9. mathematical representations predict the gravitational force between two 15-kg balls whose centers are 35 cm apart. what fraction is this of the weight of one ball?
10. gravity if earth began to shrink, but its mass remained the same, what would happen to the value of g on earth’s surface?
11. the value of g cavendish did his investigation using lead spheres. would his value of g be the same or different if he used copper spheres of equal mass? explain.
12. laws or theories? kepler’s three statements and newton’s equation for gravitational attraction are called laws. were they ever theories? will they ever become theories?
13. critical thinking picking up a rock requires less effort on the moon than on earth. how will the moons gravitational force affect the path of the rock if it is thrown horizontally?

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Accepted Answer

### Question 9
# Explanation:
## Step1: Recall Gravitational Force Formula
The gravitational force between two objects is given by Newton's law of universal gravitation: $ F = G\frac{m_1m_2}{r^2} $, where $ G = 6.67\times 10^{-11}\, \text{N}\cdot\text{m}^2/\text{kg}^2 $, $ m_1 $ and $ m_2 $ are the masses of the two objects, and $ r $ is the distance between their centers. The weight of an object is $ W = mg $, where $ g = 9.8\, \text{m/s}^2 $.

## Step2: Convert Units
The mass of each ball $ m_1 = m_2 = 15\, \text{kg} $. The distance $ r = 35\, \text{cm} = 0.35\, \text{m} $.

## Step3: Calculate Gravitational Force
Substitute the values into the gravitational force formula:
$$
\begin{align*}
F &= (6.67\times 10^{-11})\frac{(15)(15)}{(0.35)^2}\
&= (6.67\times 10^{-11})\frac{225}{0.1225}\
&\approx (6.67\times 10^{-11})(1836.74)\
&\approx 1.225\times 10^{-7}\, \text{N}
\end{align*}
$$

## Step4: Calculate Weight of One Ball
The weight of one ball $ W = mg = 15\times 9.8 = 147\, \text{N} $.

## Step5: Find the Fraction
The fraction is $ \frac{F}{W} = \frac{1.225\times 10^{-7}}{147} \approx 8.33\times 10^{-10} $.

# Answer:
The gravitational force is approximately $ 1.23\times 10^{-7}\, \text{N} $, and the fraction of this force to the weight of one ball is approximately $ 8.33\times 10^{-10} $.

### Question 10
# Brief Explanations:
The acceleration due to gravity on the surface of a planet is given by $ g = \frac{GM}{r^2} $, where $ G $ is the gravitational constant, $ M $ is the mass of the planet, and $ r $ is the radius of the planet. If Earth shrinks (decreases $ r $) but its mass $ M $ remains the same, from the formula, since $ g $ is inversely proportional to $ r^2 $, as $ r $ decreases, $ g $ will increase.

# Answer:
The value of $ g $ on Earth’s surface would increase.

### Question 11
# Brief Explanations:
The gravitational constant $ G $ is a universal constant, meaning it does not depend on the material of the objects used in the experiment. Cavendish’s experiment to determine $ G $ relies on the gravitational force between two masses, which depends on the product of their masses, the distance between them, and $ G $. Since $ G $ is universal, using copper spheres of equal mass (same $ m_1 $, $ m_2 $, and $ r $) would give the same value of $ G $ because $ G $ is independent of the material of the spheres.

# Answer:
His value of $ G $ would be the same. Because $ G $ is a universal constant, independent of the material of the objects (lead or copper) used, as long as the masses and distance are the same.

### Question 12
# Brief Explanations:
- **Laws vs. Theories**: A scientific law describes a phenomenon (what happens), often in mathematical terms, based on repeated observations. A scientific theory explains why a phenomenon happens, integrating multiple laws and observations into a coherent framework.
- **Kepler’s Laws and Newton’s Gravitational Equation**: Kepler’s three laws described planetary motion (what planets do) based on observations. Newton’s gravitational equation described gravitational attraction (what the force of gravity is). Initially, before there was a comprehensive explanation, they were descriptions (laws). A theory explains the underlying mechanism. These are laws because they describe phenomena, and they are unlikely to become theories because theories explain "why," while these are mathematical descriptions of "what." However, Newton’s law of gravitation was later explained (in part) by Einstein’s theory of general relativity, but the law itself (the mathematical relation) remains a law, and the explanation is the theory.

# Answer:
- Kepler’s three statements and Newton’s equation for gravitational attraction were (and are) laws. A law describes a phenomenon (what happens), while a theory explains why. These are mathematical descriptions of observed gravitational/planetary motion phenomena.
- They were never "just" theories (they started as laws from observations). They are unlikely to become theories in the traditional sense because theories explain the underlying mechanism, and these are descriptive laws. (Note: Newton’s law was later related to Einstein’s theory of general relativity, but the law itself remains a law, with the theory explaining the mechanism behind gravity.)

### Question 13
# Brief Explanations:
When a rock is thrown horizontally on the Moon, the Moon’s gravitational force (which is weaker than Earth’s, $ g_{\text{Moon}} \approx \frac{1}{6}g_{\text{Earth}} $) acts vertically downward (toward the Moon’s center), causing the rock to accelerate downward. This downward acceleration will curve the rock’s path into a parabola (similar to projectile motion on Earth, but with a smaller downward acceleration, so the rock will travel farther horizontally and take longer to fall the same vertical distance compared to Earth).

# Answer:
The Moon’s gravitational force will act vertically downward on the rock, causing it to accelerate downward (with $ a = g_{\text{Moon}} \approx \frac{1}{6}g_{\text{Earth}} $). This downward acceleration will curve the rock’s horizontal motion into a parabolic path (projectile motion), but since $ g_{\text{Moon}} $ is smaller than $ g_{\text{Earth}} $, the rock will travel a greater horizontal distance and take longer to fall vertically compared to being thrown on Earth with the same horizontal velocity.

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Explanation:

Question 9

Step1: Recall Gravitational Force Formula

Step2: Convert Units

Step3: Calculate Gravitational Force

Step4: Calculate Weight of One Ball

Step5: Find the Fraction

Answer:

Question 10