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Answer :
Answer:
a) The magnitude of the net gravitational force exerted by these objects on a 59.0 kg is [tex] 4.3390\times 10^{-7} N[/tex].
b) By placing object of 59.0 kg 1.976 m from 245 kg object and 1.324 m from 545 kg object.
Explanation:
Mass of an object-1= [tex]M_1=245 kg[/tex]
Mass of an object-2= [tex]M_2=545 kg[/tex]
Distance between [tex]M_1 \&M_2[/tex] = 3.30 m
Mass of an object-3 =[tex]M_3=59.0 kg[/tex]
If [tex]M_3[/tex] placed midway between them [tex]M_1 \&M_2[/tex].
Force on [tex]M_3 [/tex] due to [tex]M_1[/tex]
Distance between [tex]M_1 \&M_3,r=\frac{3.30 m}{2}=1.65 m[/tex]
[tex]F=G\times \frac{M_1\times M_3}{r^2}[/tex]
[tex]F=6.674\times 10−11 m^3/kg s^2\frac{245 kg\times 59.0 kg}{(1.65 m)^2}[/tex]
[tex]F=3.5435\times 10^{-7} N[/tex]
Force on [tex]M_3 [/tex] due to [tex]M_2[/tex]
Distance between [tex]M_2 \&M_3,r'=\frac{3.30 m}{2}=1.65 m[/tex]
[tex]F'=G\times \frac{M_1\times M_3}{r^2}[/tex]
[tex]F'=6.674\times 10−11 m^3/kg s^2\frac{545 kg\times 59.0 kg}{(1.65 m)^2}[/tex]
[tex]F'=7.8825\times 10^{-7} N[/tex]
Net force on [tex]M_3[/tex]:
[tex]F'+ F=7.8825\times 10^{-7} N+(-3.5435\times 10^{-7} N)[/tex]
[tex]=4.3390\times 10^{-7} N[/tex]
The magnitude of the net gravitational force exerted by these objects on a 59.0 kg is [tex] 4.3390\times 10^{-7} N[/tex].
b) Distance between [tex]M_1 \&M_2[/tex] = r = 3.30 m
Let the distanced between [tex]M_1 \&M_3[/tex] be r
Force experienced by [tex]M_3[/tex] due to [tex]M_1[/tex]:
[tex]f=G\times \frac{M_1\times M_3}{r^2}[/tex]
Let the distanced between [tex]M_2 \&M_3[/tex] be r'
Force experienced by [tex]M_3[/tex] due to [tex]M_2[/tex]:
[tex]f'=G\times \frac{M_2\times M_3}{r'^2}[/tex]
Net force equal to zero, then :
[tex]f=f'[/tex]
[tex]G\times \frac{M_1\times M_3}{r^2}=G\times \frac{M_2\times M_3}{r'^2}[/tex]
[tex]\frac{245 kg\times 59.0 kg}{r^2}=\frac{545 kg\times 59.0 kg}{r'^2}[/tex]
[tex]r=1.4914\times r'[/tex]
R = r+r'
3.30 m = 1.4914\times r' + r'
r' = 1.324 m
r = 1.976 m
By placing object of 59.0 kg 1.976 m from 245 kg object and 1.324 m from 545 kg object.
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