To determine
a)
To find:
The work to move the other body from r=a to r=b if one of the bodies is fixed
Answer
W=Gm1m21a-1b
Explanation
1) Concept:
The work is calculated by using the formula W=∫abf(x)dx
2) Given:
F=Gm1m2r2
m1, m2 are masses, G is the gravitational constant and r is the distance between bodies.
3) Calculation:
The work is calculated by using the formula:
W=∫abf(x)dx
The gravitational force is given by F(r)=Gm1m2r2 We need to find the work done when the position changes from r=a to r=b
Therefore,
W=∫abGm1m2r2dr
=Gm1m2∫ab1r2dr
=Gm1m2-1rab
=-Gm1m21b-1a
W=Gm1m21a-1b
Conclusion:
Thework done is W=Gm1m21a-1b
To determine
b)
To compute:
The work required to launch the satellite
Answer
W=8.496×109
Explanation
1) Concept:
The required work is calculated by using the formula W=Gm1m21a-1b
2) Given:
The mass of the earth is M=5.98×1024kg
Radius of the earth is R=6.37×106m
G=6.67×10-11 N·m2/kg2
Height of the satellite from earth’s surface is 1000km=1000000 m
The weight of the satellite m=1000kg
3) Calculation:
By using part a), the work done by the satellite is calculated by
W=Gm1m21a-1b
The satellite is 1,000,000 m from the earth’ssurface; therefore, the distance from the centre of the earth to the satellite is R+1,000,000 where R is the radius of the earth
W=GMm1R-1R+1,000,000
W=(6.67×10-11)(5.98×1024)(1000)16.37×106-16.37×106+1,000,000
W=(6.67×10-11)(5.98×1024)(103)16.37×106-16.37×106+106
W=(6.67×5.98×1016)16.37×106-17.37×106
W=6.67×5.98×10100.021
W=0.8496 ×1010
W=8.496×109
Conclusion:
The work required to launch the satellite is W=8.496×109