Newton and Gravitation

Problem : If a space shuttle is launched from the equator, what eastward speed (relative to the ground) is required to propel it into a low-earth orbit ( r r _e )? Take into account the rotation of the earth in this problem.

Solution for Problem 1 >>

The earth rotates from west to east. The circumference of the earth is approximately c _e 2Πr _e = 4.00×10⁷ m. The earth rotates through one twenty-fourth of this distance every hour, and the tangential velocity can be calculated to be 464 m/s. What speed is required for a low-earth orbit?

= âá’v =

Plugging in the radius and mass of the earth yields 7.91×10³ m/s. If the shuttle is to be launched into an eastward orbit, the rotation of the earth helps out: the eastward speed required is the speed required for a low earth orbit, minus the speed of rotation. That is 7.44×10³ m/s.

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Problem : What if the space shuttle, instead of being launched from the equator, is launched from Melbourne, Australia, at a latitude of 38 ^o below the equator? Again, calculate the required eastwards speed.

Solution for Problem 2 >>

The shuttle requires the same velocity to achieve low-earth orbit, but now in a direction 38 ^o northwards of due east. At this latitude, the radius of the earth is given by r _ecos(38 ^o ) = 5.03×10⁷ m and hence the tangential speed is = 366 m/s. But the eastwards speed required by the shuttle is 7.91×10³cos(38 ^o ) = 6.23×10³ m/s. Again we can subtract the eastwards speed of the earth, which we calculated for this latitude, and thus we have a required eastwards speed of: 6230–366 = 5.87×10³ m/s. Of course, the shuttle also needs to be given some velocity in the northwards direction.

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Problem : Calculate the value for G which would be given by a Cavendish apparatus (in an imaginary universe) set up as shown:

Cavendish apparatus.

Solution for Problem 3 >>

The force on the fiber is given by F = 0.5×10^-11 Newtons. The force due to the masses is given by the Universal Law of Gravitation:

F = = 4000G

Therefore we have 4000G = 0.5×10^-11 which implies that G = 1.25×10^-15 N m ² /kg ² .

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Problem : Calculate g on the moon due to the moon's gravity. Compare this to the gravitational force exerted on a mass m on the moon due to the earth (ie. what is g for the earth at the height of the moon?)(The mass of the moon is M _m = 7.35×10²² and its radius r _m = 1700 kilometers, the earth-moon distance is r = 3.84×10⁸ meters, the mass of the earth is 5.98×10²⁴ kilograms).

Solution for Problem 4 >>

The value for g is given by: g = = 1.7 . The g value of the earth at the moon distance is given by: g _e = = 0.003 . The gravity on the moon is therefore approximately 6 times less than that on the earth.

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Problem : A ball is dropped from a height of 1000 kilometers above the earth. Calculate the time taken for it to hit the ground as given by i) assuming g takes the value at the surface of the earth; ii) assuming g takes the value at the top of the ball's path. The mass of the earth is 5.98×10²⁴ kilograms.

Solution for Problem 5 >>

In the first case, we can just use the standard kinematical equation: x = 1/2gt ²âá’t = = = 452 secs. In the second case, we must replace r _e by r _e + h : t = = 523 secs. As we would expect, the time is longer, since the gravitational force is weaker further away from the earth, accelerating the ball more slowly.

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