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Chapter 4: Problem 19

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

Short Answer

Expert verified
(a) Mass on Earth: 4.49 kg; (b) Mass on Io: 4.49 kg, Weight on Io: 8.13 N.

Step by step solution

01

Determine the mass from weight on Earth

To find the mass of the watermelon on the Earth's surface, use the formula for weight: \( W = m \cdot g \), where \( W \) is the weight, \( m \) is the mass, and \( g \) is the acceleration due to gravity on Earth (\(9.81 \text{ m/s}^2\)). Rearranging the formula gives \( m = \frac{W}{g} \). Substitute \( W = 44.0 \text{ N} \) and \( g = 9.81 \text{ m/s}^2 \) to calculate the mass: \( m = \frac{44.0}{9.81} \approx 4.49 \text{ kg} \).
02

Find the mass on Io

The mass of the watermelon is constant irrespective of location. Thus, the mass on Io's surface is the same as on Earth's surface. Therefore, the mass on Io is \( 4.49 \text{ kg} \).
03

Calculate the weight on Io

The weight of the watermelon on Io can be calculated using the formula \( W = m \cdot g_{Io} \), where \( g_{Io} = 1.81 \text{ m/s}^2 \). Substituting the mass \( m = 4.49 \text{ kg} \) and \( g_{Io} = 1.81 \text{ m/s}^2 \), calculate the weight: \( W = 4.49 \cdot 1.81 \approx 8.13 \text{ N} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Acceleration Due to Gravity
Acceleration due to gravity is a force that pulls objects towards the center of a celestial body, such as Earth or any planet or moon. It is different for each celestial body because it depends on two main factors: the mass of the body and the distance from its center to its surface.
This force, often denoted by the symbol \( g \), explains why objects fall and have weight on different planets. On Earth, the value of \( g \) is approximately \( 9.81 \text{ m/s}^2 \). Meanwhile, on Jupiter's moon Io, \( g \) is just \( 1.81 \text{ m/s}^2 \).
Even though the value of \( g \) changes, the mass of an object remains constant no matter where it is in the universe.
Calculating Weight with Different Gravities
Weight calculation involves multiplying an object's mass by the acceleration due to gravity for the specific location. On Earth, we use the formula \( W = m \cdot g \), where \( W \) is weight, \( m \) is mass, and \( g \) is gravity. For instance, a watermelon weighing 44 N on Earth has a mass of about 4.49 kg calculated by \( m = \frac{44.0}{9.81} \).
To find the weight on a different celestial body, such as Io, you substitute the mass into the formula using Io's gravity: \( W = m \cdot g_{Io} \). Since the gravity on Io is less, the watermelon's weight is also less, only about 8.13 N. This is due to the lower gravitational pull on Io compared to Earth.
Exploring Mass vs. Weight Concept
Mass and weight, though related, are not the same. Mass is a measure of how much matter is in an object, and it remains constant across the universe. It doesn't change whether you are on Earth, the Moon, or even in space. This is why the watermelon's mass is 4.49 kg both on Earth and Io.
On the other hand, weight is a force resulting from gravity acting on an object's mass. It depends on both the mass and the local gravitational pull. Thus, on Io, the watermelon weighs less than it does on Earth, although its mass has not changed.
In everyday terms:
  • Mass: Invariant, measured in kilograms (kg).
  • Weight: Variable, measured in newtons (N), and calculated as \( W = m \cdot g \).
Understanding this distinction is crucial when studying different celestial environments and their effects on objects.

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Most popular questions from this chapter

BIO Human Biomechanics. The fastest served tennis ball, served by "Big Bill" Tilden in \(1931,\) was measured at 73.14 \(\mathrm{m} / \mathrm{s} .\) The mass of a tennis ball is \(57 \mathrm{g},\) and the ball is typically in contact with the tennis racquet for \(30.0 \mathrm{ms},\) with the ball starting from rest. Assuming constant acceleration, (a) what force did Big Bill's racquet exert on the tennis ball if he hit it essentially horizontally? (b) Draw free-body diagrams of the tennis ball during the serve and just after it moved free of the racquet.

A crate with mass 32.5 \(\mathrm{kg}\) initially at rest on a warehouse floor is acted on by a net horizontal force of 140 \(\mathrm{N}\) . (a) What acceleration is produced? (b) How far does the crate travel in 10.0 \(\mathrm{s} ?\) (c) What is its speed at the end of 10.0 \(\mathrm{s} ?\)

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?

Two crates, \(A\) and \(B,\) sit at rest side by side on a friction- less horizontal surface. The crates have masses \(m_{A}\) and \(m_{B}\) . A horizontal force \(\vec{\boldsymbol{F}}\) is applied to crate \(A\) and the two crates move off to the right. (a) Draw clearly labeled free-body diagrams for crate A and for crate \(B\) . Indicate which pairs, if any, are third- law action- reaction pairs. (b) If the magnitude of force \(\vec{F}\) is less than the total weight of the two crates, will it cause the crates to move? Explain.

CP A 4.80 -kg bucket of water is accelerated upward by a cord of negligible mass whose breaking strength is 75.0 \(\mathrm{N}\) . If the bucket starts from rest, what is the minimum time required to raise the bucket a vertical distance of 12.0 \(\mathrm{m}\) without breaking the cord?
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