Chapter 3: Problem 38
You find yourself stranded on planet Alpha, which is half as dense as Earth but which has a radius three times that of Earth's. What is your weight on Alpha compared to your weight on Earth? (A) \(2 / 3\) (B) The same (C) \(3 / 2\) (D) 3 (E) 6
Short Answer
Expert verified
Based on the analysis and solution, your weight on planet Alpha would be 1/2 of your weight on Earth. This means you would weigh 2/3 as much on planet Alpha as you do on Earth (Option A).
Step by step solution
01
Find the volume of planet Alpha
We are given that the radius of planet Alpha is three times that of Earth's. Let's denote the radius of Earth as R_earth and the radius of Alpha as R_alpha. Then, R_alpha = 3 * R_earth. The volume of a sphere can be found using the formula V = (4/3) * pi * r^3.
Volume of Earth (V_earth) = (4/3) * pi * (R_earth)^3
Volume of Alpha (V_alpha) = (4/3) * pi * (R_alpha)^3
02
Find the mass of planet Alpha
Next, we need to find the mass of the planet Alpha. We are given that planet Alpha is half as dense as Earth. Density is mass divided by volume, so we can find the mass of planet Alpha by multiplying its density by its volume.
Let's denote the density of Earth as D_earth and the density of Alpha as D_alpha. Then, D_alpha = (1/2) * D_earth.
Mass of Earth (M_earth) = D_earth * V_earth
Mass of Alpha (M_alpha) = D_alpha * V_alpha
03
Find the gravitational forces
Now, we can find the gravitational forces on Earth and Alpha using the gravitational force formula: F = (G * M * m) / r^2, where G is the gravitational constant, M is the mass of the planet, m is the mass of the object (your weight), and r is the distance from the center of the planet.
Force on Earth (F_earth) = (G * M_earth * m) / (R_earth)^2
Force on Alpha (F_alpha) = (G * M_alpha * m) / (R_alpha)^2
04
Find the ratio of the forces
By dividing F_alpha by F_earth, we can find the ratio of your weight on planet Alpha compared to Earth:
(F_alpha / F_earth) = [(G * M_alpha * m) / (R_alpha)^2] / [(G * M_earth * m) / (R_earth)^2]
The mass terms (m) and the gravitational constant (G) will cancel out, leaving us with:
(F_alpha / F_earth) = [(M_alpha * (R_earth)^2) / (M_earth * (R_alpha)^2)]
Substitute the expressions for M_alpha, R_alpha and R_earth:
(F_alpha / F_earth) = [((1/2) * D_earth * (4/3) * pi * (3 * R_earth)^3 * (R_earth)^2) / (D_earth * (4/3) * pi * (R_earth)^3 * (3 * R_earth)^2)]
After simplifying:
(F_alpha / F_earth) = 1/2
So the weight on planet Alpha compared to the weight on Earth is 1/2, which corresponds to the option (A) \(2 / 3\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
SAT Physics Preparation
Preparing for the SAT Physics subject test requires a clear understanding of key concepts in classical physics, including gravity. The concept of gravitational force is fundamental and often tested. In the provided exercise, students must apply their knowledge of gravity to determine how a difference in planetary mass and size affects weight.
Students should start by reviewing concepts such as the gravitational force formula \( F = \frac{G \cdot M \cdot m}{r^2} \), where \( G \) is the gravitational constant, \( M \) is the mass of the planet, \( m \) is the mass of the object, and \( r \) is the distance from the center of the planet. Mastering these concepts not only helps to solve problems on the SAT Physics test but also enhances overall problem-solving skills crucial for the exam.
Students should start by reviewing concepts such as the gravitational force formula \( F = \frac{G \cdot M \cdot m}{r^2} \), where \( G \) is the gravitational constant, \( M \) is the mass of the planet, \( m \) is the mass of the object, and \( r \) is the distance from the center of the planet. Mastering these concepts not only helps to solve problems on the SAT Physics test but also enhances overall problem-solving skills crucial for the exam.
AP Physics Review
For those reviewing AP Physics content, understanding gravitational forces within different planetary conditions is integral. The exercise we're looking at involves variations in density and radius, which are directly related to AP Physics topics like gravitation and circular motion.
Students should be comfortable manipulating equations related to mass, density, and volume, such as \( V = \frac{4}{3} \pi r^3 \) for volume and \( \rho = \frac{M}{V} \) for density. A thorough review of these topics not only aids in problem-solving but also cultivates a deeper conceptual understanding, which is essential for success in AP Physics.
Students should be comfortable manipulating equations related to mass, density, and volume, such as \( V = \frac{4}{3} \pi r^3 \) for volume and \( \rho = \frac{M}{V} \) for density. A thorough review of these topics not only aids in problem-solving but also cultivates a deeper conceptual understanding, which is essential for success in AP Physics.
Planetary Mass and Weight
The notion of mass versus weight is sometimes confusing. Planetary mass refers to the amount of matter within a planet, while weight is the force exerted by gravity on that mass. In our exercise, the key was understanding how the mass of a planet affects the weight of an object.
By acknowledging that mass can be derived from density and volume, and that weight on a planet varies with the square of the radius, we tie in the intricate relationship between these concepts. This knowledge is not only relevant for physics but also for enriching our understanding of how we would fit within different cosmic environments.
By acknowledging that mass can be derived from density and volume, and that weight on a planet varies with the square of the radius, we tie in the intricate relationship between these concepts. This knowledge is not only relevant for physics but also for enriching our understanding of how we would fit within different cosmic environments.
Physics Problem Solving
Physics problem-solving is a skill refined through practice and the application of critical thinking. In our example, it involved a methodical approach that required breaking down the problem into manageable steps. Each step, from finding the volume and mass of planet Alpha to comparing gravitational forces, contributes to a systematic solution.
For successful physics problem-solving, always start by identifying the known quantities and required outcomes. Then, apply the appropriate physical laws and formulas. Finally, analyze your results in the context of the problem, ensuring they make sense both mathematically and physically. It's through this meticulous process that students sharpen their problem-solving abilities crucial for both SAT and AP Physics.
For successful physics problem-solving, always start by identifying the known quantities and required outcomes. Then, apply the appropriate physical laws and formulas. Finally, analyze your results in the context of the problem, ensuring they make sense both mathematically and physically. It's through this meticulous process that students sharpen their problem-solving abilities crucial for both SAT and AP Physics.