Problem 17 - College Physics

Question

An astronaut's pack weighs 17.5 \(\mathrm{N}\) when she is on earth but only 3.24 \(\mathrm{N}\) when she is at the surface of an asteroid. (a) What is the acceleration due to gravity on this asteroid? (b) What is the mass of the pack on the asteroid?

Step-by-Step Solution

Answer

The acceleration due to gravity on the asteroid is about 1.817 m/s², and the mass of the pack is 1.784 kg on the asteroid.

Step 1: Understand the Relationship between Weight and Gravity

Weight is the force exerted on an object due to gravity and is calculated as: \( W = m \cdot g \), where \( W \) is the weight, \( m \) is the mass, and \( g \) is the acceleration due to gravity. The weight of the pack on Earth is given as 17.5 N and on the asteroid as 3.24 N.

Step 2: Find the Mass of the Pack on Earth

On Earth, the standard acceleration due to gravity \( g_{earth} \) is approximately 9.81 m/s². We can rearrange the weight formula to find mass: \( m = \frac{W}{g} \). Substituting the given values: \( m = \frac{17.5 \, \mathrm{N}}{9.81 \, \mathrm{m/s^2}} \approx 1.784 \, \mathrm{kg} \).

Step 3: Calculate the Acceleration due to Gravity on the Asteroid

The mass of the pack remains the same on the asteroid. Using the rearranged formula for gravitational acceleration \( g_{asteroid} = \frac{W_{asteroid}}{m} \), where \( W_{asteroid} = 3.24 \, \mathrm{N} \). Substitute the known values: \( g_{asteroid} = \frac{3.24 \, \mathrm{N}}{1.784 \, \mathrm{kg}} \approx 1.817 \, \mathrm{m/s^2} \).

Step 4: Confirm the Mass of the Pack on the Asteroid

The mass of the pack remains constant regardless of location, so the mass continues to be \( m \approx 1.784 \, \mathrm{kg} \) on the asteroid, as calculated from the weight on Earth.

Key Concepts

Weight and Mass

Weight and mass are two distinct but related concepts that are often confused. It's important to clarify the difference:

• Mass is the amount of matter in an object. It is a scalar quantity with units of kilograms (kg). Mass is constant and does not change regardless of where the object is located.
• Weight is the gravitational force acting on an object’s mass. It is a vector quantity and depends on the local gravitational field. Weight is measured in newtons (N) and can be calculated using the formula: \( W = m \cdot g \), where \( W \) is weight, \( m \) is mass, and \( g \) is the acceleration due to gravity at that particular location.

For instance, in the given exercise, the astronaut's pack weighs 17.5 N on Earth due to Earth’s stronger gravitational pull compared to the asteroid, where the pack weighs just 3.24 N. However, the mass of the pack remains the same at approximately 1.784 kg in both places because mass is unaffected by gravity.

Acceleration Due to Gravity

The acceleration due to gravity (

• Earth's Gravity: On Earth, the acceleration due to gravity is approximately 9.81 m/s². This is a measure of how fast the velocity of an object increases as it falls under Earth's gravity.
• Asteroid's Gravity: On smaller celestial bodies like asteroids, gravitational acceleration is much weaker. This is because gravity is dependent on the mass and radius of the celestial body. In the example, the calculated gravitational acceleration on the asteroid was 1.817 m/s², which is significantly lower than that of Earth.

The difference in gravitational acceleration explains why objects weigh less on an asteroid. In calculations, \( g \) can be determined using the equation: \( g = \frac{W}{m} \), where \( W \) is weight and \( m \) is mass, allowing us to solve for the acceleration due to gravity on different planetary bodies.

Newton's Laws

Newton's laws of motion provide a foundation for understanding the relationship between forces and the motion of objects, and they are essential when discussing gravity and weight:

• First Law (Inertia): An object will remain at rest or in uniform motion unless acted upon by a force. This means that without gravity, the astronaut's pack would not "weigh" anything, as there would be no force exerting on it to give it weight.
• Second Law (F=ma): This law states that the force on an object is equal to its mass multiplied by its acceleration (\( F = m \cdot a \)). In the context of gravity, this becomes \( W = m \cdot g \). This law helps us understand why the pack's weight changes depending on the gravitational pull of the body it is on, despite the mass remaining the same.
• Third Law (Action-Reaction): For every action, there is an equal and opposite reaction. As the astronaut's pack presses down on the ground with its weight, the ground pushes back with an equal force. This concept helps explain how scales work to measure weight.

In summary, understanding these laws aids in grasping how the same mass can have different weights under different gravitational conditions.