Question

two isolated point masses m and M are separated by a distance l.The moment of inertia of the system about an axis passing through a point where gravitational field is zero and perpendicular to the line joining the two masses,is

Answer 1

Answer on Question#37475, Physics, Mechanics

Question:

Two isolated point masses $m$ and $M$ are separated by a distance $l$ . The moment of inertia of the system about an axis passing through a point where gravitational field is zero and perpendicular to the line joining the two masses, is?

Answer:

Gravitational field equals

$$g = \frac {F}{m _ {t}}$$

where $F$ is the gravitational force, $m_{t}$ is the mass of the test particle

$F, f$ are gravitational forces, $x, l$ - distances

Gravitational field is zero if gravitational forces $f$ and $F$ are the same:

$$f = F$$

Gravitational force equals:

$$F = \frac {G m _ {1} m _ {2}}{r ^ {2}}$$

Therefore:

$$\frac {m}{x ^ {2}} = \frac {M}{(l - x) ^ {2}}$$ $$\left(\frac {l - x}{x}\right) ^ {2} = \frac {M}{m}$$ $$\frac {l}{x} - 1 = \sqrt {\frac {M}{m}}$$

Therefore $x$ equals:

$$x = \frac {l}{1 + \sqrt {\frac {M}{m}}}$$

Moment of inertia of small body equals:

$$I = m r ^ {2}$$

where $r$ is perpendicular distance to the specified axis

Total moment of inertia equals sum of moments of inertia:

$$\begin{array}{l} I = I _ {m} + I _ {M} = m x ^ {2} + M (l - x) ^ {2} = m \left(\frac {l}{1 + \sqrt {\frac {M}{m}}}\right) ^ {2} + M \left(l - \frac {l}{1 + \sqrt {\frac {M}{m}}}\right) ^ {2} \\ = \frac {m l ^ {2}}{\left(1 + \sqrt {\frac {M}{m}}\right) ^ {2}} + M \left(\frac {l \sqrt {\frac {M}{m}}}{1 + \sqrt {\frac {M}{m}}}\right) ^ {2} = \frac {m ^ {2} + M ^ {2}}{(\sqrt {m} + \sqrt {M}) ^ {2}} l ^ {2} \\ \end{array}$$

Answer: $\frac{m^2 + M^2}{(\sqrt{m} + \sqrt{M})^2} l^2$