Companion to the reading passage The Shoulders of Giants: Newton, Hooke, and the Rivalry That Forged Modern Physics. Answer each question, then click Check. Multiple-choice and number questions grade themselves; for written answers, compare yours with the model answer and checklist, then score yourself honestly.
In the 1670s Robert Hooke studied how springs stretch when they are loaded. A group of students repeated a version of his investigation. They hung a spring vertically from a clamp and attached a hanger to its lower end. They added masses to the hanger one at a time, waited for the hanger to come to rest, and measured how far the spring had stretched from its unloaded length. Their results are shown in Table 1.
Table 1: Stretch of a Spring with Hanging Masses
| Mass Hung on Spring (kg) | Stretch of Spring (m) |
|---|---|
| 0.00 | 0.000 |
| 0.10 | 0.025 |
| 0.20 | 0.050 |
| 0.30 | 0.075 |
| 0.40 | 0.100 |
Data are representative values for practice.
1. Which graph best represents the relationship between the magnitude of the gravitational force on the hanging mass and the stretch of the spring, based on Table 1?
2. Substitute numerical values into a mathematical representation that can be used to determine the magnitude of the gravitational force on the 0.30-kg mass hanging from the spring. Then calculate the force.
In a 1679 letter, Hooke suggested to Isaac Newton that the force keeping a planet in orbit is a pull toward a central body, and that this pull follows an inverse-square pattern with distance. Newton later provided the mathematics. A student used a gravity simulation to test this idea. In the simulation, a 1000-kg space probe was placed at different distances from Earth’s center, measured in Earth radii (RE). The simulation reported the magnitude of the gravitational force of Earth on the probe at each distance, as shown in Table 2.
Table 2: Gravitational Force on a 1000-kg Probe
| Distance from Earth’s Center | Gravitational Force of Earth on Probe (N) |
|---|---|
| 1 RE (Earth’s surface) | 9800 |
| 2 RE | 2450 |
| 3 RE | 1089 |
| 4 RE | 613 |
Simulation values, rounded to the nearest newton.
3. The student makes the following claim.
Place a check mark in either the Support or Refute box. Justify your response using a mathematical relationship.
4. The probe is moved to a distance of 5 RE from Earth’s center. Which row correctly describes the magnitude of the gravitational force of Earth on the probe and the magnitude of the gravitational force of the probe on Earth at this distance?
Newton checked the inverse-square idea by comparing the Moon’s motion with the acceleration of a falling apple at Earth’s surface. The Moon moves in a nearly circular orbit around Earth. Its distance from Earth’s center is about 60 times Earth’s radius. Table 3 lists modern values for the Moon’s orbit.
Diagram 1: Earth is a circle at the center of the page. The Moon is a small circle at the far right side of its circular orbit, and at that instant the Moon is moving straight toward the top of the page.
Table 3: Data for the Moon’s Orbit
| Quantity | Value |
|---|---|
| Radius of the Moon’s orbit (from Earth’s center) | 3.84 × 108 m |
| Orbital speed of the Moon | 1.02 × 103 m/s |
| Radius of Earth | 6.37 × 106 m |
| Acceleration due to gravity at Earth’s surface | 9.8 m/s2 |
Diagram 1
5. Using Diagram 1 (top of the page is up), which arrow starting on the Moon represents the direction of the gravitational force of Earth on the Moon, Fg?
6. Now choose the direction of the Moon’s velocity, v, at the instant shown in Diagram 1 (top of the page is up, right is right), and enter its magnitude from Table 3.
7. Using one or more mathematical relationships and the data in Table 3, support the claim that the Moon’s centripetal acceleration is about 1/3600 (that is, 1/602) of the acceleration due to gravity at Earth’s surface. First calculate the Moon’s centripetal acceleration, then calculate how many times larger g is.