Suppose for an unexplained reason a star near Earth started shrinking. At what stage is the gravitational force between Earth and the star the greatest?
| Stage | Mass of Earth (kg) | Mass of Star (kg) |
|---|---|---|
| 1 | 5.97 x 10^{24} | 1.8 x 10^{30} |
| 2 | 5.97 x 10^{24} | 2.7 x 10^{27} |
| 3 | 5.97 x 10^{24} | 5.0 x 10^{25} |
| 4 | 5.97 x 10^{24} | 4.5 x 10^{23} |
A. Stage 1
B. Stage 2
C. Stage 3
D. Stage 4
Answer (1)
Calculate the product of the masses of Earth and the star at each stage.
Stage 1 has the largest product: 5.97 × 1 0 24 × 1.8 × 1 0 30 = 1.0746 × 1 0 55 .
Comparing the products, Stage 1 has the greatest value.
The gravitational force is greatest at Stage 1: St a g e 1 .
Explanation
Understanding the Problem We are given the masses of Earth and a star at four different stages. Our goal is to determine at which stage the gravitational force between them is the greatest. The gravitational force is directly proportional to the product of the masses of the two objects. Since the mass of Earth remains constant, we only need to compare the product of Earth's mass and the star's mass at each stage.
Calculating the Products Let's calculate the product of the masses at each stage:
Stage 1: 5.97 × 1 0 24 k g × 1.8 × 1 0 30 k g = 1.0746 × 1 0 55 k g 2
Stage 2: 5.97 × 1 0 24 k g × 2.7 × 1 0 27 k g = 1.6119 × 1 0 52 k g 2
Stage 3: 5.97 × 1 0 24 k g × 5.0 × 1 0 25 k g = 2.985 × 1 0 50 k g 2
Stage 4: 5.97 × 1 0 24 k g × 4.5 × 1 0 23 k g = 2.6865 × 1 0 48 k g 2
Comparing the Results Comparing the products, we have:
Stage 1: 1.0746 × 1 0 55
Stage 2: 1.6119 × 1 0 52
Stage 3: 2.985 × 1 0 50
Stage 4: 2.6865 × 1 0 48
Since 1 0 55 is the largest power of 10 among these, Stage 1 has the greatest product of masses, and therefore the greatest gravitational force.
Final Answer The gravitational force is greatest at Stage 1.
Examples
Understanding gravitational forces is crucial in astrophysics. For example, when studying binary star systems, astronomers analyze the gravitational interactions between the stars to determine their masses and orbital parameters. Similarly, the gravitational force between a planet and its star dictates the planet's orbit and habitability. By applying these principles, scientists can predict the movements of celestial bodies and explore the potential for life on other planets.
