The gravitational force between two objects, [math: M1] and [math: M2] , a distance of [math: R] apart is found to be [math: F]. Both masses were multiplied by a factor of 15, and the distance between them by a factor of 3. One of the masses was then doubled. What would be the ratio of the new force compared to the initial force [math: FnewFinitial]?

Question

BlackTom AI · Accepted Answer

Newton’s law of universal gravitation: F = G M1 M2 / R^2. Use this to compare initial and new forces.

1) Relevant concepts:
- Gravitational force is proportional to the product of the two masses and inversely proportional to the square of the separation: F ∝ (M1·M2) / R^2.
- Multiplicative changes in masses and distance scale the force by the product of the mass factors divided by the square of the distance factor.

2) Step-by-step reasoning and intermediate calculations:
- Initial force: F_initial = G·M1·M2 / R^2.
- After multiplying both masses by 15:
  M1' = 15·M1, M2' = 15·M2 → product factor = 15·15 = 225.
- After increasing distance by factor 3:
  R' = 3·R → denominator factor = (3·R)^2 = 9·R^2.
- One mass is then doubled. Doubling either mass multiplies the product by 2 (choice of which mass doesn’t matter because product is symmetric). So overall mass factor becomes 225·2 = 450.
- Combine effects: F_new = G·(450·M1·M2) / (9·R^2).

3) Complete arithmetic:
- Mass factor: 15 × 15 × 2 = 450.
- Distance-squared factor: 3^2 = 9.
- Ratio F_new / F_initial = (450 / 9) = 50.00.

4) Why the correct answer is correct:
- Because gravitational force scales as product of masses divided by square of distance, the combined multiplicative change is exactly (15·15·2)/(3^2) = 450/9 = 50. So the new force is 50 times the original.

5) Why the other options are incorrect:
- 25.00 would be obtained if the final doubling were omitted (225/9 = 25), so it misses the factor of 2.
- 3.33 (≈10/3) and 0.30 (≈3/10) do not follow the correct mass-product divided by distance-squared scaling and result from incorrect combinations or inversions of the factors.

Conclusion: The new force is 50 times the initial force, matching option a. 50.00.

类似问题

A person of mass 53.8 kg weighs 807 N on Planet A. The person's mass is now doubled. If Planet B's gravitational acceleration is 2 times the gravitational acceleration of Planet A, what is the weight of the same person on Planet B's surface correct to the nearest whole number?

What event will produce the greatest increase in the gravitational force between the two masses?

A rocket moves away from the surface of Earth. As the distance r from the center of the planet increases, what happens to the force of gravity on the rocket?

A spacecraft starts on Earth is moving to Mars. Which of the following is correct about the gravitational force on the spacecraft due to Earth’s attraction?

What is the force of gravity acting on an astronaut of mass 74.3 kg on a planet of mass 6.9 × 1024 kg and radius 6000 km? Give your answer to the nearest whole number. G is 6.67 × 10-11 in SI units.

Suppose the gravitational force between two spheres is 30 N. If the magnitude of each mass doubles, what is the force between the masses?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单