Problem 20 - A History of Mathematics: An Introduction

Question

Suppose in a simplified solar system that all planets revolved uniformly in circles with the sun at the center. If the centripetal force is inversely as the square of the radius,| show that the squares of the periodic times of the planets are as the cubes of the radii. (This is a special case of Kepler's third law.)

Step-by-Step Solution

Answer

Question: Demonstrate that the squares of the periodic times of the planets are proportional to the cubes of the radii in a simplified solar system. Answer: By analyzing the given conditions involving centripetal force, we have derived the relationship T^2 ∝ r^3, which shows that the square of the periodic times is proportional to the cube of the radii, a special case of Kepler's third law.

Step 1: Express centripetal force formula

The centripetal force acting on a planet of mass m moving around in a circle of radius r at a velocity v is given by: F_c = \frac{mv^2}{r}

Step 2: Express the formula for velocity in terms of radius and period

The relationship between velocity, radius, and period (T) is given by: v = \frac{2\pi r}{T}

Step 3: Substitute the velocity formula into the centripetal force formula

Insert the expression for velocity into the centripetal force formula: F_c = m\left(\frac{2\pi r}{T}\right)^2 \frac{1}{r}

Step 4: Simplify the equation

Simplify the equation by expanding the term in parentheses and reducing the fraction: F_c = \frac{4\pi^2 r^2 m}{T^2}

Step 5: Derive a relationship between F_c, r, and T

We are given that the centripetal force is inversely proportional to the square of the radius, meaning F_c * r^2 = constant. Let's call this constant k: F_c * r^2 = k

Step 6: Substitute the centripetal force formula into the relationship

Substitute the centripetal force formula (derived in step 4) into the relationship between F_c, r, and T: \frac{4\pi^2 r^2 m}{T^2} * r^2 = k

Step 7: Rearrange the equation to express the relationship between T^2 and r^3

To express the relationship between the square of the periodic times (T^2) and the cube of the radii (r^3), rearrange the equation: T^2 = \frac{4\pi^2 r^5 m}{k}

Step 8: Express the proportionality

Now, we can relate the square of the periodic times (T^2) to the cube of the radii (r^3) through a constant (let's call it K): T^2 \propto r^3 Specifically, T^2 = K r^3

Step 9: Conclusion

By manipulating the equations involving centripetal force, velocity, radius, and periodic time, we have shown that the squares of the periodic times of the planets are proportional to the cubes of the radii, a special case of Kepler's third law.

Key Concepts

Centripetal Force

Centripetal force plays a crucial role when analyzing the motion of planets around the sun. It is the force that keeps a planet moving in a circular path, preventing it from flying off into space. For a planet of mass \( m \) moving in a circle of radius \( r \) with a velocity \( v \), the force is calculated using the equation:

• \( F_c = \frac{mv^2}{r} \)

However, in this exercise, the centripetal force is inversely proportional to the square of the radius. This means that as the radius increases, the required centripetal force decreases quickly.

• This inverse-square relationship is written as: \( F_c \propto \frac{1}{r^2} \).

This fundamental idea helps us connect this force to other elements of planetary motion, such as the periodic time.

Periodic Time

Periodic time, denoted as \( T \), is the time it takes for a planet to complete one full orbit around the sun. It can be linked to other concepts like velocity and radius. Since the planet moves in a circular path, we can relate its velocity \( v \), to the period using:

• \( v = \frac{2\pi r}{T} \)

Here, \( 2\pi r \) is the circumference of the orbit, and dividing by \( T \) gives the speed at which the planet travels around its path. By substituting this expression for \( v \) into the centripetal force formula, we can examine how changes in radius \( r \) influence \( T \). Eventually, through a series of mathematical manipulations, we find that \( T^2 \), the square of the periodic time, is proportional to \( r^3 \), the cube of the radius, illustrating a clear, predictable relationship in planetary motion.

Planetary Motion

Planetary motion refers to the cyclical paths that planets trace around the sun. In our solar system, the planets orbit in patterns that can be described mathematically. This exercise helps break down some of those descriptions, showing how the laws of physics apply to celestial bodies. Kepler's Third Law of Planetary Motion states that the square of a planet's periodic time \( T^2 \) is proportional to the cube of its orbital radius \( r^3 \). This intrinsic relationship, discovered centuries ago, helps to simplify our understanding of how planets move. Extending beyond simple observations, these equations allow us to predict the relationship between the properties of a specific planet's orbit in a uniform and quantifiable way.

Inversely Proportional Relationships

In physics, understanding inversely proportional relationships can help elucidate how two variables relate to each other. When the centripetal force \( F_c \) and the square of the radius \( r^2 \) are inversely proportional, it means that increasing the radius causes the force to decrease by a similar magnitude. Particularly in this context, if we double the radius \( r \), the force \( F_c \) becomes one quarter of its original value, demonstrating how sensitive the relationship is.

• This is expressed as \( F_c \cdot r^2 = \text{constant} \).

Inversely proportional relationships are fundamental in astronomy and physics, showing how changing one variable affects another, often in predictable patterns that can reveal deeper understanding of complex systems such as planetary orbits.