a) The centripetal force is [tex]2.99\cdot 10^{22} N[/tex]
b) The angular acceleration is [tex]4.03\cdot 10^{-15} rad/s^2[/tex]
c) The number of complete rotations over 7 years is [tex]1.57\cdot 10^{31}[/tex]
Explanation:
a)
First of all, let's convert all the given quantities into SI units:
[tex]r = 58 mil. km = 58 \cdot 10^6 km = 58\cdot 10^9 m[/tex] is the radius of the circular path of the body
[tex]T=58 days = 58 \cdot (24\cdot 60 \cdot 60 )=5.01 \cdot 10^6 s[/tex] is the period of revolution
[tex]m=3.3\cdot 10^{23}kg[/tex] is the mass of the body
From these data, we can calculate first the angular velocity of the body, which is given by:
[tex]\omega=\frac{2\pi}{T}=\frac{2\pi}{5.01\cdot 10^6}=1.25\cdot 10^{-6}rad/s[/tex]
And now we can calculate the centripetal force, which is given by:
[tex]F=m\omega r^2 = (3.3\cdot 10^{23})(1.25\cdot 10^{-6})^2(58\cdot 10^9)=2.99\cdot 10^{22} N[/tex]
b)
We already calculated the initial angular velocity of the body, which was
[tex]\omega = 1.25\cdot 10^{-6} rad/s[/tex]
Here we are said that the new period of revolution is
[tex]T'=34 days = 34 \cdot (24\cdot 60 \cdot 60 )=2.94 \cdot 10^6 s[/tex]
This means that the new angular velocity is
[tex]\omega'=\frac{2\pi}{T'}=\frac{2\pi}{2.94\cdot 10^6}=2.14\cdot 10^{-6}rad/s[/tex]
And this change occurs in a time interval of
[tex]t=7 years = 7 \cdot 365 \cdot 24 \cdot 60 \cdot 60 = 2.21\cdot 10^8 s[/tex]
Therefore, the angular acceleration of the body is
[tex]\alpha = \frac{\omega'-\omega}{t}=\frac{2.14\cdot 10^{-6} -1.25\cdot 10^{-6}}{2.21\cdot 10^8}=4.03\cdot 10^{-15} rad/s^2[/tex]
c)
To solve this part, we have to use the following suvat equation for rotational motions:
[tex]\theta= \omega t + \frac{1}{2}\alpha t^2[/tex]
where:
[tex]\theta[/tex] is the angular displacement covered in a time t
[tex]\omega=1.25\cdot 10^{-6}rad/s[/tex] is the initial angular velocity
[tex]t=2.21\cdot 10^8 s[/tex] is the time interval
[tex]\alpha=4.03\cdot 10^{-15} rad/s^2[/tex] is the angular acceleration
Substituting into the equation,
[tex]\theta= (1.25\cdot 10^{-6})(2.21\cdot 10^8) + \frac{1}{2}(4.03\cdot 10^{15})(2.21\cdot 10^8)^2=9.84\cdot 10^{31} rad[/tex]
And converting into revolutions,
[tex]\theta=\frac{9.84\cdot 10^{31} rad}{2\pi rad/rev}=1.57\cdot 10^{31} rev[/tex]
Learn more about circular motion:
brainly.com/question/2562955
brainly.com/question/6372960
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