**Problem1 :**What is the force exerted by Big Ben on the Empire State building? Assume that Big Ben has a mass of 10

^{8}kilograms and the Empire State building10

^{9}kilograms. The distance between them is about 5000 kilometers and Big Ben is due east of the Empire State building.

F === 2.67×10^{-7} N |

**Problem 2:**What is the gravitational force that the sun exerts on the earth? The earth on the sun? In what direction do these act? (

*M*

_{e}= 5.98×10

^{24}and

*M*

_{s}= 1.99×10

^{30}and the earth-sun distance is 150×10

^{9}meters).

F === 3.53×10^{22} |

**Problem 3:**

Figure %: alignment of Mercury, Venus and the Sun.

*r*

_{v}= 108×10

^{9}meters, Sun-Mercury distance

*r*

_{m}= 57.6×10

^{9}meters, mass of Sun

*M*

_{s}= 1.99×10

^{30}kilograms, mass of Mercury

*M*

_{m}= 3.3×10

^{23}kilograms, mass of Venus

*M*

_{v}= 4.87×10

^{24}kilograms).

F _{s} == 5.54×10^{22} |

*r*

_{mv}== 1.08×10

^{11}meters. The magnitude of the force from Mercury, then, is:

F _{m} == 9.19×10^{15} |

**Problem 4:**It is possible to simulate "weightless" conditions by flying a plane in an arc such that the centripetal acceleration exactly cancels the acceleration due to gravity. Such a plane was used by NASA when training astronauts. What would be the required speed at the top of an arc of radius 1000 metres?

^{2}. Centripetal acceleration is given by

*a*

_{c}=. We have been given

*r*= 1000 meters, so

*v*==99 m/s.

**Problem5 :**Show using Newton's Universal Law of Gravitation that the period of orbit of a binary star system is given by:

T ^{2} = |

*m*

_{1}and

*m*

_{2}are the masses of the respective stars and

*d*is the distance between them. Notice that we derived the same result in a problem in the previous section, using the reduced mass and Kepler's Third Law.

*m*

_{1}

*r*

_{1}-

*m*

_{2}

*r*

_{2}= 0 where

*r*

_{1}and

*r*

_{2}are the radii of orbit. Since

*r*

_{2}+

*r*

_{1}=

*d*âá’

*r*

_{2}=

*d*-

*r*

_{1}, we can write

*m*

_{1}

*r*

_{1}=

*m*

_{2}

*r*

_{2}=

*m*

_{2}(

*d*-

*r*

_{1}) . Rearranging, we can solve for

*r*

_{1}:

*r*

_{1}=

*d*. Now the force acting between the two masses is given by Newton's Law:

F = |

*m*

_{1}:

== |

*r*

_{1}, we have:

T ^{2} ==d ^{3} = |