Problem 16 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

(a) The acceleration due to gravity on the moon is approximately 1.817 m/s². (b) The mass of the pack on the moon is 1.784 kg, the same as on Earth.

Step 1: Understanding Weight on Earth

On Earth, the weight of the astronaut's pack is given as 17.5 N. The weight of an object can be calculated using the formula \( W = m imes g \), where \( W \) is the weight, \( m \) is the mass, and \( g \) is the acceleration due to gravity. The standard acceleration due to gravity on Earth \( g \) is approximately \( 9.81 \text{ m/s}^2 \).

Step 2: Calculating Mass on Earth

Using the weight equation, \( W = m imes g \), and substituting \( W = 17.5 \text{ N} \) and \( g = 9.81 \text{ m/s}^2 \), we can solve for the mass \( m \) of the pack on Earth.\[ m = \frac{W}{g} = \frac{17.5}{9.81} \approx 1.784 \text{ kg} \]

Step 3: Understanding Weight on the Moon

On the moon, the weight of the astronaut's pack is 3.24 N. We already solved for the mass \( m = 1.784 \text{ kg} \), which remains unchanged regardless of location. The key difference is the change in gravity. On the moon, weight is calculated as \( W_{\text{moon}} = m \times g_{\text{moon}} \).

Step 4: Calculating Gravity on the Moon

We use the moon's weight equation, \( W_{\text{moon}} = m \times g_{\text{moon}} \) where \( W_{\text{moon}} = 3.24 \text{ N} \) and \( m = 1.784 \text{ kg} \). Substitute these values to find the acceleration due to gravity on the moon.\[ g_{\text{moon}} = \frac{W_{\text{moon}}}{m} = \frac{3.24}{1.784} \approx 1.817 \text{ m/s}^2 \]

Step 5: Verifying Mass on the Moon

The mass of the pack does not change with location, so it remains \( 1.784 \text{ kg} \). Even though the weight is different on the moon, the mass calculated on Earth is valid elsewhere.

Key Concepts

Mass Calculation

To determine the mass of an object, like the astronaut's pack, you'll need to understand how weight relates to mass and gravity. Weight is the force exerted by gravity on an object. It can be expressed by the equation:

• \( W = m \times g \)

Where \( W \) is weight, \( m \) is mass, and \( g \) is the gravitational acceleration. Mass is a measure of the amount of matter in an object and remains constant regardless of its location. In this exercise, the pack's mass was calculated on Earth using the weight of 17.5 N. By rearranging and solving the equation for mass:

• \( m = \frac{W}{g} \)

By inputting Earth's gravitational acceleration of \( 9.81 \, \text{m/s}^2 \), the mass comes out to be approximately \( 1.784 \, \text{kg} \). This mass remains the same whether on Earth or elsewhere in the universe.

Weight on Celestial Bodies

When we talk about weight on celestial bodies, we refer to the force due to gravity acting on an object on planets, moons, or other space locations. Weight varies because gravitational force is different on different celestial bodies. Even though the mass doesn’t change, the weight does, because it depends on the local gravitational acceleration. For the moon, where the pack weighs 3.24 N, the gravitational acceleration \( g_{\text{moon}} \) is different compared to Earth. This is crucial for astronauts and equipment being used in missions beyond Earth, as it helps in designing and selecting the appropriate materials and tools for specific gravity fields.

Gravitational Force

Understanding gravitational force is key to figuring out how weight changes across different celestial bodies. Gravitational force is the attractive pull two masses exert on each other. On Earth's surface, this force is what we experience as weight. This force is calculated by:

• \( F = m \times g \)

Where force \( F \) is equal to the product of mass and the gravitational acceleration at that point. When the astronaut's pack was on the moon, we used similar logic to calculate \( g_{\text{moon}} \) by rearranging the force equation:

• \( g_{\text{moon}} = \frac{W_{\text{moon}}}{m} \)

Here, \( W_{\text{moon}} = 3.24 \, \text{N} \), demonstrating how differences in gravity affect the force experienced by objects in space.