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Chapter 41: Newton’s Law of Universal Gravitation

- All objects falling near the surface of Earth have the same speed.
- The period of a pendulum on the mass of the bob is easy to observe.
- There is no "theory" of gravity.

- The magnitude of the force of gravity due to Earth that is acting on the object is represented by the weight of the object.
- We know that gravity causes a projectile to assume a path.
- This might be the reason why the Moon is in the sky.

- The centripetal acceleration of the Moon was determined by observations from astronomy.
- He was aware of the relationship between centripetal acceleration and period.

- We know that r is 3.8 x 10 8 m, and that T is 2.3 x 10 6 s, but we don't know the magnitude of ac.
- The centripetal force on the Moon was acting as universal gravity.

- The force that Earth exerts on the Moon should be the same as the force that the Moon exerts on Earth.
- The relationship implied that the force should be dependent on the mass of Earth, and the interaction implied that the force of gravity should be proportional to the product of the two planets.

- The inverse square law is referred to as the law of universal gravitation.
- The force between them is one-fourth if the distance is doubled.

- In this case, the force of gravity is directed toward the center of mass for the system.
- The Moon and Earth are not the same since Earth is larger.

- Henry Cavendish determined the value of G in 1795.

- The mass of the falling object does not affect the acceleration due to gravity.
- This is consistent with Galileo's observations of falling bodies.

- The value of the known acceleration is accounted for by a theory.
- The value of g on the Moon is about one-sixth of the value on Earth.
- The objects on the Moon weigh less than those on Earth.
- The formula allows us to solve for gravity at any distance above Earth's surface.

- The attraction between a car and a truck is determined by the force of attraction.

- Only h values that are small when compared to rE are good for this formula.

- The square of the distance from the center of Earth affects the value of g.

- The function of distance is shown in a graph of universal gravitational potential energy.

- As an object gets higher and higher above Earth's surface, the potential energy increases.
- The negative values of energy show that the object is bound to Earth and can't be free if it doesn't have enough mechanical energy.

- The "escape velocity" from the force of Earth's gravity can be determined by considering the situation where an object has just reached zero final velocity and is moving away from Earth.

- The surface of Earth can be left with any speed.
- If the spaceship coasted, it wouldn't fall back to Earth because of gravity.

- We can determine the orbital velocity by assuming that we have a circular trajectory.

- The mass of the object doesn't affect the escape velocity or the orbital velocity.

- We can substitute in 2p r / T if we assume the circle is circular.
- We can derive one of the original Laws of Planetary Motion by setting these two expressions equal to each other.
- The third law of planetary motion states that all objects in the same central mass have the same ratio of T 2 /R 3.
- Although he could not explain why the relationship was the same for all of them, he did know what the value was.

- We have M E, rE, h, and g, as well as the known value for G given earlier in the chapter.

- There is a loss of weight at that height.

- Three spheres of mass are placed at the corners of a right triangle.
- The positions relative to the system are shown.

- We have to determine the force of attraction between the smaller mass and the larger mass.
- We have to do a sum of these forces.

- The direction is positive because it is to the right.

- The magnitude of the force is given by the Pythagorean theorem since F 1-4 is horizontal and F 2-4 is vertical.

- Thomas is looking at motion.
- He is trying to figure out what new speed a satellite would have if its altitude was doubled and its mass doubled.

- Replacing M with 2 M and r with 2 r would leave him with the same expression for speed.
- All satellites have the same speed.

- Justify your answer.

- Thomas made two mistakes in his analysis.
- The M in the equation is for the mass of the central object.
- The formula is no longer relevant due to the cancellation of the object's mass.
- The conclusion that the mass of the satellite does not affect speed is correct.
- The r in the equation is from the center of the planet to the center of the satellite.
- A doubling of altitude would be less than a doubling of the original circle.
- Satellites at higher altitudes are at lower speeds.
- They do this as a function of altitude.

- The magnitude of the force between the two masses is proportional to the square of the distance between them, according to the law of gravitation.

The inverse square law is sometimes referred to as the "Newton's law of gravitation."

- Keep in mind that gravity is a force.
- If an object is above the surface of Earth, you need to add the height to the radius.
- You may be able to deduce an answer from logic if you remember that the force of gravity is an inverse-square-law relationship.

- A planet has half the mass of Earth.

- A black hole has an escape velocity that is greater than or equal to the light's speed.

- A synchronized satellite is a satellite that circles the Earth for 24 hours.
- The Earth has a mass of around 10 to 24 kilograms.

- Alternative units are newtons per kilogram.

- The value will increase by four times if we decrease the radius by half.
- An increase of two times is achieved by combining both effects.

- The planet is a sphere of mass, radius, and density, where V is the volume.
- V is the volume of the planet.

- We need to add 300 km to the Earth's radius.
- The given values are 6.7 x 10 6 m and 7.7 x 10 3 m/s.

- v esc is 2,324 or 2.32 x 10 3 m/s.

- To find r, we need to square both sides.
- The yield is 2 GM / v 2.
- The theoretical size for the black hole is 3,000 or 3 x 10 3 m.

- The force is decreased when the distance is tripled.
- 4 N is one-ninth of 36 N.

- The top mass causes the field strength at P to be g 1 and the bottom mass causes the field strength at P to be g 2.
- Both of the field strengths are accelerations.
- The line connecting the mass is connected by a distance of r and b.
- The distance from point P to each mass is given by the Pythagorean theorem.

- Each mass is directed from point P by the direction of the acceleration g. The angles formed by the vectors to the x - axis are the same since each mass is identical.
- Each angle has a magnitude of b / r.

- The x component is directed to the left and the y component is directed upward because the direction of g 1 is toward the upper left.
- Field strength g 2 has an x component that is directed to the left and equal in magnitude to the x component of g 1 The y component of g 2 is directed downward and is equal in magnitude to the y component of g 1 The net field will not be contributed to by the two vectors since they are both equal and opposite.

- The expression for the x -component of g 1 or g 2 needs to be determined and then multiply by 2.

- The mass of an object does not affect this acceleration.

- We know that F is for ma and F is for gravity.

- The amount of force per unit mass is measured by a gravitational field.
- The strength of the field is referred to as g.

- There are objects in the sky.
- All objects fall together.
- A person in free fall doesn't feel his or her weight and doesn't experience contact forces.

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