Question

The gravitational acceleration on the Moon is a sixth of that on Earth. The weight of an apple is \(1.00 \mathrm{~N}\) on Earth. a) What is the weight of the apple on the Moon? b) What is the mass of the apple?

Short answer

Answer: The weight of the apple on the Moon is approximately 0.167 N, and its mass is approximately 0.102 kg.

Step-by-step solution

1. Finding the Gravitational Acceleration on the Moon

We are given that gravitational acceleration on the Moon is a sixth of that on Earth, so we can write that as: \(g_{moon} = \frac{1}{6}g_{earth}\) The gravitational acceleration on Earth is approximately \(9.81 \mathrm{~m/s^2}\). Therefore, the gravitational acceleration on the Moon is: \(g_{moon} = \frac{1}{6}(9.81 \mathrm{~m/s^2}) = 1.635 \mathrm{~m/s^2}\)

2. Finding the Weight of the Apple on the Moon

To find the weight of the apple on the Moon, we will use the given weight on Earth, which is \(W_{earth} = 1.00 \mathrm{~N}\), and the relation between the gravitational accelerations on Earth and the Moon: \(W_{moon} = \frac{g_{moon}}{g_{earth}} * W_{earth}\) Using the values for \(g_{moon}\), \(g_{earth}\), and \(W_{earth}\): \(W_{moon} = \frac{1.635 \mathrm{~m/s^2}}{9.81 \mathrm{~m/s^2}} * 1.00 \mathrm{~N} = 0.167 \mathrm{~N}\) So the weight of the apple on the Moon is approximately \(0.167 \mathrm{~N}\).

3. Finding the Mass of the Apple

To find the mass of the apple, we can use the definition of weight: \(W = mg\). We are given the weight of the apple on Earth and the gravitational acceleration on Earth, so we can write: \(W_{earth} = m_{apple} * g_{earth}\) Solving for \(m_{apple}\): \(m_{apple} = \frac{W_{earth}}{g_{earth}}\) Now, we will substitute the values for \(W_{earth}\) and \(g_{earth}\): \(m_{apple} = \frac{1.00 \mathrm{~N}}{9.81 \mathrm{~m/s^2}} \approx 0.102 \mathrm{~kg}\) Thus, the mass of the apple is approximately \(0.102 \mathrm{~kg}\). In summary, the weight of the apple on the Moon is \(0.167 \mathrm{N}\), and its mass is \(0.102 \mathrm{kg}\).

Key concepts

Weight Calculation

Weight calculation is fundamental to understanding the effect of gravity on an object. In physics, the weight of an object is the force exerted on the object due to gravity. It is calculated by the formula:
\( W = mg \),
where \(W\) is the weight, \(m\) is the mass of the object, and \(g\) is the gravitational acceleration.
For instance, if you have an object with a mass of 1 kilogram on Earth, it would weigh approximately \(9.81 \text{ Newtons} (N)\), since Earth's standard gravity is \(9.81 \text{ m/s}^2\). However, this value for \(g\) changes depending on where in the universe the object is because different celestial bodies exert different gravitational forces. An understanding of how to calculate weight is critical for tasks ranging from designing architecture on Earth to planning missions to other planets.

Moon Gravity

Let's talk about moon gravity, which is distinct from Earth's gravity. The gravitational acceleration on the Moon is only about \(\frac{1}{6}\) of that on Earth, resulting in a much weaker gravitational pull. This difference is due to the Moon's smaller mass and radius compared to Earth.
As the exercise illustrated, if the gravitational acceleration on Earth is \(9.81 \text{ m/s}^2\), then on the Moon it would be:
\(g_{moon} = \frac{1}{6}g_{earth} \),
Leading to \(g_{moon} = 1.635 \text{ m/s}^2\). This means that objects on the Moon weigh less than they do on Earth. This variance in gravitational acceleration affects everything from the escape velocity needed for a spacecraft to leave the lunar surface, to the way astronauts move when they're on the Moon. Understanding moon gravity is crucial when planning lunar missions or studying lunar geology and helps us comprehend how gravity varies across celestial bodies.

Mass of an Object

The mass of an object is a measure of how much matter it contains and is commonly measured in kilograms (kg). Unlike weight, mass is not affected by gravity or location, hence it remains constant regardless of where the object is in the universe.
Following our earlier example, the mass of the apple from the exercise was determined using its weight on Earth and the gravitational acceleration on Earth, resulting in a calculated mass of approximately \(0.102 \text{ kg}\). This mass reflects the amount of matter in the apple, which would be the same whether the apple is on Earth, the Moon, or floating in space. One way to comprehend mass is to think of it as the 'stuff' that makes up an object. Mass is a key concept in physics and plays a central role in mechanics, thermodynamics, and many other areas of science.