A group of engineers is preparing a satellite to land by moving it 10% closer to Earth in each rotation. Which statement is correct about the rotational inertia of the satellite? O Rotational inertia first decreases and then increases as the satellite is ready to land.

O Rotational inertia decreases proportional to the decrease in the radius of rotation.

O Rotational inertia increases proportional to the decrease in the radius of rotation.

O Rotational inertia does not change because it is conserved

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Ответ:

tristanart5749

22.09.2021

Physics

see explanation

Explanation:

Magnetic flux is defined by $\Phi = \int \vec{B} \cdot d\vec{A}$

we have a disk with cross-section area $A = \pi r^2$ , where is the radius of the disk ( r = 0.01 [m] ).

a) disk is perpendicular to the magnetic field

We assume the magnetic field is coming from the bottom, therefore $\vec{B}$ and $d\vec{A}$ are parallel and the dot product is maximum because the angle between the vectors is 0.

the magnetic flux takes the following form:

$\Phi = \int \vec{B} \cdot d\vec{A}=\int B*dA*cos(0)=\int B*dA$

now the magnitude of B is constant, we have:

$B\int dA$

remember what is? then we just derivate it with respect to the radius and we get

$dA = \frac{d }{dr}(\pi r^2)=2\pi rdr$

the flux now is $\Phi=\B\int dA=B \int 2\pi rdr=2\pi B\int rdr= 2\pi B(\frac{r^2}{2})=\pi Br^2$

we just demonstrated that the flux of a magnetic field whose direction and magnitude are constant is equal to B times the area A of the surface the magnetic field is passing through:

now we replace values

$\Phi = 0.01 [T] * \pi *(0.01)^2=\pi *10^{-6}$

b) Now if the surface is oriented at 45° we go back a few steps and we just have a small difference this time:

$\Phi = \int \vec{B} \cdot d\vec{A}=\int B*dA*cos(45)=\pi*B*r^2*cos(45)$

therefore

$\Phi = 0.01 [T] * \pi *(0.01)^2*cos(45)=\pi *10^{-6}*\frac{1}{\sqrt{2}}$

the magnetic flux has decreased because of the incidence angle

c) if the surface vector is perpendicular to the magnetic field then the expression takes the following form:

$\Phi = \int \vec{B} \cdot d\vec{A}=\int B*dA*cos(90)=\pi*B*r^2*0 = 0$

there is no magnetic flux because the thin disk is perpendicular to the magnetic field

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