Question

1. a student proposes to measure the gravitational constant G by suspending two spherical objects from the ceiling of a tall cathedral and measuring the deflection of the cables from the vertical. draw a free-body diagram of one of the objects. if two 100.0kg objects are suspended at the lower ends of cables 45.00m long and the cables are attached to the ceiling 1.00m apart, what is the separation of objects?

Answer 1

$$\vec {T} + \overline {{m g}} + \vec {F} = \vec {0}$$

In projection on horizontal axis:

$$T \sin \alpha = F = G \left(\frac {m}{R}\right) ^ {2}$$

In projection on horizontal axis:

$$T \cos a = m g \quad \Rightarrow \quad T = \frac {m g}{\cos \alpha}$$ $$m g \tan \alpha = G \left(\frac {m}{R}\right) ^ {2}$$ $$\tan \alpha \approx \sin \alpha$$ $$g \sin \alpha = g \frac {d - R}{2 l} = G \frac {m}{R ^ {2}}$$ $$R ^ {2} (d - R) = 2 l G \frac {m}{g}$$ $$R \approx d \Rightarrow R ^ {2} \approx d ^ {2}$$ $$d ^ {2} (d - R) = 2 l G \frac {m}{g}$$ $$R = d - \frac {2 l G m}{g d ^ {2}}$$ $$R = 1 - \frac {2 * 4 5 * 6 . 6 7 * 1 0 ^ {- 1 1} * 1 0 0}{9 . 8 * 1 ^ {2}} = 0. 9 9 9 9 9 9 9 4 (m)$$

So, in this experiment it's unreal to measure G