Question

The weight of an object on an orbiting space vehicle is measured to be \(42 \mathrm{~N}\) based on an artificial gravitational acceleration of \(6 \mathrm{~m} / \mathrm{s}^{2}\). What is the weight of the object, in \(\mathrm{N}\), on earth, where \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\) ?

Short answer

Approximately 68.67 N.

Step-by-step solution

- Understanding the Given Values

Identify the given values from the problem.

- Calculate the Mass of the Object

Use the formula for weight, which is given by: \[ W = m \times g \] Rearrange to solve for mass: \[ m = \frac{W}{g} \] Substitute the given values of weight (42 N) and artificial gravitational acceleration (6 \mathrm{~m} / \mathrm{s}^{2}): \[ m = \frac{42}{6} = 7 \mathrm{~kg} \]

- Calculate the Weight on Earth

Use the calculated mass and Earth's gravitational acceleration to find the weight on Earth. \[ W_{earth} = m \times g_{earth} \] Substitute the mass (7 \mathrm{~kg}) and Earth's gravity (9.81 \mathrm{~m} / \mathrm{s}^{2}): \[ W_{earth} = 7 \times 9.81 \approx 68.67 \mathrm{~N} \]

Key concepts

artificial gravitational acceleration

Artificial gravitational acceleration is a concept used in space to create a force similar to gravity, allowing astronauts to stay healthy under long periods in space. This is achieved by rotating the spacecraft or a part of it to simulate gravity. The artificial gravitational accelation in this problem is given as 6 m/s². This force can be adjusted based on the needs of the mission and the health of the astronauts. It's important to understand that artificial gravity is not real gravity; it's a result of rotational forces acting on the mass of objects. It helps in keeping muscles and bones strong, preventing them from becoming weaker over time.

mass calculation

Mass is a fundamental property of an object that does not change, regardless of location. To determine mass when a gravitational force (weight) and gravitational acceleration are known, you can use the equation: \[ W = m \times g \] Rearranging gives: \[ m = \frac{W}{g} \] In this exercise, the weight measured under artificial gravity is 42 N, and the artificial gravitational acceleration is 6 m/s². Using the formula: \[ m = \frac{42}{6} = 7 \text{ kg} \] The calculated mass of the object is 7 kg. Once we know the mass, we can find the weight under different gravitational forces by using the same mass with the new gravitational acceleration.

gravitational force

Gravitational force, often referred to as weight, is the force with which an object is pulled towards the center of a planet, such as Earth, due to gravity. On Earth, gravity accelerates objects downward at 9.81 m/s². To calculate the weight of an object on Earth, you use the object's mass and Earth's gravity: \[ W_{earth} = m \times g_{earth} \] Given the mass calculated earlier as 7 kg, we substitute this value into the equation: \[ W_{earth} = 7 \times 9.81 \approx 68.67 \text{ N} \] The weight of the object on Earth is approximately 68.67 N, demonstrating how the gravitational force changes based on the gravitational acceleration at different locations.