Two point masses are a distance $x$ apart in a vacuum. State an expression for the force $F$ between masses $M$ and $m$. State the name of any other symbol used.
A small sphere $S$ is fixed to one end of a rod, as shown in Fig. 1.1. The rod is suspended from a vertical thread and remains horizontal. The distance from the centre of sphere $S$ to the thread is $8.0\ \text{cm}$. A large sphere $L$ is placed close to sphere $S$, as shown in Fig. 1.2.
An attractive force between spheres $S$ and $L$ makes sphere $S$ move through a distance of $1.2\,\text{mm}$. The line joining the centres of $S$ and $L$ is perpendicular to the rod. Show that the angle $\theta$ through which the rod turns is $1.5 \times 10^{-2}\,\text{rad}$.
The rod’s rotation twists the thread. The torque $T$ (in $\text{N m}$) needed to twist the thread through an angle $\beta$ (in rad) is given by $T = 9.3 \times 10^{-10} \times \beta$. Calculate the torque in the thread when sphere $L$ is in the position shown in Fig. 1.2.
The centres of spheres $S$ and $L$ are $6.0\,\text{cm}$ apart. Sphere $S$ has mass $7.5\,\text{g}$ and sphere $L$ has mass $1.3\,\text{kg}$. By equating the torque from part (b)(ii) to the moment about the thread caused by the gravitational attraction between the spheres, calculate a value for the gravitational constant.
Suggest why the total force between the spheres might not be the same as the force predicted by Newton’s law of gravitation.