

Mass and weight are two related but distinct concepts. Mass is a measure of the amount of matter in an object and is measured in kilograms. It is intrinsic and remains constant regardless of its location in the universe. On the other hand, weight is the force exerted by gravity on that mass. Weight varies depending on where the object is in the universe due to differing gravitational accelerations. The weight of an object can be calculated using the formula:

• \( W = mg \)

Where \( W \) is the weight, \( m \) is the mass, and \( g \) is the acceleration due to gravity. For example, if a piece of ice weighs 44.0 \(\mathrm{N}\) on Earth, its mass can be calculated as approximately 4.49 \(\mathrm{kg}\) using Earth's gravity. Since mass doesn’t change, it remains 4.49 \(\mathrm{kg}\) on Jupiter's moon Io as well. However, its weight on Io will be only 8.12 \(\mathrm{N}\), demonstrating how weight depends on gravity.

Gravitational Force on Different Planets

Gravitational force varies across different planets and moons because of their unique mass and radii. These differences result in diverse values of acceleration due to gravity. In our solar system, planets like Jupiter have a much stronger gravitational pull compared to Earth, whereas smaller celestial bodies like Io have considerably weaker gravity.The gravitational force exerted on an object can cause significant variations in how heavy it seems. A 10 kg object on Earth will weigh around 98.1 \(\mathrm{N}\), while only weighing around 18.1 \(\mathrm{N}\) on Io. This is crucial for understanding not just theoretical physics but also practical tasks such as engineering and planning space missions, where the equipment must function under different gravity conditions.In summary, by studying the gravitational force of planets and moons, we can predict how objects interact and are pulled within different celestial environments. Each body in space not only captivates with its beauty but also presents a unique physics paradigm to explore.