means that the period increases as the radius increases. For the Earth, the period is exactly one Earth year! Planets with orbital radii **greater than Earth's** will have **periods** that are also **greater than Earth's**. Mars, Jupiter, and Saturn have periods **greater** than one Earth year. Planets with orbital radii **smaller than Earth's** will have **periods** that are also **smaller than Earth's**. Mercury and Venus have periods **smaller** than one Earth year.

**Extra Exercise**

** Ex 9.X If the mass of the Earth somehow increased, with all other factors remaining the same, would your weight also increase?**

**(Hint: Let the equation for gravitational force guide your thinking.)**

Increasing the mass of the Earth would increase the force of gravity on me and "the force of gravity on me" is my weight. That is, my weight would **increase**.

Calculational Problem

** Pb 9.3 The value of g at Earth's surface is about 9.8 m/s2. What is the value of g at a distance from Earth's center that is 4 times Earth's radius?**

The force of gravity is an inverse-square force. Earth's surface is one Earth radius away from Earth's center. So being 4 times Earth's radius away from the center means the force of gravity -- and the acceleration due to gravity -- will be 1/16 as great as it is on Earth's surface. g' = (1/16) (9.8 m/s2) g' = 0.61 m/s2

**Extra Problem**

**Pb 9. X Saturn is approximately ten times as far from the Sun as the Earth's distance from the Sun. By way of Kepler's Third Law, estimate the approximate time in Earth years for Saturn to take to make one revolution about the Sun.**

R S = 10 R E T2 / R3 = const

TS2 / RS3 = TE2 / RE3

TS2 / (10 RE)3 = TE2 / RE3

TS2 / (1,000 RE3) = TE2 / RE3

TS2 / (1,000) = TE2

TS2 = 1,000 TE2

TS = [SQRT (1,000) ] TE

TS = [31.6] TE

The period for Saturn is about 31.6 Earth years.

"SQRT" simply means "square root of"

but it is **far** easier for me to type!

**| ****Back to Calendar**** | ****ToC, Ch 9** | **Ch 10, Projectile Motion** |

**Typical or possible multiple-choice questions:**

1. According to Kepler's laws, the paths of planets about the Sun are

A) straight lines B) parabolas

C) ellipses

D) hyperbolas

2. According to Newton, the greater the masses of interacting objects, the

A) greater the force of gravity, by the product of the masses B) less the force of gravity

C) greater the force of gravity, by the square of the masses

D) less the force of gravity, inversely as the square of the masses

3. According to Newton, the greater the distance between masses of interacting objects, the

A) greater the force of gravity, proportional to the distance B) less the force of gravity, inversely as the distance

C) greater the force of gravity, proportional to the square of the distance

D) less the force of gravity, inversely as the square of the distance

4. What is the force of gravity on a 500-newton woman standing on the earth's surface?

A) 9.8 N B) 50 N

C) 500 N

D) 5,000 N

5. If the mass of the earth somehow increased with no change in radius, your weight would

A) increase B) decrease

C) stay the same

D)

6. If the radius of Earth somehow decreased with no change in mass, your weight would

A) increase B) decrease

C) stay the same

D)

7. If Earth's mass decreased to one-half its original mass with no change in radius, then your weight would

A) decrease to one-quarter its original value B) decrease to one-half its original value

C) remain the same

D) increase to twice its original value

8. The force of gravity acting on the Space Shuttle in orbit is nearly

A) zero B) equal to the weight of the Space Shuttle at Earth's surface

C) about one-tenth its weight at Earth's surface

D) about one-one hundredth its weight at Earth's surface

9. A woman who normally weighs 400 N stands on top of a very tall ladder so she is one earth radius above the earth's surface. How much would she weigh there?

A) zero B) 100 N

C) 200 N

D) 400 N

10. The force of gravity acts on all apples on an apple tree. Some apples are twice as far from the ground as others. These twice- as-high apples, for the same mass, have practically

A) one-fourth the weight B) one-half the weight

C) the same weight

D) twice the weight

11. The planet Jupiter is about 300 times as massive as Earth, yet on its surface you would weigh only about 3 times as much. This is because

A) your mass is 100 times less on Jupiter. B) Jupiter is significantly farther from the sun.

C) Jupiter's radius is 10 times Earth's radius.

D) you are 100 times more weightless there.

**| ****Back to Calendar**** | ****ToC, Ch 9** | **Ch 10, Projectile Motion** |

**Answers to typical or possible multiple-choice questions:**

1. According to Kepler's laws, the paths of planets about the Sun are

A) straight lines B) parabolas

**C) ellipses**

D) hyperbolas

2. According to Newton, the greater the masses of interacting objects, the

**A) greater the force of gravity, by the product of the masses** **F = G M m / d2**

B) less the force of gravity

C) greater the force of gravity, by the square of the masses

D) less the force of gravity, inversely as the square of the masses

3. According to Newton, the greater the distance between masses of interacting objects, the

A) greater the force of gravity, proportional to the distance B) less the force of gravity, inversely as the distance

C) greater the force of gravity, proportional to the square of the distance

**D) less the force of gravity, inversely as the square of the distance**

**F = G M m / d2**

4. What is the force of gravity on a 500-newton woman standing on the earth's surface?

A) 9.8 N B) 50 N

**C) 500 N**

D) 5,000 N

5. If the mass of the earth somehow increased with no change in radius, your weight would

**A) increase** **F = G M m / d2**

B) decrease

C) stay the same

D)

6. If the radius of Earth somehow decreased with no change in mass, your weight would

**A) increase** **F = G M m / d2**

B) decrease

C) stay the same

D)

7. If Earth's mass decreased to one-half its original mass with no change in radius, then your weight would

A) decrease to one-quarter its original value **B) decrease to one-half its original value**

**F = G M m / d2**

C) remain the same

D) increase to twice its original value

8. The force of gravity acting on the Space Shuttle in orbit is nearly

A) zero **B) equal to the weight of the Space Shuttle at Earth's surface**

**F = G M m / d2** **In a low-Earth orbit, the Space Shuttle's distance from the center of Earth is nearly the same as it was at Earth's surface.**

C) about one-tenth its weight at Earth's surface

D) about one-one hundredth its weight at Earth's surface

9. A woman who normally weighs 400 N stands on top of a very tall ladder so she is one earth radius above the earth's surface. How much would she weigh there?

A) zero **B) 100 N**

**F = G M m / d2** **If d is made twice as large, then d2 will be four times as large. Since we are dividing by d2, this means the Force -- her weight -- will be only one-fourth as much.**

C) 200 N

D) 400 N

10. The force of gravity acts on all apples on an apple tree. Some apples are twice as far from the ground as others. These twice- as-high apples, for the same mass, have practically

A) one-fourth the weight B) one-half the weight

**C) the same weight**

**F = G M m / d2** **Being twice as far above the ground has hardly any effect on d, the distance from the center of Earth to the apple.**

D) twice the weight

11. The planet Jupiter is about 300 times as massive as Earth, yet on its surface you would weigh only about 3 times as much. This is because

A) your mass is 100 times less on Jupiter. B) Jupiter is significantly farther from the sun.

**C) Jupiter's radius is 10 times Earth's radius.**

D) you are 100 times more weightless there.

**| ****Back to Calendar**** | ****ToC, Ch 9** | **Ch 10, Projectile Motion** |

(C) 2003, Doug Davis; all rights reserved